an overview : peak to average power ratio (papr) in ofdm system using some new papr techniques (with...

An Overview : Peak to Average Power Ratio (PAPR) in OFDM system using some new PAPR techniques (with matlab code)

Zainab S. H. AL-Hashmi

An Overview : Peak to Average Power Ratio (PAPR) in OFDM system using some new PAPR techniques (with matlab code)

By Zainab Saad Hadi AL-Hashmi

A graduate of University of Baghdad, College of Engineering

Electronic & Communications Engineering Department

i

تغى هللا انشد انشدى

د ﴿ م انث جظ أ كى انش ة ػ إا شذ هللا نز

شا ﴾ طشكى ذط

صذق هللا انؼه انؼظى

{ }االدضاب /

ii

االذاء

بسم هللا الرحمن الرحيم

ػهى انث يلئكر صه هللا ﴿إ ا ػه آيا صه ا أا انز

ا﴾ ا ذغه عه ذ ذ آل يذ ى صم ػهى يذ .انهـ

دثة لهتا شفغ رتا عذ أذا انى ث انشدح شفغ االيح

انخهك اجؼ انثؼز سدح نهؼان ات انماعى يذذ صم هللا

صم هللا ػه عهى أذا انى تاب يذح انؼهى سعل هللا ػه آن

عهى ات انغثط ايش انؤي انى ػرشخ سعل هللا آن

.فرمثها ي ا آل غ أرى انكشاو ال ذشد انذاا

أدة أ أذا انى ي ستا صغشا انى جذذ جذي انشدو

أه أخص تانزكش انغذ انغذ دغ ػه ػثاط ص انى أي

لاعى دغ ص انغذ يصى دغ ص انغذ فاظم دغ ص

.انغذ ػادل دغ ص

ادة اعا اذا نكم غانة ظه انششف ا ي يغؤل ػ

ا اخز دم تاخرصاس اذا نكم يظهو الل صادة انذك

عهطا فذافؼا ػ دملكى

iii

Acknowledgments

praise belongs to God who showed favour to us through His

religion, singled us out for His creed, and directed us onto the

roads of His beneficence, in order that through His kindness we

might travel upon them to His good pleasure, a praise which He

will accept from us and through which He will be pleased with

us. !Allah send peace and blessings upon Mohammed and his

progeny (S.A.W.)

Finally I would like to thank my family,

Especially my grandfather Mr. Hassan Ali Zwain,

my mother, Mr. Qasim Hassan Zwain and Mr. Maythem Hassan

Zainab saad hadi

2015

iv

Abstract

The Orthogonal frequency division multiplexing (OFDM) is multicarrier

modulation scheme which has recently become comparatively popular in

both wireless and wired communication systems for transfer the

multimedia data. OFDM could be used at the core of well-known systems

like Asymmetric digital subscriber line (ADSL) internet, digital

television/radio broadcasting, wireless local area network (LANs), and

Long Term Evolution (LTE).

High PAPR is the major drawback of OFDM, which results in lower

power efficiency hence impedes in implementing OFDM. The PAPR

problem is more significant in the uplink because the efficiency of power

amplifier is critical because a mobile terminal has a limited battery

power.

High peak-to-average power ratio (PAPR) occurs due to large envelope

fluctuations in OFDM signal this requires a highly linear high power

amplifier (HPA). Power amplifiers with large linear range are expensive,

bulky 50% of the size of a transmitter lies and difficult to manufacture.

In order to reduce the PAPR, several techniques have been proposed in

this thesis, primarily the repeated frequency domain filtering and clipping

(RFC) has been proposed and compared with the existing method

repeated clipping and frequency domain filtering (RCF). The RFC is

better than RCF in performance especially when I ≥ 2, although they have

the same complexity and cost.

The proposed method is not only improving PAPR but also improving

BER. Best case for the bit error rate (BER) is at I =4 and CR =4, where

Signal to Noise Ratio (SNR) at BER ( ) improved by (5.7601 dB)

and Complementary Cumulative Distribution Function (CCDF) of PAPR

was improved by (4.775 dB) and PAPR was improved by (11.4177 dB).

The best one improvement in PAPR and CCDF of PAPR So as not to

BER deteriorate is at I =4 and CR =1.75. The improvement in PAPR by =

(18.2789 dB), CCDF of PAPR = (8.0187 dB), and the SNR at

BER( ) by = (0.6101 dB).

In addition to (RFC) six new types of companding have been proposed

and compared with the μ-law and A-law compandings. all these proposed

methods have better performance than the μ-law and A-law compandings,

and the best one is Absolute Exponential (AEXP) companding and the

v

best one improvement in PAPR and CCDF of PAPR is at d= 1.1. The

improvement in PAPR by = (17.6492 dB), and CCDF of PAPR = (7.2405

dB), while the SNR at BER( ) deteriorated by = (-3.4186 dB).

Five types of pre-coding are used in this work and then compared them

with each other. The best type of precoding in term of reduced PAPR and

BER is the Discrete Fourier Transform (DFT) pre-coder, while the least is

the Walsh Hadamard Transform (WHT) pre-coding.

Also four new types of hybrids PAPR reduction techniques have been

proposed. These methods are:

1. RCF with precodings (WHT, Discrete Cosine Transform (DCT),

Discrete Sine Transform (DST),and Discrete Hartley Transform (DHT)).

2. RCF with compandings (the all proposed compandings, μ-law and A-

law compandings).

3. RFC with compandings (the all proposed compandings, μ-law and A-

law compandings).

4.and finally precodings (WHT, DCT, DST,and DHT), with compandings

(the all proposed compandings, μ-law and A-law compandings).

The best one improvement is at (RFC with AEXP) because the PAPR,

CCDF of PAPR, and BER. This improvement in PAPR and CCDF of

PAPR is at d = 0.6 and CR =4. The improvement in PAPR by

(21.0509dB), CCDF of PAPR = (8.7178 dB), and the SNR at

BER( ) by (0.0116 dB).

The DHT with tangent Rooting (tanhR) have acceptable results where the

PAPR and CCDF of PAPR were improved while BER was degarded.

The best one improvement for this case is at k=15, y=.8 and DHT. The

improvement in PAPR by = (22.7711 dB), and CCDF of PAPR = (8.9691


All methods are simulated using matlab.

vi

Contents

Chapter One: Introduction 1

1.1 Introduction 1

1.2 Literature survey 1

Chapter Two: LTE and OFAM 7

2.1. Introduction 7

2.2. LTE Requirements 7

2.3. LTE Architecture 8

2.4. Air interface in LTE 9

2.5 History of OFDM 10

2.6 OFDM 12

2.6.1 Orthogonality of the subcarriers and OFDM 15

2.6.2 Guard Interval 17

2.6.3 One-tap equalizer 18

2.7 OFDM based Multiple Access 19

2.8 Orthogonal Frequency Division Multiple Access 20

2.9 SC-FDMA 24

Chapter Three: Peak To Average Power Ratio Reduction 27

3.1Definitions of PAPR 27

3.2 PAPR of OFDM signal 28

3.3 Oversampling discrete OFDM symbols to find true (continuous) peaks 29

3.4 Distribution of PAPR 29

3.5 Identification of the Problem 32

3.5.1 Nonlinear HPA and DAC 32

3.5.2 Power Saving 35

3.6 Factors influencing the PAPR 35

3.6.1 The number of sub carriers 35

3.6.2 The order of Modulation 35

3.6.3 Constellation shape 36

3.6.4 Pulse Shaping 36

vii

3.7 The gauge for judgment of the PAPR reduction in OFDM systems 36

3.8 Fitness function-based approach for determining an appropriate Algorithm 37

Chapter Four: PAPR Reduction Techniques 39

4.1There are three different way to divide the PAPR 39

4.1.1The first way is 39

4.1.2 The second way 43

4.1.3The third way 45

4.1.4 And finally there is Hybrid techniques 45

4.2 Clipping and Filtering 46

4.3 Peak Windowing Method 47

4.4 Envelope Scaling 48

4.5 Peak Reduction Carrier 48

4.6 Companding Technique 49

4.6.1 Square-Rooting Companding Technique ( SQRT) for PAPR Reduction in

OFDM Systems

50

4.6.2 Exponential Companding Algorithm 51

4.6.3 Trapezoidal power companding 53

4.6.4 Hyperbolic tangent ( ) companding 53

4.6.5 Error Function ( ) Companding 54

4.6.6 Logarithm Function (log) Companding 54

4.7 Coding techniques 54

4.8 Selective Mapping (SLM) 56

4.9 Partial Transmit Sequence (PTS) 57

4.10 Tone Reservation 59

4.11 Tone Injection 60

4.12 Interleaving 61

4.13 Active Constellation Extension (ACE) 61

4.14 Dummy Sequence Insertion (DSI) 62

Chapter Five: Simulation Results and Analysis 63

5.1 OFDM System model 63

5.2 PAPR techniques used 65

viii

5.2.1 Repeated clipping and frequency domain filtering (RCF) 65

5.2.2 Repeated frequency domain filtering and clipping RFC 72

5.2.3 The OFDM System with discrete time companding 77

5.2.3.1 A companding 77

5.2.3.2 Companding 81

5.2.3.3 Rooting Companding Technique (RCT) 85

5.2.3.4 New error function Companding (NERF) 87

5.2.3.5 Absolute Exponential companding (AEXP) 89

5.2.3.6 Cos companding 91

5.2.3.7 tangent Rooting Companding (tanhR) 95

5.2.3.8 Logarithmic Rooting Companding (logR) 101

5.2.4 OFDM System with pre-coding 104

5.2.4.1 Pulse Shaping or Pre-coding 104

5.2.4.2 Discrete Hartley transform (DHT) 105

5.2.4.3 Walsh-Hadamard Transform (WHT) 105

5.2.4.4 Discrete Cosine Transform (DCT): 106

5.2.4.5 Discrete Sine Transform (DST) Precoding Technique 107

5.2.4.6 The Discrete Fourier Transform (DFT) Precoding 107

5.2.4.7 Simulation results and analysis of OFDM system with pre-coding 108

Chapter six: Simulation Results and Analysis of Hybrid PAPR techniques 110

6.1 Hybrid pre-coding with RCF 110

6.2 Hybrids RCF with companding 119

6.2.1 RCF + A companding 119

6.2.2 RCF + 121

6.2.3 RCF + RCT 123

6.2.4 RCF + AEXP 126

6.2.5 RCF + cos 128

6.2.6 RCF + NERF 130

6.2.7 RCF + tanhR 131

6.2.8 RCF +logR 132

6.3 Hybrid RFC with companding 134

6.3.1 RFC + A companding 134

ix

6.3.2 RFC + companding 137

6.3.3 RFC + RCT 139

6.3.4 RFC + AEXP 141

6.3.5 RFC + cos 143

6.3.6 RFC + NERF 145

6.3.7 RFC + tanhR 146

6.3.8 RFC +logR 147

6.4 Pre-coding + companding 148

6.4.1 Pre-coding + A companding 149

6.4.2 Pre-coding + 152

6.4.3 Pre-coding + RCT 154

6.4.4 Pre-coding + AEXP 156

6.4.5 Pre-coding + cos 159

6.4.6 Pre-coding + tanhR 161

6.4.7 Pre-coding + logR 162

6.4.8 Pre-coding + NERF 163

Chapter seven : Conclusions and future work 165

7.1Conclusions 165

7.2Future work 167

References 168

Appendices

Appendix A : Table of Results A.1

Appendix B : MATLAB Code B.1

Chapter One Introduction

1

Chapter One

Introduction

1.1 Introduction:

During the last two decades, the demand for multimedia wireless communication

services have grown tremendously and this trend are expected to continue in the near

future. Orthogonal frequency division multiplexing (OFDM) is one of such multi-

carrier techniques which have attracted vast research attention from academics,

researchers and industries since last two decades. It has become part of new emerging

standards for broadband wireless access [1].

Energy efficiency, particularly matters in future mobile communications networks. A

key driving factor is the growing energy cost of network operation which can make up

as much as 50% of the total operational cost nowadays [2].

The transmitted signal of OFDM exhibits a high Peak-To-Average Power Ratio

(PAPR). This high PAPR reduces the efficiency of high power amplifier and degrades

the performance of the system [3].

A major source for reducing energy costs is to increase the efficiency of the high

power amplifier (HPA) in the radio frequency (RF) front end of the base stations [4].

However, the efficiency of the HPA is directly related to the PAPR of the input signal.

The problem, especially, becomes serious in OFDM multicarrier transmission, which

is applied in many important wireless standards such as the third Generation

Partnership Project (3GPP) Long Term Evolution Advanced (LTE-A). The PAPR

problem still prevents OFDM from being adopted in the uplink of mobile

communication standards, and, besides from power efficiency, it can also place severe

constraints on output power and therefore coverage in the downlink. In the past, there

have been many efforts to deal with the PAPR problem resulting in numerous papers

and several overview articles, e.g., [5], [6], [7].

PAPR has a deleterious effect on battery lifetime in mobile applications. As handy

devices have a finite battery life, it is significant to find ways of reducing the PAPR

allowing for a smaller, more efficient HPA, which in turn will mean a longer lasting

battery life.

In many low-cost applications, the problem of high PAPR may outweigh all the

potential benefits of multicarrier transmission systems [6]. A number of promising

approaches or techniques have been proposed & implemented to reduce PAPR with

the expense of increase transmit signal Power, bit error rate (BER) & computational

complexity and loss of data rate, etc. So, a system trade-off is required [8].

1.2 Literature survey:

In 1996 Robert [9]. The selected mapping was used for the reduction of PAR. The

selected mapping can be used for arbitrary numbers of carriers and any signal

constellation. The selected mapping provides significant gains at moderate additional

complexity. Actually, it is appropriate for all kinds of multiplex techniques, which

transform data symbols to the transmit signal. Even in single carrier systems where


2

PAR grows as the roll of factor of the pulse shaping filter decreases, selected mapping

can be applied advantageously.

The first nonlinear companding transform (NCT) for PAPR reduction was given by

Wang et.al in 1999 [10]. It was based on the speech processing algorithm µ-law and it

has found better performance than that of clipping technique. The µ-law companding

transform mainly focuses on enlarging small amplitude signals while keeping peak

signals unchanged, and thus it increases the average power of the transmitted signals

and may lead to overcome the saturation region of the HPA to make the performance

of the system worse. In order to overcome the problem of µ-law companding

(increasing average power) and to have an efficient PAPR reduction. [10]

In 2000 Myonghee et.al [11] Hadamard transform is an effective technique to reduce

the PAPR of an OFDM system. The PAPR can be reduced in OFDM system without

any power increase and side information. The increase of system complexity is not

much. As further study, the equalization problem combining with Hadamard

transform, which is induced to reduce PAPR, over multipath fading channel, is

considered.

In 2001 J. Armstrong [12] the clipping and frequency domain filtering PAPR

reduction technique has been described in which an interpolated version of the

baseband signal is clipped and then filtered with a new form of filter. The filter

consists of a forward and an inverse fast Fourier transform (IFFT). It is designed to

remove the out-of-band (OOB) noise without distorting the in-band discrete signal. It

is shown that significant PAPR reduction can be achieved without any increase in

OOB power. Some in-band distortion results, but this will have negligible effect on

the overall BER in most systems.

In 2002 J. Armstrong [13] the repeated clipping and frequency domain filtering of an

OFDM signal can significantly reduce the PAPR of the transmitted signal. This

method causes any increase in OOB power. Considerable PAPR reduction can be

obtained with only moderate levels of clipping noise.

In 2004 Ryu, et al. [14] The Dummy Sequence Insertion (DSI) technique reduces

PAPR through increased the average power of the signal. Herein, after switchting the

input data stream into parallel through the serial to parallel converter a, dummy

sequence is inserted in the input signal. Thus, the average value is raised and the

PAPR is reduced later.

In 2005 Tao Jiang et.al [15] “exponential companding”. It can adjust the amplitudes

of both large and small input signals, while maintaining the average power unchanged

by properly choosing transform parameters, so as to make the output signals have a

uniform distribution (with a specific degree). The exponential companding schemes

can efficiently reduce PAPR for various modulation formats and sub-carrier sizes.

http://www.mathworks.com/help/matlab/ref/ifft.html


3

The exponential companding schemes make less spectrum side-lobes than µ-law

companding. Simulation results have shown that exponential companding schemes

could provide better system performance in terms of PAPR reduction, power

spectrum, BER, and phase error than the µ -law companding scheme.

In 2007 Wisam et.al [16] square rooting companding (SQRT) companding a simple

method of reducing the PAPR value of OFDM symbol by changing the statistical

characteristics of the output signals . This was achieved by applying a non-linear

square rooting operation of the OFDM signals. The process changed also the

describing parameters of power signals: average and peak power values, and as a

result the PAPR value is reduced. This companding more suitable for OFDM

applications that do not have sophisticated processor, since it allows significant

reduction in PAPR value with very low cost of computational complexity, and only

slight performance degradation.

In 2008 Pisit et.al [17] the simple PAPR reduction method by using the dummy sub-

carriers. The features of proposed method is to decide the frequency data for dummy

subcarriers theoretically by using the certain number of larger amplitude levels

detected in the time domain signal and to achieve the better PAPR performance with

less computational complexity.

In 2008 Carole et.al [18] they present an incipient PAPR reduction technique which

exploits the utilization of used carriers in addition to the phase information of pilot

signals in OFDM systems to limit the PAPR while not degrading channel estimation

or frequency offset. Compared to conventional techniques like clipping and

windowing, this technique introduces significantly lower OOB distortions and

provides a lower BER since there is no interference to the original data signals. There

is additionally no requisite for side information to be transmitted to the receiver.

In 2009 Kazuki and Fumiyuki [19] A tone injection (TI) has been suggested which

exploits the property of a nonlinear modulo function. The TI is identically equivalent

to the one that superimposes a quadrature amplitude modulation (QAM) signal on the

data symbol to reduce the PAPR. Without the transmission of the side information,

the TI dramatically reduces the PAPR level. Albeit the TI-OFDM reduces the 1%

PAPR level by about 3~4.5dB, the BER performance remarkably degrades. However,

the utilization of antenna diversity reception can reduce the BER performance

degradation.

In 2010 Zhongpeng et.al [20] a combined μ companding transform and hadamard

transform technique is suggested to reduce PAPR of OFDM signal .Simulation results

shows that the PAPR reduction performance is improved compared with companding

transform used only. On the other hand, the BER of system using proposed PAPR

reduction scheme is not degraded.


4

In 2010 Imran and Varun [21] the PAPR of discrete hartley transform (DHT)-

Precoded OFDM system for M-ary Quadrature Amplitude Modulation (M-QAM)

(where M=16, 32, 64, 256). The Matlab simulation shows that DHT-Precoded OFDM

System shows better PAPR gain as compared to OFDM-Original system, Walsh

Hadamard transformation (WHT)-Precoder Based OFDM system and selective

mapping (SLM) OFDM (with V=2) system respectively. Thus, it is concluded that

DHT Precoder Based OFDM System shows better PAPR reduction then WHT-

Precoder Based OFDM System, SLM-OFDM System and OFDM-Original system for

MQAM. Additionally, the DHT-Precoded OFDM system does not require any power

increase, complex optimization and side information to be sent for the receiver.

In 2011 Zhongpeng [22] a combined reduction in PAPR of the

OFDM signals based on the combination of the discrete cosine transform (DCT) with

μ companding. While taking both BER performance and PAPR performance into

account, a united DCT and companding scheme to reduce the PAPR of OFDM

signals.

In 2011 Hem [23] a combinational method of pre-coding and clipping is proposed to

reduce the PAPR of an OFDM system. The proposed technique is better than

conventional because it does not require any increase in roll-off factor to reduce

PAPR. Thus, it reduces the overhead in comparison to conventional pre-coding

method. Increasing the roll-off factor degrades the BER as given in [24]. The clipping

after pre-coding reduces PAPR but degrades BER. However, this degradation in not

significant in comparison to degradation obtained by increasing roll off factor.

In 2012 Malhar and Prof.Abhishek [25] tone reservation includes no of set of

reservation of tones. By using this technique reserved tones can be utilized to

minimize the PAPR. This method is used for multicarrier transmission and also

demonstrated the reserving tones to limit the PAPR. Advantage of this tone

reservation is very positive that no process is needed at receiver end. Furthermore

there is no need to transmit the side information combined with the transmitted signal.

In 2012 Eugen [26] The PAPR reduction technique based on combination of a WHT

and a new signal companding function. The numerical results show that the hybrid

scheme realizes an improved PAPR reduction than the component methods. The

computation complexity increases linearly with the number of considered signal

variants because of several signal variants are applied to the precoding block. This

problem can be solved, by using few subcarriers as markers.

In 2012 Chau, and Hsuan [27] presents a combination scheme, which using a

combination of precoding by utilizing least null subcarriers in the frequency domain

and nonlinear companding technique by applying proper -Law characteristic in time

domain, for reducing PAPR. Simulation results indicate that the proposed scheme


5

achieves a advantageous trade-off between OOB power emission in OFDM systems

and the reduction of PAPR.

In 2013 Sroy et.al [28] an Iterative Clipping and Filtering (ICF) Technique for PAPR

Reduction of OFDM Signals: System Using DCT/ inverse discrete cosine transform

(IDCT) Transform. The OFDM symbol enters the ICF block with DCT/IDCT

technique, then clipping and filtering is iteratively performed. Although we

demonstrate that significant PAPR reduction is obtained through Iterative clipping

and filtering using fast Fourier transform (FFT)/IFFT transform, but better results are

observed applying DCT/IDCT in the classical ICF technique.

In 2013 Zihao et.al [29] a trapezoidal power companding method which could

significantly reduce the PAPR for a complex OFDM or Filterbank Based Multicarrier

Transmission (FBMC) system. The proposed approach provides a convenient way for

designing a compander where the trade-offs among several system performances

(such as PAPR, power spectral density (PSD) and BER) can be made.

In 2013 Mohit et.al [30] the performance of tanh and erf companding is

approximately. Log companding is better than the hyperbolic tangent and error

function companding . μ-law and A-law companding give the same performance and

the μ-law and A-law companding is better than the tanh, log and erf companding.

Some more non-linear transform have been suggested in the paper [31, 32, 33, 34, and

35]

In 2013 Jaspreet et.al [36] the performance analyzed in terms of PAPR in Orthogonal

Frequency Division Multiple Access (OFDMA) by utilizing some pre-coding

techniques, called Zadoff-Chu Transform (ZCT) and WHT with the µ-law

companding to limit the PAPR of the OFDM signals .These pre-coding techniques

produced the lower PAPR as compared to the conventional OFDM system.

Furthermore ZCT is better than WHT because it produced the lowest PAPR than

WHT. μ -law companding further reduces PAPR of OFDM signal and as with

increasing the value the PAPR reduces. The obtained results approved that the

proposed method have gotten better results than conventional OFDM.

In 2013 Navneet and Lavish [37] The PAPR reduction method is based on combining

clipping with WHT. Combined technique is simple to implement and has no

limitations on the system parameters such as number of subcarriers modulation order,

and constellation type. This system produces the lowest PAPR and is efficient, signal

independent, distortion less and do not require any complex optimizations

representing better PAPR reduction methods than others existing techniques because

it does not require any power increment, complex optimization and side information

to be sent to the receiver.


6

In 2013 Mohit et.al [38] To reduce the PAPR of OFDM has been proposed Hybrid

Clipping-Companding techniques for PAPR Reduction. the performance of hybrid

PAPR reduction scheme with either tanh or erf as companding function is

approximately same .Hybrid PAPR reduction scheme with log companding function

is better than the last two. Hybrid PAPR reduction scheme with either μ-law or A-law

companding gives the same performance and the Hybrid PAPR reduction scheme

with either μ-law or A-law companding is best.

In 2013 K. muralibabu et.al [39] In the proposed scheme, a combined reduction in

PAPR of the OFDM system by grouping the sub carrier on the basis of the

combination of joining the Discrete Cosine Transform (DCT) with companding

technique. The simulation results indicat that the proposed scheme can yield good

tradeoff between computational complexity and PAPR reduction performance

In 2014 Jijina et.al [40] a comparative study is made on the three typical linear

precoding techniques: Hadamard transform precoding, Discrete Sine Transform

(DST) precoding and Square root raised cosine function precoding used in OFDMA

system. The performance of these different schemes in terms of PAPR reduction is

analyzed with the conventional Random Interleaved OFDMA system. Linear

precoding schemes are efficient, signal independent, distortion less and do not require

complex optimization when compared to the other reduction schemes.

Chapter Two LTE and OFAM

7

Chapter Two

LTE and OFAM

2.1. Introduction: The growth in data intensive mobile services and applications like Web browsing,

social networking, video streaming and music has become a driving force for

development of the next generation of wireless standards. Thus, new standards are

being developed to provide the data rates and network capacity needful to support

worldwide delivery of these kinds of rich multimedia application. LTE have been

developed to respond to the requirements of this generation and to achieve the aim of

realizing global broadband mobile communications [41].

2.2. LTE Requirements: The demand for high speed and widespread network access in mobile

communications increases every day as the number of users‟ increases and

applications are constantly developed with greater demand for network resources. As

a result of this trend, mobile communications has experienced significant

developments within the last two decades, which is the result of tremendous research

that has been carried out. [42]

Requirements and objectives for the LTE Discuss the main requirements for the new

LTE system Resulted in a the creation of a formal

„Study Item‟ in 3GPP with the specific aim of „evolving‟ the 3GPP radio access

technology to guarantee competitiveness over a ten-year time-frame. Depending on

the study of this Study Item, the requirements for LTE Release 8 were revised and

crystallized. They can be summed up as follows [41,43, and 44]:

High peak data rates and diminished delays, in both connection establishment

and transmission latency. These improvements are to be realized through the

simplification of the overall system, the decrease of complexity and the

automated process of system management (i.e. optimization).

greater flexibility of spectrum usage, in each of the new and pre-existing bands;

Seamless integration with existing systems (Universal Mobile

Telecommunications System (UMTS), Wireless Fidelity (Wi-Fi), etc.).

Infrastructure-building economy. Although the implementation of every new

system brings construction and building costs, LTE should be realized through

minimal investment and use as much of the existing mobile communication

infrastructure as possible.

Multi-antenna support.

Improved system capacity and coverage

Reasonable power consumption for the mobile terminal. The mobile terminal is

being associated with mobile phones and similar devices which have limited

battery capacities. Therefore a flexible bandwidth system (with lower

frequencies used for uplink transmission) and automated signal power-level

optimization have to be included into LTE [45].

Seamless mobility, including between different radio-access technologies;

Simplified network architecture;

Increased cell-edge bit-rate, for unification of service provision;

Increased user data rates;

Reduced cost per bit, implying an enhanced spectral efficiency;


8

Packet switched domain utilization. To eliminate additional system complexity,

introduced through the support of both the circuit switched and packet switched

domain, the circuit switched domain will not be included into the LTE system.

The traditional voice and text messaging services must be replaced with system-

external subsystems (e.g. Information Management System (IMS)).

High-level security and mobility. As the mobile communication system is now

similar to a data network (e.g. internet), additional emphasis will be set on new

security measures in combination with IP (Internet Protocol)-security functions.

Mobility efficiency is provided through the use of evolved base stations, i.e.

eNodeBs (E-UTRAN Node-B or Evolved Node-B).

These main targets resulted in the creation of additional requirements and spin-off

functionalities, whose realizations were researched, developed and evolved by 3GPP

and hence introduced in LTE‟s specifications and standardization upgrades.

These improvements were further evolved and enhanced in Release 9, which

contained additional techniques, functionalities and technology approaches to enable a

quick, efficient and low-cost implementation of the LTE system. The following

techniques are included:

introduction to Self-Organizing Networks (SON),

improved approach to emergency calls, as they oppose the system‟s security

policy,

multiple-eNodeB broadcast signal combination (LTE MBMS),

further improvement of Frequency Division Duplex (LTE-FDD) and Time

Division Duplex (LTE-TDD),

improvement of SON technologies and mechanisms, and

Minimization of system drive-tests (MDT).

The LTE system and its standardization are 3GPP‟s most significant milestone

achieved so far, triggering an increase of participation in their further projects and

worldwide acknowledgement of their existing work. Takahiro Nakamura, the 3GPP

RAN Chairman, states: “Operators need to work on issues that have been created in

signaling and the volume of data being carried. Therefore, further improvements to

the 3GPP system are being driven by that data explosion”. A continued evolution of

the system is given in Releases 10, 11 and 12, introducing an improved mobile

communication standard named LTE-Advanced [45].

2.3. LTE Architecture: The LTE architecture was highly simplified and flattened, as shown in Figure 2.1. The

system contains only two types of nodes named Mobility Management Entity/System

Architecture Evolution Gateway (MME/SAE GW) and evolved Node-B (eNB) [46,

47].

All LTE network interfaces are based on IP protocols and therefore two major

changes were made compared to previous cellular radio architectures. The first

significant modify is that the Radio Network Controller (RNC) is removed from the

data path and its functions are now situated in eNB [46]. The main benefits of this

type of single node access network are the diminished latency and the distribution of

the RNC processing overhead into multiple eNBs. The second major change is that

there are no nodes for Circuit Switched (CS) domain, such as the Mobile Switching


9

Centre (MSC). Therefore speech services are handled as Voice over IP (VoIP) calls in

the LTE network [47, 48].

The eNBs are connected to each other via X2 interface and to Evolved Packet Core

(EPC) through S1 interface, as also shown in Figure 2.1. The S1 interface supports in

addition many-to-many relations between MMEs / SAE Gateways and eNBs [46].

SAE Gateway contains two logical gateway entities named as the Serving Gateway

(SGW) and the Packet Data Network Gateway (P-GW). The S-GW is responsible for

receiving and forwarding IP packets. Therefore, it can be seen as a local mobility

anchor to the eNBs [48]. The P-GW, on the other hand, is responsible for handling the

internet protocol functions, like routing, packet filtering, policy enforcement and

address allocation [47].

The new system architecture was designed so that it will reduce the overhead from

increased traffic. This is achieved because only the MME is responsible for signaling

and therefore the user IP packets do not go over MME. This way the network capacity

stays on a good level as the signaling and the traffic can grow separately [49]. The

main responsibilities of MME are idle-mode User Equipment (UE) reachability

including the control and execution of paging retransmission, different type of

authentication procedures with Non-Access Stratum (NAS) signaling, roaming, P-

GW/S-GW selection, tracking area list management and bearer management

including dedicated bearer establishment [47,48].

2.4. Air interface in LTE: The air interface and communication environment used in LTE mobile

communication systems is called the LTE Radio Access Network. [45]

The LTE air interface is based on OFDMA for the downlink. OFDMA is an extension

of OFDM for the implementation of a multi-user communication system. For the

uplink, a single-carrier frequency-division multiple access (SC-FDMA) technique has

been selected. Advantages of this method include the relatively low adjacent channel

power, even if the power amplifier is not 100% linear. With SC-FDMA, no exacting

requirements are imposed on the linearity of the power amplifier in the mobile

handset. As a result, power consumption can be kept within limits. [50]

The utilization of OFDM provides considerable advantages over alternative multiple-

access techniques and signals severe departure from the past. Among the benefits are

adaptability for broadband data transmission and high spectral efficiency, impedance

to Inter Symbol Interference (ISI) resulting from the multipath fading, naturally

provide MIMO (Multiple Input Multiple Output) schemes, and provide frequency-

domain techniques like frequency-selective scheduling [51].

The design of the time-frequency representation of OFDM to provide high levels of

flexibility in allocation of each of the time frames for transmission and the spectra.

The spectrum flexibility in LTE supports not only a scalable set of bandwidths, but

also a variety of frequency bands. LTE also supplies a small frame size of 10 ms in

order to reduce latency. By designate short frame sizes, LTE allows better channel

estimation to be carried out the mobile, allowing timely feedbacks needful for link

adaptations to be supplied to the base station.[41]


10

Figure 2.1: System architecture for LTE Rel-8 network [47].

2.5 History of OFDM: The initial development of multi-carrier communication system was basically done by

military systems in the late 1950s and mid-1960s. KINEPLEX, ANDEFT and

KATHRYN etc. are the few OFDM based systems utilized by US military systems for

high frequency applications [10].

In 1966, the concept of multicarrier communication was first introduced by Chang

[60] .He suggested a multicarrier scheme utilizing the parallel data transmission by

means of 10 frequency division multiplexing (FDM) with overlapping subcarriers. It

was found to be an efficient scheme for bandwidth utilization and to mitigate the

effect of multipath propagation. It also eliminated the requirement of high-speed

equalization technique. He gave the basic concept of OFDM and outlined a theoretical

way to transmit simultaneous data stream trough linear band limited channel without

Inter Symbol Interference (ISI) and Inter Carrier Interference (ICI) [61] [62].


11

These systems are called classical Multicarrier Modulation (MCM) system and

transmitted data on non-overlapped band-limited orthogonal signals. These systems

require analog oscillator and filter of much wider bandwidth and sharp cut-off.

Therefore, the concept of OFDM was not gained so much attention or popularity

because of the difficulty in subcarrier recovery without inter-subcarrier interference

by means of analog filters. Due to this reason only, a number of studies in the 1960s

were dedicated for MCM employing overlapped band-limited orthogonal signals [63,

64, and 65]. In the year 1967, B. R. Saltzberg suggested a MCM system employing

Orthogonal time-staggered Quadrature Amplitude Modulation (O-QAM) on the

carriers [63]. The concept of MCM scheme employing time-limited orthogonal

signals, which is similar to OFDM, was first given by H. F. Marmuth [66] in 1960.

[10]

The KINEPLEX system was developed by Collins Radio Company for data

transmission at high frequency over multipath fading channel, in this system, 20 tones

are modulated by DQPSK without filtering, which resulted in overlapping channels.

Initially the implementation of an OFDM system with large number of subcarriers

was very complex and expensive because it requires the array of sinusoidal generators

and coherent demodulators for parallel operations. In order to avoid the crosstalk

between the subcarriers, the system should be free from frequency and timing offsets

[62].

A major breakthrough in the history of OFDM came in 1971 when Weinstein and

Ebert used Discrete Fourier Transform (DFT) to perform baseband modulation and

demodulation which eliminated the need of bank of subcarrier oscillators thus making

the operation efficient and simpler [1,67].

In 1979, after evolutionary growth and development in signal processing and VLSI

technologies, high speed chips can be built around special-purpose hardware

performing the large size Fast Fourier Transform (FFT) (efficient algorithm for DFT)

at affordable price [68], [69].

All the proposals of OFDM systems used guard spaces in frequency domain and

a raised cosine windowing in time domain to combat ISI and ICI. Another milestone

for OFDM history was when Peled and Ruiz introduced Cyclic Prefix (CP) or cyclic

extension in 1980 [67,70] .This solved the problem of maintaining orthogonal

characteristics of the transmitted signals at severe transmission conditions. The

generic idea that they placed was to use cyclic extension of OFDM symbols instead of

using empty guard spaces in frequency domain. This effectively turns the channel as

performing cyclic convolution, which provides orthogonality over dispersive channels

when CP is longer than the channel impulse response [56,70].

Since 1990s, OFDM has been utilized for many broadband communication systems, it

includes high-bit-rate digital subscriber lines (HDSL) [71], asymmetric digital

subscriber lines (ADSL) [72], very high-speed digital subscriber lines (VHDSL) [72],

high definition television (HDTV) terrestrial broadcasting etc. It has also been utilized

by many wireless standards like Digital Audio Broadcasting (DAB) [73] The DAB

standard was in fact the first OFDM-based standard (project started in 1988 by ETSI

and completed in 1995), Digital Video Broadcasting (DVB) [74].

Many standards have been proposed for wireless local area networks (WLANs)

operating in ISM band, which are based on spread-spectrum technology. A number of

studies regarding the commercial applications of OFDM were made during 1990s like

High Bit rate Digital Subscriber Lines (HDSL; 1.6 Mbps), Asymmetric Digital

Subscriber Lines (ADSL; 6 Mbps), Very High Speed Digital Subscriber Lines


12

(VDSL; 100 Mbps), DAB and High Definition Television (HDTV) terrestrial

broadcasting [75].

In 1997, first OFDM-based WLAN standard, IEEE 802.11, was released. IEEE

802.11 can support a data rate up to 2 Mbps. Later on, in 1999, IEEE approved an

OFDM based standard 802.11a for supporting a data rate up to 54 Mbps. During this

period ETSI has also standardized the HiperLAN/2 standard, which has adopted

OFDM for their PHY standards [1].

In 2001, the FCC came with some new rules for modulations scheme operating in the

2.4 GHz range, which allow IEEE to extend 802.11b to 802.11g standard. Now days,

it has also been used in WiMAX (IEEE 802.16), and mobile broadband wireless

access (MBWA) IEEE 802.10. It is 11 also utilized by 4G wireless communication

systems, such as IMT-A. It is also been considered for 3GPP Long Term Evolution,

which is under deployment [62].

2.6 OFDM: With the ever growing require of this generation, the necessity for high speed

communication has become a top priority. Different multicarrier modulation

techniques have developed to meet these demands, a few prominent among them

being OFDM and Code Division Multiple Access (CDMA) [52].

The fundamental principle of OFDM is a division of high data rate streams into a

number of lower data rate streams and then transmitted these streams in parallel using

several orthogonal sub-carriers (parallel transmission). Due to this parallel

transmission, the symbol duration increases, thus decrease the prorated amount of

dispersion in time resulting from the multipath delay spread. OFDM can be seen as

either a modulation technique or a multiplexing technique [10].

OFDM communication systems do not depend on increased symbol rates for

achieving higher data rates. That makes the task of managing ISI much easier.

Because data is transmitted in parallel instead of serially, OFDM symbols are

basically much longer than symbols on single carrier systems of equivalent data rate

[53].

The concept of OFDM is very much similar to the well-known and extensively used

technique of Frequency Division Multiplexing (FDM). OFDM uses the principles of

FDM to allow multiple messages to be sent over a single radio channel. It is however

in a much more controlled manner, allowing an improved spectral efficiency [54].

In conventional broadcast, each radio station transmits on a different frequency,

effectively using FDM to maintain a separation between the stations. Due to non-

orthogonal nature of carrier frequencies in FDM, a large band gap is required to avoid

inter-channel interference, which reduces the overall spectral efficiency. The

difference between FDM and OFDM is shown in Figure 2.2 [55].


13

Figure 2.2: Comparison of FDM and OFDM [55]

The sub-carriers are mutually orthogonal (The principle of orthogonality is discussed

in next sub-section.) in the frequency domain which alleviates the effects of ISI as

indicated in the Figure 2.3. All of these sub-carriers experiences „flat fading‟ because

they have a bandwidth less than the Mobile channel coherence bandwidth [56].

Figure 2.4 shows a baseband transceiver structure for OFDM utilizing the Fourier

transform for modulation and demodulation. Here the serial data stream is mapped to

complex data symbols (Phase Shift Keying (PSK), QAM, etc.) with a symbol rate

of

. The data is then demultiplexed by a serial to parallel converter resulting in a

block of N complex symbols, .The parallel samples are then passed

through an N point IFFT (in this case no oversampling is assumed) with a rectangular

window of length N.Ts, resulting in complex samples

.Assuming the incoming complex data is random it follows that the IFFT

is a set of independent random complex sinusoids summed together. The

samples, are then converted back into a serial data stream producing a

baseband OFDM transmit symbol of length T=N.Ts [57].

A Cyclic Prefix (CP), which is a copy of the final part of the samples, is appended to

the front of the serial data stream before RF up conversion and transmission. The CP

combats the disrupting effects of the channel which introduce ISI.

In the receiver the whole process is reversed to recover the transmitted data, the CP is

removed prior to the FFT which reverses the effect of the IFFT [58]. The complex

symbols at the output of the FFT, are then decoded and the original bit

steam recovered.

Thus, the IFFT and FFT blocks at the transmitter and at the receiver, respectively, are

important components in an OFDM system. A lot of work has gone into the

optimization of the FFT implementations and the design community has leveraged this

trend to advantage leading to the popularity of OFDM based systems. The time-


14

frequency view of an OFDM signal is shown in Figure 2.5, where the important

parameters like subcarrier spacing and OFDM symbol period are shown [59].

Figure 2.3 OFDM subcarrier spacing [56].

Figure 2.4 a block diagram of a basic OFDM system.

Signal

Mapper

Signal

demapper

Equalizer

And

P/S

DFT

OR

FFT

S/P

D/A Add

CP

IDFT

OR

IFFT

P/S S/P

Multipath

Fading Ch.

&

noise

A/D Remove

CP

Input

output


15

Figure 2.5 Time-Frequency view of OFDM signal [59]

2.6.1. Orthogonality of the subcarriers and OFDM: Two functions or signals are said to be orthogonal if they are mutually independent of

each other. Orthogonality is a feature that lets multiple information signals to be

transmitted skillfully over a common channel with the successful detection [24 and

76].

The subcarrier spacing is chosen so that the waveforms transmitted on different sub

carriers are orthogonal in time, but overlap in frequency. The orthogonality is

achieved by making the peak of each subcarrier signal coincide with the null of the

other subcarrier signals resulting in a perfectly aligned and spaced subcarrier signal

[77].

The principle of orthogonality state that if the time-averaged integral of the product of

any two functions from a set of functions { ( ) ( ) ( ) ( ) }, over a

joint existence time interval [ ] is equal to zero, irrespective of the limit of

existence of the functions, then the functions are told to be orthogonal to each other

within this time-interval [16]. Mathematically, it can be expressed as –

∫ ( ) ( )

(2.1)

The orthogonality property of OFDM signals can be shown with the help of its

spectrum. In the frequency domain every OFDM subcarrier has a ( )

frequency response, as shown in Figure 2.6 [10].

One of the key advantages of OFDM is its efficient use of the frequency band as the

subcarriers are allowed to overlap each other in the frequency domain. The N equally

spaced subcarriers will be orthogonal if the frequency separation between subcarriers

is f =

, where N.Ts is symbol duration, and rectangular windowing of the

IFFT is performed. Under these conditions the subcarriers will have a waveform

frequency response [78].

Simple rectangular pulse of the length is used and rectangular shape in time

domain corresponds to a -square shaped spectrum in frequency domain as

illustrated in Figure 2.6. The LTE sub-carrier spacing is set to Δf= 15 KHz [62].


16

Figure 2.6 Per-subcarrier pulse shape and spectrum of basic OFDM transmission [48]

Figure 2.7 shows the frequency response of a 5 carrier system where it is seen that

because of the orthogonal relationship the maximum of a particular sample

corresponds to a null in all other carriers, therefore eliminating the effects of

interference.

Figure 2.7: Frequency spectrum of 5 orthogonal subcarriers of an OFDM transmit

signal [78].

The orthogonality among sub carriers can be viewed in time domain as shown in

Figure 2.8. Each curve represents the time domain view of the wave for a subcarrier.

As seen from Figure 2.3, in a single OFDM symbol duration, there are integer

numbers of cycles of each of the subcarriers [62]


17

Figure 2.8: Time domain representation of the signal waveforms to show

orthogonality among the subcarriers [62]

2.6.2. Guard Interval: Individual sub channels can be perfectly separated by the FFT at the receiver when

there are no ISI and Inter-channel Interference (ICI) introduced by channel distortion.

Practically these conditions cannot be acquired. Since the spectra of an OFDM signal

is not precisely band limited, linear distortion like multipath fading caused sub

channel to spread energy in the adjacent channels [79, 80].

Figure 2.9 illustrates the CP insertion technicality, the Guard Interval or CP is a

periodic addition of the final part of an OFDM symbol that is added to the front of the

symbol in the transmitter, and at the receiver the CP is removed before demodulation

[81].

It serves as a recurrence of the end of the symbol, so allowing the linear convolution

of a frequency selective multipath channel to be modeled as circular convolution

which in turn might be transformed to the frequency domain utilizing a discrete

Fourier transform (DFT). This process allows for simple frequency domain processing

like channel estimation and equalization [82].

CP insertion, therefore, increases the size of the data symbol from to ,

being the duration of the guard-period containing the CP. The standard length of

the guard-period in LTE is defined to be 4.69 μs, allowing the system to tolerate path

variations up to 1.4 km (considering the standard LTE symbol length of 66.7 μs).

When a cyclic extension longer than a channel impulse response is added, the

negative effect of the previous symbol can be avoided by simply removing that

extension. CP insertion implies the copying of the last part of the OFDM data symbol

and attaching it to the timing at the beginning of the symbol, creating a break between

signals (hence: guarding-period). The receiver can then sample the incoming

waveform at optimum time, as time-dispersion problems (i.e. delays caused by

reflections of the signal) up to the length of the guarding-period are ignored [45].


18

Figure 2.9 the CP insertion mechanism [83]

2.6.3 One-tap equalizer [10]: The tap-delay line model with path is considered for multipath fading channel.

After Considering the effect of the multipath fading channel, the samples of The

received signal can be expressed as:

( ) ∑ ( ) ( ) ( ) (2.2)

where, ( ) is the impulse response of multipath fading channel with path gains

{ ( ) }, is the path delay of path, and ( ) is a zero-

mean, unit variance complex Gaussian noise.

After discarding first G sample of the received signal and taking Z-point FFT, the

output of FFT block is ( ) given as :

(2.3)

Where, the term is the channel response to the subcarrier frequency and is

the Additive white Gaussian noise (AWGN) term in the frequency domain. To

compensate the fading effect of the channel, one-tap equalizer is used and each

element of the vector is multiplied by an equalized gain factor the output of

equalizer may be written as –

(2.4)

Where, is defined as –

(| | (

)) . (2.5)


19

2.7 OFDM based Multiple Access: Various multiple access schemes can be combined with OFDM transmission and they

include orthogonal frequency division multiplexing-time division multiple access

(OFDM-TDMA), OFDMA, and multicarrier code division multiple access (MC-

CDMA). In OFDM-TDMA, time-slots in multiples of OFDM symbols are used to

separate the transmissions of multiple users as shown in figure. 2.10. This means that

all the used subcarriers are allocated to one of the users for a finite number of OFDM

symbol periods.

The only difference from OFDM-TDMA is that the users capture the channel and use

it for certain duration, i.e., the time dimension is used to separate the user signals [84]

Figure 2.10: Time – Frequency view of an OFDM-TDMA Signal

In OFDMA systems, both time and/or frequency resources are used to separate the

multiple user signals. Groups of OFDM symbols and/or groups of subcarriers are the

units used to separate the transmissions to/from multiple users. In figure 2.11, the

time, frequency view of a typical OFDMA signal is shown in a case where there are 3

users. It can be seen from figure 2.11 that users‟ signals are separated either in the

time-domain by using different OFDM symbols and/or in the subcarrier domain.

Thus, both the time and frequency resources are used to support multiuser

transmissions. We shall discuss this technique in more detail in the subsequent

sections and also compare it with OFDM-TDMA [85].


20

Figure 2.11: Time – Frequency view of an OFDMA Signal [85]

2.8 Orthogonal Frequency Division Multiple Access: The approach used in LTE‟s access techniques consists of using OFDMA for the

downlink (DL) and SC-FDMA for the uplink (UL).

The main reason that justifies different access techniques for the UL and DL is the

fact that SC-FDMA optimizes range and power consumption at the UE, while

OFDMA minimizes receiver complexity and enables frequency domain scheduling

with flexibility in resource allocation. OFDMA is a multi-carrier transmission scheme

in opposition to SC-FDMA. Both allow multiple user access, depending on the

available bandwidth, by dynamically allocating each user to a specific time-frequency

resource, depending on which duplexing is deployed. OFDM requires a large dynamic

range due to PAPR [86 and 87].

The main difference between an OFDM system and an OFDMA one is represented in

Figure 2.12. The different colors represent different users using resources. In OFDM,

users are assigned to resources in the time domain only, while in OFDMA, users can

be assigned also in the frequency domain, optimizing resource usage.

In OFDMA systems, the multiple user signals are separated in the time and/or

frequency domains. OFDMA has been developed with multi-user operation as its

purpose, allowing a flexible assignment of bandwidth to users according to their

needs.

Typically, a burst in an OFDMA system will consists of several OFDM symbols. The

subcarriers and the OFDM symbol period are the finest allocation units in the

frequency and time domain, respectively. Hence, multiple users are allocated different

slots in the time and frequency domain, i.e., different groups of subcarriers and/ or

OFDM symbols are used for transmitting the signals to/from multiple users. For

instance, we illustrate an example in figure 2.13 wherein the subcarriers in an OFDM

symbol are represented by arrows and the lines shown at different times represent the

different OFDM symbols. We have considered 3 users and we have shown how

resources can be allocated by using the different subcarriers and OFDM symbols [88

and 89].


21

Figure 2.12 Difference between OFDM and OFDMA resource by user allocation [86].

Figure 2.13: Example allocation of resources to users in an OFDMA system [85].

Figure 2.14 is a detailed block diagram of OFDMA. The LTE PHY (Physical Layer)

specification has been designed to adapt bandwidths from 1.25 MHz to 20 MHz

OFDM was selected as the main modulation scheme due to its robustness with a

severe multipath fading. Downlink multiplexing is achieved through the OFDMA.

OFDM is the modulation scheme for the DL. The primary subcarrier spacing is 15

kHz, with lower subcarrier spacing of 7.5 kHz available for some MB-SFN

(Multicast-broadcast single-frequency network) scenarios. OFDM modulation

parameters summarizes in Table 2-1 [90]


22

Table 2-1 Downlink OFDM Modulation Parameters [90]

Transmission

BW

1.25 MHz 2.5

MHz

5 MHz 10 MHz 15 MHz 20 MHz

Sub-frame

duration

0.5 ms

Sub-carrier

spacing

15 kHz

Sampling

frequency

192 MHz

(1/2 x 3.84

MHz)

3.84

MHz

7.68

MHz

(2

x 3.84

MHz)

15.36

MHz

(4 x

3.84

MHz)

23.04

MHz (6

x 3.84

MHz)

30.72 MHz

(8 x 3.84

MHz)

FFT size 128 256 512 1024 1536 2048

No. of

occupied

subcarrier

76 151

301 601 901

1201


23

Figure 2.14 Complete block diagram of an OFDMA transmitter and receiver [91]


24

2.9 SC-FDMA: In cellular systems, the wireless communication service in a certain geographical area

is supplied by multiple base stations. The downlink transmissions in cellular systems

are one-to-many, whilst the uplink transmissions are many-to-one. A one-to-many

service means that a base station transmits concurrent signals to multiple users‟

equipment‟s in its coverage area. This demands that the base station has very high

transmission power ability; as a result of the transmission power is involved for

transmissions to multiple users‟ equipment‟s [92]. On the other hand, in the uplink, a

single user‟s equipment has all its transmission power available for its uplink

transmissions to the base station. In the uplink, the design of an effective multiple

access and multiplexing scheme is more challenging than on the downlink because of

the many-to-one nature of the uplink transmissions. Another consequential requisite

for uplink transmissions is the low signal peakiness by means of the limited

transmission power at the user‟s equipment [92].

One of the main parameters that affect all mobile UE devices is their battery life. It is

therefore necessary to ensure an economic and efficient power use in the transmission

and reception of signals. With the RF power amplifier (i.e enhancer of mixed signals)

and the transmitter being the parts with the highest energy consumption within the

mobile UE; it is essential to establish a transmission model with near constant

operating power level [45].

The downlink physical layer of LTE is depending on OFDMA. Thus, in spite of its

many advantages, OFDMA has specific drawbacks like high sensitivity to frequency

offset (Doppler spread by cause of mobility and Arising from the instability of

electronics) and PAPR. PAPR occurs due to the random constructive addition of sub-

carriers and results in spectral spreading of the signal which leads to adjacent channel

interference. It is a problem that could be insurmountable with high compression point

power amplifiers and amplifier linearization techniques. While these approaches may

be utilized on the base station, they become costly on the UE [93 and 94].

In LTE, a new concept is used for the access technique of the uplink, called SC-

FDMA. Its characteristics combine lower PAPR of single-carrier systems because

there is only a single carrier unlike N carriers. (Which allows maintaining a lower

operating power level than OFDMA) with immunity to multipath interference, as well

as flexible subcarrier frequency allocation (as a crucial part of OFDM) [45]. Figure

2.15 shows the concepts of OFDMA and SC-FDMA.


25

Figure 2.15 frequency domain description of downlink and uplink LTE access

technologies

SC-FDMA differs from OFDMA in one additional transmission step, caused by the

single-path transmission of single-carrier systems. That transmission step, called

resource element mapping (and its counterpart, resource element selection), shifts all

symbols obtained through the FFT to the desired center frequency and passes them on

to the IFFT for further conversion Figure 2.16.

Since the power of the modulation signals used in this process is constant (QPSK

(Quadrature Phase Shift Keying), 16QAM and 64QAM) and the result of the resource

element mapping step is a waveform similar to the original, on another center

frequency; the required result of a constant-power signal is achieved [45].

For practicality, SC-FDMA is implemented in LTE utilizing a Discrete Fourier

Transform Spread OFDM transmission (DFTS-OFDM) which is repeatedly referred

to as a frequency-domain generalization of SC-FDMA. The DFT is used to multiplex

uplink transmissions in definite frequency allocation blocks within the general system

bandwidth in accordance with eNodeB scheduler instructions. The bandwidth of the

single carrier is specified based on the desired data rate by the user. Data remains

serial and not parallelized as done on the downlink with OFDMA (i.e. one

information bit is being transmitted at a time). This results in similar link performance

parameters for the uplink and downlink. Nevertheless, there would be comparatively

high ISI for the uplink because of the single carrier modulation. Thus, the eNodeB

receiver requires a low-complexity equalizer to rectify for the distorting impacts of

the radio channel. SC-FDMA is not as sensitive to Doppler Effect and frequency

instability the as OFDM by cause of its single carrier nature [93].


26

Figure 2.16 Block diagram of an SC-FDMA transmitter and receiver [37]

Chapter Three Peak-to-Average Power Ratio Reduction

27

Chapter Three

Peak-to-Average Power Ratio Reduction:

High PAPR of transmitted signals is one of the major issues of the OFDM system. A

large dynamic range of input data symbols is the main cause of getting high PAPR.

An OFDM signal consists of independent data symbols modulated on N orthogonal

subcarriers, and when these signals are added to the same phase, higher peak

amplitude is observed. The value of this peak may be times of the average

amplitude [10].

3.1 Definitions of PAPR: For a continuous time baseband OFDM signal, the PAPR of any signal is defined as

the proportion of the maximum instantaneous power of the signal and its average

power. If x (t) is a transmitted baseband OFDM signal, then PAPR is defined as:

, ( )- , ( ) -

(3.1)

Where, is the average power of x (t) and can be computed in frequency domain

because IFFT is a unitary transformation is useful duration of an OFDM symbol

[95].

For a discrete OFDM signal, the PAPR is computed from time oversampled

OFDM signal as:

, ( )- [ ( ) ]

[ ( ) ] (3.2)

The , ( )- at (dB) =

[ ( ) ]

[ ( ) ] (3.3)

Where, , - denotes the expectation operator and is the total number of sub-

carriers. The PAPR of pass band OFDM signal is approximately twice that of

baseband PAPR [95].

The above power characteristics can also be described in terms of their magnitudes

(not power) by defining the crest factor (CF), which is defined as the ratio between

maximum amplitude of OFDM signal ( ) and root-mean-square (RMS) of the

waveform. The CF is defined as:

| ( )|

,|| ( )| |- √ (3.4)

In most cases, the peak value of signal ( ) is equals to a maximum value of its

envelope | ( )| However, it can be seen from Figure 3.1 that the appearance of peak

amplitude is very rare, thus it does not make sense to use max | ( )| to represent the


28

peak value in real application. Therefore, the PAPR performance of OFDM signals is

commonly measured by certain characterization constants which relate to probability

[96].

Figure 3.1: High PAPR when sub-carriers are modulated by same symbols [96]

3.2 PAPR of OFDM signal [62]: The discrete time baseband OFDM signal is defined in (6). The PAPR of the discrete

time OFDM signal determines the complexity of the digital circuitry in terms of the

number of bits necessary to achieve the desired signal to quantization noise ratio

during signal digitization and recovery. To better approximate the PAPR of a

continuous time OFDM signal, the discrete time OFDM signal is to be obtained by L

times oversampling. The oversampled discrete time OFDM signal can be obtained by

performing LN point IFFT on the data block with (L-1) N zero padding as follows:

, ( )-

√ (

) , 0≤ n ≤NL-1 (3.5)

PAPR of the oversampled OFDM signal of becoming

, ( )- , ( ) -

, ( ) - (3.6)

where, E[. ] denotes the expectation operator and N is total number of sub-carriers.

The PAPR of passband OFDM signal is approximately twice that of baseband PAPR.

Complementary Cumulative Distribution Function (CCDF) for an OFDM signal can

be written as:

P (PAPR > PAP )= ( ) (3.7)

Where PAP is the clipping level.

This equation can be read as the probability that the PAPR of a symbol block exceeds

some clip level PAP .


29

3.3 Oversampling discrete OFDM symbols to find true (continuous)

peaks: The PAPR for the discrete-time baseband signal x [n] may not be the same as that of

the continuous-time baseband signal ( ) In fact, the PAPR for , - is lower than

that for ( ), simply because , - may not have all the peaks of ( ) In practice, the

PAPR for the continuous-time baseband signal can be measured only after

implementing the actual hardware, including digital-to-analog convertor (DAC). In

other words, measurement of the PAPR of the continuous-time baseband signal is not

straightforward. Therefore, there must be some means of estimating the PAPR from

the discrete-time signal , -. Fortunately, it is known that , - can show almost the

same PAPR as ( ) if it is L-times interpolated (oversampled) as shown in Figure 3.2

where L ≥ 4 [97 and 98].

Figure 3.2 Block diagram of L time‟s interpolator [83]

3.4 Distribution of PAPR: To design and develop an effective PAPR reduction technique, it is very important to

accurately identify the distribution of PAPR in OFDM systems. The distribution of

PAPR plays an important role in the design of the whole OFDM system. The

distribution of PAPR can be used in determining the proper output back-off of the

HPA to minimize the total degradation. It can be used directly to calculate the BER

and to estimate the achievable information rates [10].

For the OFDM system, if we assume that the input data stream is statistically

independent and identically distributed (i.e.) then the real and imaginary parts of x[n]

are uncorrelated and orthogonal. From central limit theorem, it follows that, for large

values of N, the real and imaginary parts of x[n] are independent and identically

distributed (i.e.) Gaussian random variables, each with zero mean and variance

,| , - | - . (3.8)

The probability distribution of complex OFDM signals with large N is a complex

Gaussian distribution given by following relation:

* , -+

√ .

, -

/ (3.9)

Where Pr{.} denotes the probability distribution function. Where, is the variance

of , -.The amplitude of OFDM signal has a Rayleigh distribution and its

probability density function (PDF) is given by:

* , -+ | , - |

.

| , - |

/ (3.10)


30

The histogram plots for the real part, imaginary part and the absolute value of a time

domain OFDM signal are shown in Figure 3.3(a), (b) and (c) respectively. The plots

shown in Figures 3.3(a) and (b) are obtained after performing the computer

simulations of an OFDM system having N=256 QPSK modulated subcarriers as

shown in Fig. 2.4. The signal obtained from IFFT block of Figure 2.4 is complex

OFDM signal. After that real, imaginary and absolute values of OFDM signal (x[n])

are calculated and their histograms are plotted [62].

The power of OFDM signal has chi-square distribution. The distribution of PAPR is

often expressed on the one hand Complementary Cumulative Distribution Function

(CCDF). In probability theory and statistics, the CCDF describes the probability that a

real-valued random variable X with a given probability distribution will be found at a

value greater than or equal to x [99 and 10].

The Cumulative Distribution Function (CDF) of the PAPR of the amplitude of a

signal sample is given by

( ) ( ) (3.11)

The CCDF of the PAPR of the data block is desired in our case is to compare outputs

of different reduction techniques. This is given by:

( ) ( ) (3.12)

( ) (3.13)

( ( ) (3.14)

Where, is the given reference level.

Figure 3.3 (a)


31

Figure 3.3 (b)

Figure 3.3 (c)

Figure 3.3: Histogram of (a) Real part of OFDM signal amplitude (b) Imaginary part

of OFDM signal amplitude (c) OFDM signal magnitude [63].


32

3.5 Identification of the Problem: Multi-carrier phenomena is considered to be one of the major development in wireless

communication and among them OFDM is becoming the important standard.

However, high PAPR is the major drawback of OFDM, which results in lower power

efficiency hence impedes in implementing OFDM. To overcome the low power

efficiency requires not only large back off and large dynamic range DAC but also

highly efficient HPA and linear converters. These demands result in costly hardware

and complex systems. Therefore to lessen the difficulty of complex hardware design it

has become imperative to employ efficient PAPR reduction techniques [100 and 101].

The drawback of a large dynamic range is that it places pressure on the design of

components such as the word length of the IFFT/FFT pair, mixer stages, and most

importantly the HPA, which must be designed to handle irregularly occurring large

peaks, decreases the SQNR (Signal-to-Quantization Noise Ratio) of ADC (Analog-to-

Digital Converter) and DAC, The PAPR problem is more important in the uplink

since the efficiency of power amplifier is critical due to the limited battery power in a

mobile terminal. Failure to design components with a sufficiently large linear range

result in saturation of the HPA [98, 78]. Saturation creates both in band distortion,

increasing the BER and out of band distortion, or spectral splatter, which causes

Adjacent Channel Interference (ACI). One obvious solution is to design the

components to operate within large linear regions, however this is impractical as the

components will be operating inefficiently and the cost becomes prohibitively high.

This is especially apparent in the HPA where much of the cost and ~50% of the size

of a transmitter lies which will be explained in next sections [98, 78].

3.5.1 Nonlinear HPA and DAC: HPA are used in the transmitter of communication systems for sufficient transmission

power. To achieve maximum output power efficiency they have to be operated at or

near the saturation region. [100]

If the data on the subcarriers add up in a constructive manner at the transmitter, the

resulting signal could exhibit large PAPR. As a result, the composite transmit signal

could be severely clipped by the DAC and power amplifiers for their bounded

dynamic range as described in Figure 3.4. In this case, the reconstructed output ( )

can possess a significant amount of distortion. It can be reduce the PAPR of an

OFDM signal by modifying the signal characteristics in time-domain or frequency

domain clipping of the composite OFDM signal causes several undesirable outcomes,

such as signal distortion and spectral regrowth. For instance, clipping causes in band

noise that results in a degradation of the BER performance .Moreover, higher-order

harmonics that spill over into OOB spectrum can also result from signal clipping.

Although filtering after the HPA can be employed to remove this spectral leakage, it

is very power-inefficient, so it is an undesirable solution. Therefore, the dynamic

range of DAC should be large enough to accommodate the largest peaks of signals or

high PAPR values [102].

A high-precision DAC support high PAPR with acceptable amount of quantization

noise, but could be very costly to a certain sampling rate of the system. On the other

hand, a low-precision DAC would be cheaper, but the quantization noise will be

significant, which reduces the signal SNR (Signal to Noise Ratio) when the dynamic

range of DAC is increased to support high PAPR. Otherwise, the DAC will saturate

and clipping will occur [48, and 103].


33

Figure 3.4 An example illustrating effect of clipping.

The dynamic range of the power amplifiers should also be large enough to

accommodate large PAPR values. Otherwise, the power amplifiers may saturate and

clipping might occur. The component cost of the DAC and power amplifiers increase

with the increase in the dynamic range.


34

It is worth mentioning that the clipping of high signal peaks rarely happens, resulting

in a comparatively low incidence clipping noise. In this manner, the impact of

clipping at the transmitter on the error performance of the OFDM system liable to be

subjected frequency selective fading is minimal [102].

If an HPA with limited linear range is utilized for amplification, it may operate near

saturation and can cause OOB radiations and in-band distortion. The OOB

distortion/noise is a major concern, especially in wireless communications, where

large differences in signal strength from a mobile transmitter impose stringent

requirements on ACI [104]

Figure 3.5 demonstrates a classic input-output characteristic of a power amplifier. For

prevent or limit signal distortion input signals must be preserved below the Non-linear

area. The result is that the amplifier is not completely used [105]

IBO = 10 (

) (3.15)

OBO = 10 (

) (3.16)

IBO (Input Back-Off) or OBO (Output Back-Off)

High PAPR results in a wide variety of OFDM signal amplitudes which due to

nonlinear characteristics of HPA findings in inter-modulation among the various sub

carriers and leading to an increment in BER. To realize a low BER and minimal

signal distortion, HPA must be a large dynamic range and work in the linear amplifier

region. But, these types of HPA are expensive and smaller power efficient. The power

efficiency in wireless communication is very important for achieving efficient area

coverage and small size terminals. Thus, the power efficient process of non-linear

HPA is so important. Accordingly, it is best to target the reduction of PAPR the

OFDM signals before transmitting the signal into nonlinear DAC and HPA [100].

Figure 3.5 Typical input-output characteristics of a power amplifier showing the

Relation between Output Back-Off (OBO) and Input Back-Off (IBO) [98].


35

3.5.2 Power Saving [100]: A high dynamic range HPA has low power efficiency. The power could save by

reducing PAPR. This power saving that is implemented in such a way to provide a

direct correlation with the desired average output power.

On the assumption a linear model of HPA, the power efficiency is:

(3.17)

(3.18)

The η= HPA efficiency .

= the average of the output power.

. = A fixed amount of power regardless of their input power.

For example: an OFDM signal with 256 sub carriers that demand an IBO equal to the

PAPR at the probability level lower than 0.01%, i.e. (25.235).This makes

η = 0.5/25.235≈1.98%

The PAPR of OFDM systems has to reduce for avoiding this level of power

inefficiency.

3.6 Factors influencing the PAPR:

3.6.1 The number of sub carriers: In Multi-Carrier Systems the complex base band signal for one symbol in an OFDM

system can be expressed as follows:

( )

√ ∑

(3.19)

Where N is the modulating symbol and is the number of sub carriers. For moderately

large numbers of m-PSK (multiple phase-shift keying) sub carriers the quadrature

components of x (t) each tends towards a Gaussian distribution (giving the sum of

their power amplitude a Rayleigh distribution). Consequently, whilst the peak value

possible is N times the individual sub carrier peak, the probability of any value close

to that peak occurring is very low. For example, with only 24 sub carriers, the

probability of the PAPR exceeding 4dB is and of exceeding 8dB is only

[99].

3.6.2 The order of Modulation: High data bandwidth efficiency (in terms of b/s/Hz) this can be achieved by utilizing

higher order modulations based, for instance, on QAM. When using a higher-order

modulation such as QAM type, the PAPR of the summed OFDM signal is increased

by the PAPR of the QAM constellation utilized. Nevertheless, the probability of these

higher peaks happening is accordingly less. Furthermore, since among benefits of

OFDM is one that sub carriers could have their modulation independently varied to

adapt to channel conditions, the joined PAPR in any system utilizing this technique

might are hard to predict and control. PAPR for an unfiltered base band signal is listed

in the following Table 3.1. [100].


36

Table 3.1 PAPR for picked modulation formats

3.6.3 Constellation shape: The last entry in Table 3.1 is for a constellation obtained by modifying 256- QAM to

reduce PAPR. This modified constellation shape is shown in figure 3.6. However,

there is an additional processor load associated with encoding and decoding this

constellation.

Figure 3.6 256-QAM constellations: (a) regular and (b) modified mapping to reduce

PAPR

3.6.4 Pulse Shaping: In terrestrial communications, it is popular to use pulse shaping to the base band

signal, to decrease the bandwidth of the transmitted spectrum, but this causes

overshoot and can increase the PAPR of the modulating signal by 4-5 dB [100].

3.7 The gauge for judgment of the PAPR reduction in OFDM systems

[106, 107, 108]: Every method used to reduce the PAPR has some drawbacks and merits. There is

always a trade-off between PAPR reduction and some other factors like bandwidth,

computational complexity, average power etc. An ideal PAPR reduction technique

should have following characteristics:

1) High potential to limit the PAPR: It is a key factor to consider in the selection of

technology to reduce the PAPR with few adverse side effects like in-band distortion

and OOB radiation.

2) Low average power: even though it can reduce PAPR through the average power of

the original signals increase, it needs a bigger linear operation region in HPA and

which led in the deterioration of BER performance.

Modulation PAPR

256-QAM 4.23dB

64-QAM 3.68dB

256-QAM (modified) 2.85dB

16-QAM 2.55dB

m-PSK (reference) 0 dB


37

3) Low implementation complexity: mainly, complexity techniques viewing better

capability of PAPR reduction. Nevertheless, practically, both time and hardware

requisites for the PAPR reduction must be minimal.

4) No bandwidth expansion: The bandwidth is an infrequent resource in systems. The

bandwidth expansion has directly resulted in the data code rate loss because of side

information (like the complementary bits in Complement Block Coding (CBC) and

phase factors in PTS). Furthermore, when the side information is received in error

unless some methods of protection like channel coding employed. For that reason,

when channel coding is utilized, the loss in data rate is incremented further due to side

information. Then, the loss in bandwidth because of side information must be avoided

or at least be preserved minimal.

5) No BER performance degradation: The objective of the PAPR reduction is for the

best system performance, including BER than that of the original OFDM system. For

that reason, all the methods, which have an incrementation in BER at the receiver,

must be paid more attention in practice. Additionally, if the side information is

received in error at the receiver, which may also result in entire wrong data frame and

thus the BER performance is reduced.

6) Without the additional power required: The design of a wireless system must

always take into account the efficiency of power. If an operation of the technique

which reduces the PAPR require more extra power, it deteriorates the BER

performance when the transmitted signals are normalized back to the original power

signal [109].

7) No spectral spillage: Any PAPR reduction techniques cannot devastate OFDM

fascinating technical features like immunity to the multipath fading. Thus, the spectral

spillage must be avoided in the PAPR reduction.

8) Other factors: It must be driven greater concentration on the effect of the nonlinear

devices utilized in signal processing loop in the transmitter like DACs, mixers and

HPAs since the PAPR reduction fundamentally avoid nonlinear distortion as a result

of these memories-less devices introducing into the communication channels. At the

same time, the expense of these nonlinear devices is too the important factor to design

the PAPR reduction scheme.

3.8 Fitness function-based approach for determining an appropriate

Algorithm [110]: In order to determine an appropriate PAPR reduction algorithm for a given system, it

is desirable to consider all above-listed requirements. The number and nature of these

requirements may vary depending upon the system (or user) under consideration. For

a given scenario and requirements, we propose to use the fitness value or

appropriateness value of the algorithm, which is defined as the weighted sum of the

relative changes in the above-listed factors. The appropriateness value provides a

single metric for determining the appropriateness of a PAPR reduction algorithm.

Suppose X1 be the relative degradation in BER performance at certain SNR level, for

given channel conditions, AWGN or multipath, given by:

X1 = −10 ( ) (3.20)

Let X2 be the relative increase in system complexity given by:

X2 = −10 ( ) (3.21)


38

Let X3 be the relative PAPR reduction given by:

X3 = −10 ( ) (3.22)

Let X4 be the relative cost savings given by:

X4 = −10 ( ) (3.23)

Let X5 be the relative increase in transmit power given by:

X5 = −10

( ) (3.24)

Let X6 be the relative increase in spectral spillage given by:

X6 = −10 (O ) (3.25)

Let X7 be the relative reduction in goodput5 given by:

X7 = −10 ( ) (3.26)

The aggregate fitness value of the PAPR reduction algorithm can be computed as the

weighted sum of these factors, where the weights correspond to their relative

importance levels. These weights can be determined as per the system or user

requirements. Therefore, the fitness value of the algorithm is given by:

∑ (3.27)

Where

∑ (3.28)

Based on these fitness values, an appropriate algorithm can be chosen in order to

achieve large reduction in PAPR values as well as satisfy other system requirements.

Chapter Four PAPR Reduction Techniques

39

Chapter Four

PAPR Reduction Techniques

4.1There are three different way to divide the PAPR:

4.1.1The first way is [110]:

PAPR reduction techniques can be categorized into deterministic and probabilistic

approaches, as shown in Figure 4.1. Deterministic approaches guarantee that the

PAPR of an OFDM signal does not exceed a predefined threshold, whereas the

probabilistic approaches minimize the probability that the PAPR of an OFDM signal

exceeds a predefined threshold. These categories will be discussed in the following

sections

1) Deterministic Approach

Deterministic PAPR reduction approaches can be classified into techniques that

perform either time-domain based clipping or frequency-domain based coding. The

simplest approach for PAPR reduction is to deliberately clip the amplitude of the

signal to a predefined value before amplification [111]. However, the technique

suffers from various drawbacks, such as signal distortion and spectral regrowth.

Therefore, clipping alone is not a suitable option for PAPR reduction. Modified

clipping techniques exist, which fall under the probabilistic approach explained in the

next section.

Coding techniques are applied to OFDM signals in order to map symbols to codes

with smaller PAPR values [112] .

Each symbol has a choice of two or more codes, where the code yielding the lowest

PAPR is selected. However, this technique works well only when the number of

subcarriers is small. With the increased number of subcarriers, the search space for

finding codes with minimum PAPR increases exponentially and large lookup tables

are needed for encoding and decoding.

2) Probabilistic Approach

Probabilistic approaches attempt to minimize the number of occurrences of OFDM

symbols with PAPR values exceeding a predefined threshold, while simultaneously

minimizing the signal distortion and spectral growth. Probabilistic approaches can be

classified according to whether time domain processing or frequency domain

processing is involved:

time Domain-Based Processing:

Time domain-based processing approaches focus on manipulating the power of the

signal in the time domain. This approach can be further classified into blind and non-

blind techniques. Blind techniques imply that the receiver is oblivious to the changes

made at the transmitter side, whereas non-blind techniques imply that the receiver

requires a priori knowledge about the modifications made at the transmitter side for

correctly demodulating the received signals. Thus, non-blind techniques require

additional side information in order to operate, whereas blind techniques might

degrade the error performance of the system since the receiver is transparent to the

changes made at the transmitter side.


40

The simplest blind technique for PAPR reduction is to clip the amplitude of the signal

to a predefined value and filter the signal to suppress the out-of-band interference

[113,114, 115 ] . The clipping process might result in spectral regrowth, whereas

filtering the signal might result in some peak regrowth. Therefore, clipping may not

be an effective technique when reducing the PAPR of the OFDM signals as long as

the transmitted OFDM signal is strictly band-limited. Even though numerous

algorithms based on amplitude clipping and filtering have been proposed in the

literature, it has been shown that clipping does not improve the reduction of total

degradation [116]. Instead, an unclipped system outperforms a clipped system

because of the inter-carrier interference (ICI) caused by clipping, and offsets the gain

of the PAPR reduction [116]. Another technique called peak windowing can also

reduce the PAPR, where large signal peaks are multiplied with a certain narrowband

window such as Gaussian, Cosine, Kaiser, and Hamming windows [117].

Among the non-blind techniques, several companding4 techniques for compressing

the large peaks of an OFDM signal in time domain, including μ-law companding , and

exponential companding , have been proposed in literature. By compressing the large

peaks of an OFDM signal by companding, the dynamic range of the D/A converters

are reduced. However, the receiver needs to expand the compressed signal for correct

demodulation.

Frequency Domain-Based Processing

Frequency domain-based processing approaches focus on minimizing the correlation

of the input signals since it is known that the PAPR of an OFDM signal is high when

the input sequences are highly correlated. It has been shown that by altering the phase

and/or power of the input sequence, it is possible to lower the correlation of the input

sequence, thereby reducing the PAPR of an OFDM signal. However, some

approaches also try to directly manipulate the correlation of the input signals.

Frequency domain-based processing approaches can be further classified into blind

and non-blind techniques. In blind phase adjustment-based techniques, the phase of

the subcarriers are adjusted in order to reduce the coherence between the different

subcarriers such that the PAPR value of the OFDM signal is reduced. The phase

adjustments should be kept relatively small so as to minimize bit-error-rate (BER)

performance degradation. For example, signal set expansion technique maps original

signal set into an expanded signal set with two or more points, such as binary phase

shift keying (BPSK) into quadrature phase shift keying (QPSK), which provides more

freedom for phase selection and yields lower PAPR values for the OFDM signal

[118].

Blind power-based techniques alter the power level of the subcarriers such that the

PAPR of an OFDM signal is reduced. These techniques are suitable only for the

MPSK-based OFDM system since the receiver is unaware of the information about

the transmit power levels. For example, the input sequence envelope scaling technique

adjusts the power of the subcarriers so that the power of the individual subcarriers

becomes unequal yielding a minimized PAPR value [119]. Since the phase

information of the original signal is unchanged, the receiver can decode the received

signal without any side information.

In blind power and phase-based techniques, both the phase and the power of the

subcarriers are altered such that the PAPR of an OFDM signal is reduced. If the total

transmit power needs to be kept constant, these techniques are suitable only for low

order modulation techniques since the error robustness of the higher modulation

techniques degrades rapidly with the blind phase and power alterations at the


41

transmitter. When the order of the modulation techniques in-creases, the complexity

(and limitations) of the algorithm increases as well as transmit power level increases.

For example, the active constellation extension (ACE) [120,121] and dynamic

constellation shaping techniques allow changing the power and phase of some data

symbols without affecting the error probability of the other data symbols.

Non-blind power-based techniques, as well as power and phase-based techniques,

would be suitable for the higher modulation schemes such as MQAM. Non-blind

phase adjustment-based techniques update phases of the input sequence such that the

PAPR of an OFDM signal is reduced. The information about the phase updates is

transmitted to the receiver for correct demodulation. Several modified algorithms are

proposed in literature, which avoid the requirement of explicit side information. For

example, selective mapping (SLM)[9], partial transmit sequences (PTS) [122],

random phase updating [123] techniques add random phase factors to each subcarriers

in order to reduce PAPR with the information about the phase factors transmitted to

the receiver. The blind techniques reduce the PAPR values at the cost of slight

increase in the bit error rate of the system or increased transmit power level since the

adjustments would result into increased noise level at the receiver, whereas the non-

blind techniques reduce the PAPR values at the cost of a reduced information rate

since the information about the adjustments made at the transmitter need to be

transmitted to the receiver for the correct demodulation.

A low autocorrelation coefficient of a signal is a sufficient condition for low PAPR.

However this is not a necessary condition [124][125]. Non-blind autocorrelation

minimization techniques attempt to minimize the autocorrelation of the input

sequence `and the information about the changes is transmitted to the receiver for

correct

demodulation. For example, the selective scrambling [126] and interleaving

techniques [127] attempt to break the long correlation patterns of the input sequences

to reduce the PAPR. However, the techniques perform well only when the OFDM

signal has moderate PAPR values since interleaving alone is not effective to break the

correlation pattern when the input sequence are highly correlated.

Attempts have been made to develop OFDM signals with a constant envelope to yield

unity PAPR values [128] . The constant envelope waveforms have a constant

instantaneous power. Continuous phase modulation (CPM) is a class of signaling that

has very low side lobe power while maintaining the constant envelope property.

However, CPM increases the complexity of the receiver and has a poor performance

over frequency selective channels.


42

Figure 4.1.the first way taxonomy of PAPR Reduction techniques


43

4.1.2 The second way is : a) Distortion Based Techniques [11]-[8]-[4]

b) Scrambling Techniques [17]-[16]-[8]

As shown in figure 4.2

a. DISTORTION BASED TECHNIQUES

The schemes that introduce spectral re-growth belong to this category. Distortion

based techniques are the most straightforward PAPR reduction methods. Furthermore,

these techniques distort the spectrum, this spectrum distortion or “spectral re-growth”

can be corrected to a certain extent by using filtering operation [62 ,129]. These

methods reduce the PAPR by distorting the OFDM signal non-linearly. The methods

like clipping and filtering, peak windowing, and non-linear companding are the

example of these techniques. These techniques are applied after the generation of

OFDM signal (after the IFFT) [130].

The distortion category attempts to reduce PAPR by manipulation of signal before

amplification. Clipping of signal prior to amplification is a simplest method but it

causes increase in both out-of-band (OOB) as well as in-band interference thus

compromises upon performance of system. Amongst this category better techniques

include companding, peak windowing, peak power suppression, peak cancellation,

weighted multicarrier transmission etc. Any technique which is used to reduce PAPR

should not only have high spectral efficiency but must be compatibility with the

existing modulation schemes and at the same time must not be computational

complex [100].

b. Scrambling techniques :

Signal scrambling techniques are all variations on how to scramble the codes to

decrease the PAPR. Coding techniques can be used for signal scrambling. Golay

complementary sequences, Shapiro-Rudin sequences, M sequences, Barker codes can

be used efficiently to reduce the PAPR. However with the increase in the number of

carriers the overhead associated with exhaustive search of the best code would

increase exponentially. More practical solutions of the signal scrambling techniques

are block coding, Selective Level Mapping (SLM) and Partial Transmit Sequences

(PTS). Signal scrambling techniques with side information reduces the effective

throughput since they introduce redundancy [131] [132].


44

Figure 4.2.the second way taxonomy of PAPR Reduction techniques


45

4.1.3 The third way is [98]:

These methods are basically divided in five categories:

(1) The clipping technique

(2) Coding Methods,

(3) Probabilistic (Scrambling) Techniques

(4) Pre-distortion Methods.

1. The clipping technique employs clipping or nonlinear saturation around the peaks

to reduce the PAPR. It is simple to implement, but it may cause in-band and out-of-

band interferences while destroying the orthogonality among the subcarriers. This

particular approach includes block-scaling technique, clipping and filtering technique,

peak windowing technique, peak cancellation technique, Fourier projection technique,

and decision-aided reconstruction technique [133] [134].

2. The coding technique is to select such code words that minimize or reduce the

PAPR. It causes no distortion and creates no out-of-band radiation, but it suffers from

bandwidth efficiency as the code rate is reduced. It also suffers from complexity to

find the best codes and to store large lookup tables for encoding and decoding,

especially for a large number of subcarriers. Golay complementary sequence, Reed

Muller code, M-sequence, or Hadamard code can be used in this approach [133][134].

3. The probabilistic (scrambling) technique is to scramble an input data block of the

OFDM symbols and transmit one of them with the minimum PAPR so that the

probability of incurring high PAPR can be reduced. While it does not suffer from the

out-of-band power, the spectral efficiency decreases and the complexity increases as

the number of subcarriers increases. Furthermore, it cannot guarantee the PAPR

belowa specified level. This approach includes SLM (Selective Mapping), PTS

(Partial Transmit Sequence).

4. The pre-distortion methods are based on the re-orientation or spreading the energy

of data symbol before taking IFFT. The pre-distortion schemes include DFT

spreading, pulse shaping or precoding and constellation shaping. The methods like

Tone Reservation (TR) and Tone Injection (TI) are the example of constellation

shaping schemes [10].

The DFT-spreading technique is to spread the input signal with DFT, which can be

subsequently taken into IFFT. This can reduce the PAPR of OFDM signal to the level

of

Single-carrier transmission. This technique is particularly useful for mobile terminals

in uplink transmission. It is known as the Single Carrier-FDMA (SC-FDMA), which

is adopted for uplink transmission in the 3GPP LTE standard [135].

4.1.4 And finally there is Hybrid techniques:

Besides these different PAPR reduction techniques, some hybrid methods are also

available in the literature [136 ,137,138 ] . These methods combine two or more than

two techniques for PAPR reduction like clipping with coding, SLM with coding, pre-

coding with clipping, interleaving and companding , Selective Mapping and Binary

Cyclic Codes, combining Hadamard Transform and Hann peak windowing etc. The

hybrid methods are considered as better choice for PAPR reduction because it possess

the advantages of both techniques used in hybridization with slight increases in

complexity.


46

4.2 Clipping and Filtering : The clipping is the simplest method of PAPR reduction. Clipping limits the maximum

amplitude of OFDM signal to a pre-specified level. The implementation of clipping is

relatively easy.

The simplest and most widely used technique of PAPR reduction is to basically clip

the parts of the signals that are outside the allowed region .For example; using HPA

with saturation level below the signal span will automatically cause the signal to be

clipped. For amplitude clipping, that is [109]:

(4.1)

Where A is preset clipping level and it is a positive real number

Generally, clipping is performed at the transmitter. However, the receiver need to

estimate the clipping that has occurred and to compensate the received OFDM symbol

accordingly. Typically, at most one clipping occurs per OFDM symbol, and thus the

receiver has to estimate two parameters: location and size of the clip. However, it is

difficult to get this information. Therefore, clipping method introduces both in band

distortion and out of band radiation into OFDM signals, which degrades the system

performance including BER and spectral efficiency. Filtering can reduce out of band

radiation after clipping although it cannot reduce in-band distortion. However,

clipping may cause some peak regrowth so that the signal after clipping and filtering

will exceed the clipping level at some points [108] [109].

It has following drawbacks [98] [139]:

(a) It causes in-band signal distortion, resulting in BER performance degradation.

(b) It also causes out-of-band radiation, which imposes out-of-band interference

signals to adjacent channels. The out-of-band radiation can be reduced by filtering,

but the filtering may affect high-frequency components of in-band signal (aliasing)

when the clipping is performed with the Nyquist sampling rate.

(c) Filtering after clipping can reduce out-of-band radiation at the cost of peak re-

growth. The signal after filtering operation may exceed the clipping level specified for

the clipping operation.

To reduce overall peak re-growth, a repeated clipping and filtering can be used to

obtain a desirable PAPR at the cost of increase computational complexity . To reduce

peak regrowth, a repeated clipping-and-filtering operation can be used to obtain a

desirable PAPR at a cost of computational complexity increase. As improved clipping

methods, peak windowing schemes attempt to minimize the out of band radiation by

using narrowband windows such as Gaussian window to attenuate peak signals [140]

Some of clipping techniques:

1. Repeated Clipping [13]

The clipping technique is the simpler one which is used to cut the signal peak up to

desired threshold level. But repeated clipping and filtering technique proved to be

worthy one as it gives better result compared to earlier one. In this technique the peak

regrowth which is generated in filtering can be minimized. So the repeated clip and

filter process reduces these regrowth's in OFDM system

2. Reconstruction of Lost Clipped Signal


47

To remove the peak regrowth of signal oversampled sequence clipping is used which

can reconstruct the clipped samples and mitigate the clipping distortion in presence of

channel noise at the cost of bandwidth expansion. It is observed that by increasing

small bandwidth , the performance of OFDM system can be improved . PAPR is the

biggest problem in OFDM system. Many techniques are proposed for it. Clipping and

filtering technique is considered to be the simplest one [114][106].

3. Iterative Clipping & Filtering Technique

This technique is used to eliminate the peak regrowth due to CF technique. In each

iteration peak regrowth decreases significantly. The process of iteration undergoes

FFT/IFFT and one extra IFFT is required for conversion into time domain in OFDM

[115][106].

4. Recursive Clipping and Filtering with Bounded Distortion (rcfbd)

In RCF the signal is clipped by repeating process many times before feeding to power

amplifier. When the process of repetition exhibit on the signal the out of band spectral

density and the probability of the occurance of PAPR decreases but error rate

increases due to increase in number of repetitions. The bit error rate increases due to

increase in inband distortion. So to remove this increased error rate another improved

technique is proposed called recursive clipping and filtering with bounded distortion

(RCFBD) to achieve PAPR reduction. The idea of this technique is same as

oversampled digital clipping in time domain and removing out of band components in

frequency domains are used. But additional barrier on in band distortion of each

subcarrier is applied during the recursive process. In this way PAPR can be reduced

without producing any effect on the error rate [114][ 106].

RCFBD minimize PAPR and keeps the control on the distortion of data carried by

each subcarrier. So by using this technique side information can be eliminated and

receiver part becomes less complex and BER performance can be increased more. It is

also more robust against AWGN noise [113].

4.3 Peak Windowing Method: It is an improved clipping method. The basic aim of peak windowing is to reduce the

out-of-band radiation by using narrow band windows such as Gaussian window to

attenuate peak signals. As a matter of fact, any window which is narrow in time

domain and having good spectral properties can be used [10]. In 2008, an advance

peak windowing method has been given by S. Cha which overcomes the drawback of

normal peak windowing method. It effectively suppresses the peak signals to the

desired threshold level in case when the successive peaks occur within a half of the

window length [10].

The peak windowing method has been suggested by Van Nee and Wild [117]. This

method, proposes that it is possible to remove large peaks at the cost of a slight

amount of self-interference when large peaks arise infrequently. Peak windowing

reduces PAPRs at the cost of increasing the BER and out-of-band radiation. Clipping

is a one kind of simple introduces PAPR reduction technique which is self-

interference. The technique of peak windowing offers better PAPR reduction with

better spectral properties.

(Peak Windowing technique provides better PAPR reduction with better spectral

properties than clipping) [141][142].


48

In peak windowing method we multiply large signal peak with a specific window, for

example; Gaussian shaped window, cosine, Kaiser and Hamming window. In view of

the fact that the OFDM signal is multiplied with several of these windows,

consequential spectrum is a convolution of the original OFDM spectrum with the

spectrum of the applied window. Thus, the window should be as narrow band as

possible, conversely the window should not be too long in the time domain because

various signal samples are affected, which results an increase in bit error rate (BER).

Windowing method, PAPRs can be obtained to 4dB which from the number of

independent subcarriers. The loss in signal-to-noise ratio (SNR) due to the signal

distortion is limited to about 0.3dB. A back off relative to maximum output power of

about 5.5dB is needed in spectra distortion at least 30dB below the in-band spectral

density [141][142].

The PAPR reduction performance as well as spectral efficiency of peak windowing

technique is better as compared to clipping. The major advantage of peak windowing

is that PAPR reduction is achieved with minimal complexity for any number of sub

carriers. The disadvantages include an increase in out-of-band interference and BER

[100][143].

4.4 Envelope Scaling: The Envelope Scaling technique has been proposed by Foomooljareon and Fernando.

They anticipated a new algorithm to reduce PAPR by scaling the input envelope for

some subcarriers before they are sent to IFFT. They used 256 subcarriers with QPSK

modulation technique, so that envelopes of all the subcarrie4rs are equal. The key idea

of this scheme is that the input envelope in some sub carrier is scaled to achieve the

smallest amount of PAPR at the output of the IFFT. Thus, the receiver of the system

doesn‟t need any side information for decoding the receiver sequence. This scheme is

appropriate for QPSK modulation; the envelopes of all subcarriers are equal. Results

show that PAPR can be reduced significantly at around 4 dB [144].

In Envelope Scaling, the input envelopes of sub carriers are scaled prior to IFFT. The

base for this scheme is the facts that with PSK modulation all the sub carriers input

envelops are equal. Hence input envelop of some sub carriers is scaled in such a way

that minimum PAPR is achieved at IFFT output. The input which yields minimum

PAPR is fed into the system. The phase information of the input sequence is similar to

original however envelops are not the same. Hence decoding of sequence can be done

by receiver without any requirement for side information .The major drawback of this

method is that it can only be used when OFDM is employing PSK modulation. On the

other hand if we use this method when QAM modulation is implemented by OFDM,

then there is severe degradation in BER performance results [100] [145] .

4.5 Peak Reduction Carrier: Peak Reduction Carrier technique is proposed by Tan and Wassell. The technique is

to use the data bearing peak reduction carriers (PRCs) to reduce the effective PAPR in

the OFDM system. It includes the use of a higher order modulation scheme to

represent a lower order modulation symbol. Hence the phase and amplitude of these

carriers remains inside the constellation area which represents the data symbols being

transmitted. This method is suitable for PSK modulation; where the envelopes of all

subcarriers are the same. When the QAM modulation scheme will be implemented in

the OFDM system, the carrier envelope scaling will result in the serious BER


49

degradation. To limit the BER degradation, amount of the side information would also

be excessive when the number of subcarriers is large [141]

Amongst drawbacks of PRCs, one is that the overall data transmission efficiency of

the system is compromised if we try to achieve maximum PAPR reduction efficiency.

At the same time the BER performance is also affected adversely because of

employing constellation of higher order for carrying symbols of lower order results in

higher probability of error [100]

4.6 Companding Technique: Non-linear companding is an especial clipping technique which offers good PAPR

reduction with better BER performance, low implementation complexity, and no

bandwidth expansion [109] [145].

The difference between clipping and companding is that the clipping process

deliberately clips the large amplitude signals; therefore the signal cannot be recovered

exactly. On the other hand, the companding transform compand the original signals

using strict monotone increasing function; therefore the companded signals can be

recovered correctly through the corresponding inversion of companding transform at

the receiver. Clipping does not affect small amplitude signal, whereas companding

enlarge the small signals while compressing the large amplitude signals. A lot of

companding techniques are available. The basic concept of most of the companding

techniques is to transform the Rayleigh distributed OFDM signal into a uniformly

distributed signal [10].

It was based on the speech processing algorithm μ-law and it has shown better

performance than that of clipping method . The μ-law companding transform mainly

focuses on enlarging small amplitude signals while keeping peak signals unchanged,

and thus it increase the average power of the transmitted signals and possibly results

in exceeding the saturation region of HPA to make the system performance worse

[140]. In order to overcome the problem of μ-law companding (increasing average

power) and to have efficient PAPR reduction, some more Companding Transform

have been suggested [146,147,148,149,150, and 151] .

Figure 4.3 Block diagram of Companding of OFDM system


50

4.6.1 Square-Rooting Companding Technique ( SQRT) for PAPR

Reduction in OFDM Systems: The block diagram of a typical OFDM system using the original SQRT technique for

PAPR reduction is shown in figure 4.4. By using the SQRT technique, the original

OFDM output signals is processed by (3.21) before they are converted into analog

waveforms and amplified by the power amplifier

√| | (4.2)

is the new OFDM signal, and is the phase of During the entire signal

processing, the phases of the OFDM output signals are kept unchanged while only

the amplitudes are treated and changed [152].

For the complex Gaussian distributed signals, such as OFDM output signals, SQRT

process changes the Rayleigh distribution of these signals into a Gaussian-like, close

to Gaussian, distribution [16,152]; while the Chi-square distribution is converted,

according to the analysis of these signals given in the previous section, to Rayleigh

distribution. The latter is because the Rayleigh distribution in such signals represents

voltage, while the Chi_square distribution represents the power of the same signals.

However, not only the statistical distribution is changed by the SQRT process, but the

values of the mean and variance of the processed OFDM output signals are also

changed, and subsequently the values of the average power and peak power of these

signals are altered also. To understand the effect of SQRT process on the power

values of OFDM output signals, we assume normalized average power ( )

Figure 4.4 Block diagram of an OFDM system using SQRT technique

When the average power is normalized, the value of the peak power is diminished by

N because for the same PAPR. This assumption is applicable for all

OFDM symbols as the average power is constant and equal to ( ) Hence, the PAPR can be analyzed according to (3.21) through the peak power

only. The new value of normalized peak power is always greater than one because

is constantly greater than in all OFDM symbols. Therefore, the SQRT

process always causes a reduction in the value of the peak power of the normalized

OFDM symbols, and as a result the PAPR is reduced in all sizes of OFDM blocks, N.


51

In [16, 152] the SQRT process is applied on the signals of all OFDM output symbols;

therefore, the PAPR reduced without the need to send side information. The SQRT

process changes the distribution of the power signals to Rayleigh distribution and

reduces the value of average power from N to N1/2. The variance of the Rayleigh

distribution equals ( ) [152] which is approximately equal to half the

value of variance of the Gaussian distributed signals. The SQRT process in the SQRT

OFDM system performs this statistical transformation, and therefore results in a

constant degradation in the BER rate equal to 3 dB because of decreasing of variance

to the half of that of the conventional OFDM system (

)

4.6.2 Exponential Companding Algorithm:

A nonlinear companding algorithm, called “exponential companding”, developed to

reduce the high (PAPR) of (OFDM) signals. Exponential companding technique

adjusts both large and small signals and can keep the average power at the same level.

By transforming the original OFDM signals into uniformly distributed signals, the

exponential companding schemes can effectively reduce PAPR for different

modulation formats and sub-carrier sizes [15]. Let | | be the power of the

amplitude of companded signal, have a uniform distribution in the interval , - .The

exponent is called the degree of a specific exponential companding scheme the

CDF of | | is simply

| | ( )

(4.3)

The amplitude of the | | of companded signal has the following CDF

| |( ) *| | + (4.4)

*| | + (4.5)

(4.6)

The inverse function of | |(x)

| | ( ) √

(4.7)

On the other hand, given that ( )is a strictly monotonic\ increasing function, we

have,

| |( ) *| | +

(4.8)

* (| |) ( )+ (4.9)

| |( ( )) , ( √

) (4.10)

Considering the phase of input signals, the companding function ( ) is given by:

( ) ( ) | | | | (4.11)


52

( ) √ 0 (

)1

(4.12)

Where ( ) is sign function. „d‟ is the degree of companding scheme, is the

variance of input signal applied for companding. The positive constant determines

the average power of output signals. In order to keep the input and output signals at

the same average power level, we let

(

[| | ]

√[ ( | |

)]

)

(4.13)

At the receiver side, the inverse function ( ) of is used in the de-companding

operation

( ) ( )√ (

)

(4.14)

Figure 4.5 shows the exponential companding function ( ) with degree as a

parameter. The companded signals have uniformly distributed amplitudes and powers,

respectively for the cases and .

When , the ( ) can compress large input signals and expand small signals

simultaneously. While the -law companding scheme can only enlarge small signals

and does not change the signal peaks, which leads to a higher average power level of

output signals. As seen, the differences between exponential companding functions

are ignorable when [15] [153].

Figure 4.5 The exponential companding function h(x).


53

4.6.3 Trapezoidal power companding:

Is a nonlinear companding technique called “trapezoidal power companding” to

reduce the high PAPR in a complex OFDM by transforming the original signals into

new signals whose power is trapezoidally distributed. A flexible parameter is used to

determine the shape of the trapezium so that the trapezoidal power companding

scheme is able to meet the requirements for various conditions. Given an expected

PAPR value, the scheme provides a closed-form solution that guarantees the actual

PAPR the same as the expected [29].

A flexible trapezoidal design was introduced in [29], [154], transforming the

amplitude of the signals into a distribution of various trapezoidal shapes.

Since that scheme is based on the assumption that all signals are purely real or

imaginary, consequently, when the design is used in a complex system, the

theoretically estimated PAPR would be different than the actual value.

This companding scheme has three desired properties mentioned above. It converts

the power distribution of the original signals (as opposed to the amplitude used in

[29]) into a trapezoidal distribution while keeping the average output power the same

as the original signals. A parameter is used to determine the slope of the hypotenuse

so that the trapezium could have a different shape

The companding function ( ) is given by:

( )

{

| |

√

√.

/

(

| |

)

| |√ (

| |

) }

(4.15)

Where k is the slope of the trapezium. Is the maximum power

The decompanding function at the receiver can be given as:

( )

{

| ||√ (

| | ( | |

)|

| ||√ (

| |

)|

}

(4.16)

Notice that when , the power distribution is actually a rectangular distribution,

which is the same as the case in exponential companding .Since sometimes a

received signal is so distorted that the square root part in (3.35) would be an

imaginary or complex number, we then take the absolute value of the square root

parts to eliminate any further potential phase distortion.

When the flexible trapezoidal companding curve is then the same as the EC

curve [29].

4.6.4 Hyperbolic tangent ( ) companding [30]: The hyperbolic tangent ( ) companding function is defined by

( )= ( ) (4.17)


54

Where and are positive numbers controlling the companding level applied to

the envelope x.

4.6.5 Error Function ( ) Companding [30]: The error function ( ) is defined by

( )= ( ) (4.18)

Where and are positive numbers controlling the level of companding

4.6.6 Logarithm Function (log) Companding [30]: The logarithm ( ) companding function is defined by

( )= ( ) (4.19)

Where and are two positive numbers controlling the amount of companding.

[83, 84]

4.7 Coding techniques: Many early papers considered how standard coding techniques could be applied to

OFDM. The basic premise of coding is to insert redundant bits into the data stream

which can be used for error correction at the receiver. Their application to PAPR

reduction is in creating sequences of bits which will exhibit low PAPR after the IFFT.

There are 2 types of error detection and correction codes, block codes and

convolutional codes. Most papers relate to the block coding family for PAPR

reduction. During the encoding process k information bits are encoded into n code d

bits, therefore (n-k) redundant non information bits are added to the k information bits

[78].The block code is referred to as an (n,k) code, and the rate of the code as Rc=k/n.

Figure 4.6 is a block diagram showing where coding for PAPR reduction is located in

an OFDM transmitter.

4.6 Block diagram of OFDM transmitter showing PAPR coding

Different codes exhibit different degrees of error correction ability. Another important

property of codes is the weight of the code, which is the number of non-zero elements

in the codeword. Types of block codes are Hamming, Golay, and Reed- Solomon,

some of which are used for PAPR reduction .


55

The basic idea of all coding schemes for the reduction of PAPR is to reduce the

occurrence probability of the same phase of N signals. The coding method selects

such code words that minimize or reduce the PAPR. It causes no distortion and

creates no out-of-band radiation, but it suffers from bandwidth efficiency as the code

rate is reduced. It also suffers from complexity to find the best codes and to store

large lookup tables for encoding and decoding, especially for a large number of

subcarriers [10].

A simple block coding scheme was introduced by Jones et al.[155], and its basic idea

is that mapping 3 bits data into 4 bits codeword by adding a Simple Odd Parity Code

(SOBC) at the last bit across the channels. The main disadvantage of SOBC method is

that it can reduce PAPR for a 4-bit codeword [109]. Later, in 1996 Wulich applied the

Cyclic Coding (CC) to reduce the PAPR [156]. In 1998, Fragiacomo proposed an

efficient Simple Block Code (SBC) to reduce the PAPR of OFDM signals [157].

However, it is concluded that SBC is not effective when the frame size is large.

Subsequently, Complement Block Coding (CBC) and Modified Complement Block

Coding (MCBC) schemes were proposed to reduce the PAPR without the restriction

of frame size [158][159]. CBC and MCBC are more attractive due to their flexibility

on choosing the coding rate, frame size and low implementation complexity. CBC and

MCBC utilize the complementary bits that are added to the original information bits

to reduce the probability of the peak signals occurrence. To make comparisons, some

results of the PAPR reduction obtained with different coding schemes have been

shown in Table 4.1, in which the number of subblock is 2 and the coding rate

for MCBC.

Table 4.1 PAPR Reduction comparison with different coding schemes

About 3-dB PAPR reduction can be obtained when coding rate ( )

by using

CBC with long frame size. It is also shown that the PAPR reductions obtained with

CBC when coding rate ( )

are almost the same as that when

( )

. In

addition, when coding rate is 3/4, more than 3-dB more PAPR reduction can be

obtained using MCBC than the other schemes with any frame size. The flexibility in

coding rate choice and low complexity makes the proposed CBC and MCBC schemes

attractive for OFDM systems with large frame sizes and high coding rates [109].

The [160][161][162] authors used the Golay complementary sequences to achieve the

PAPR reduction, in which more than 3-dB PAPR reduction had been obtained. Codes


56

with error correcting capabilities has been proposed in [163] to achieve more lower

PAPR for OFDM signals by determining the relationship of the cosets of Reed-Muller

codes to Golay complementary sequences. While these block codes reduce PAPR,

they also reduce the transmission rate, significantly for OFDM systems with large

number of subcarriers. In fact, let C be a code defined over an equal energy

constellation, R denotes the rate and L denotes the length of the C, respectively, then

C has possible codewords. Therefore, it is possible to compute the codewords

with large PAPR by trying all the codewords of C and computing the peaks of the

corresponding signals at some selected time points [109].

However, it is little hope for computing the PAPR of an arbitrary code when L is

large. Even if it is possible, the complexity is still too high. Based on this motivates,

authors of [159] proposed a novel method of computation and reduction of the PAPR

and it mainly introduced a specific phase shift to each coordinate of all possible

codewords where phase shifts are independent of the codewords and known both to

transceiver, then it can be freely obtained more 4.5-dB PAPR reduction by using the

optimized phase shifts. From this viewpoint, we also consider the coding scheme of

PAPR reduction as a special phase optimization. In summarization, the inherent error

control capability and simplicity of implementation make coding method more

promising for practical OFDM systems design. However, the main disadvantage of

this method is the good performance of the PAPR reduction at the cost of coding rate

loss.

Coding techniques for PAPR reduction where redundant bits are added to the bit

stream before the IFFT. Properly chosen, these codewords ensure that the PAPR after

the IFFT is kept low. These codes can be combined with existing COFDM to reduce

the redundancy and complexity inherent in coding. A disadvantage of coding is that

the complexity becomes prohibitively high with an increase in the number of

subcarriers (>32). Various codewords were presented such as cyclic codes, Shapiro-

Rudin Sequences, Golay Complementary codes, and Reed-Muller codes. Golay codes

and their subset, second order Reed Muller codes were found to have excellent PAPR

properties restricting the PAPR to 3dB. This reduction could be traded off with

reductions in complexity and the code length. Still complexity remains a restrictive

issue in coding [78].

4.8 Selective Mapping (SLM): In SLM, the basic idea is to generate a set of OFDM signals, all of them representing

the same data block, and then transmitting the one with the lowest PAPR [9][10]. The

major drawback of SLM method is that it is more computationally complex because

more than one IFFT blocks are used. It also decreases the data rate because the

selected signal index, called side information, must also be transmitted to allow for

the recovery of the original data block at the receiver side. The eventual loss of the

side information during transmission significantly degrades the error performance of

the system since the whole data block is lost in this case. Therefore, a lot of work has

been suggested as a modified SLM to reduce the computational complexity [164] and

to reduce or to remove the side information transmitted [125].

In SLM, the input data sequences are multiplied by each of the phase sequences to

generate alternative input symbol sequences. Each of these alternative input data

sequences are then applied to IFFT operation, and then the one with the lowest PAPR

is selected for transmission [165]. A block diagram of SLM techniques is shown in


57

Figure 4.7. The input data is partitioned into a data block Y of length N. Then these

data block is multiplied element by element with phase sequence ( )

( ( ) , - (4.20)

resulting into U modified data blocks ( )( ( ) , - where

(4.21)

After that, the N-point IFFT of each data block ( ) is taken, the resulting OFDM

signal is given as –

( )

∑

.

/ (4.22)

Among the OFDM data blocks ( ) , only one with the lowest

PAPR is selected for transmission and the corresponding selected phase

factor also transmitted to receiver as side information. For implementation

of SLM OFDM systems, the SLM technique needs U- IFFT operation and the number

of required bits as side information is , - for each data block. Therefore, the

ability of PAPR reduction in SLM depends on the number of phase factors and the

design of the phase factors. The major drawback of SLM method is that it is

more computationally complex and less bandwidth efficient (side information is

required). Therefore, a lot of work has been suggested as a modified SLM to reduce

the computational complexity and to reduce or to remove the side information

transmitted [10].

Figure 4.7 Block diagram of selective mapping (SLM) technique for PAPR reduction

4.9 Partial Transmit Sequence (PTS) : In PTS, the original data block is divided into multiple non-overlapping sub-blocks.

Then these sub-blocks are rotated with rotation factors which are statistically

independent. After that, the signal with the lowest PAPR is chosen for transmission.


58

There are several ways for the partition of the data sequence into multiple sub-blocks,

including adjacent partition, interleaved partition and pseudorandom partition [122].

Among them, pseudo-random partitioning has been found to be the best choice.

Similar to SLM, the major drawback of PTS is also the computational complexity

(search complexity for optimal phase factor, and more than one IFFT blocks) and low

data rate (required side information). Several techniques have been proposed in the

literature to reduce the search complexity and overhead (by reducing/avoiding the

usage of side information) [166]. The complexity of PTS is less than SLM [167].

In PTS method, the original frequency-domain data sequence is divided into multiple

disjoint sub-blocks, which are then weighted by a set of phase sequences to create a

set of candidates Finally, the candidate with the lowest PAPR is chosen for

transmission [122]. A block diagram of PTS techniques is shown in Figure 4.8

The input data block in Y is divided in to M disjoint sub-blocks, which are

represented by the vectors { ( ) + The input data block Y can be

written in terms of ( )as

∑ ( ) for (4.23)

Where, ( )

with

= or 0

After that, the sub-blocks ( ) are transformed into M, time-domain partial transmit

sequences by taking the IFFT of length N. These partial transit sequences can be

written as:

( )

[ ( )] for (4.24)

These partial sequences ( )are then independently rotated by phase factors

* , for The rotated partial sequences are then

optimally combined to obtain the OFDM signals with lowest PAPR[10] . The time

domain signal after combining is given by

∑ ( ) (4.25)

There are two main issues of any PTS scheme: to reduce the computational

complexity for searching the optimal phase factors and to reduce the overhead by

minimizing the side information. Suppose that there are W phase angles to be

allowed, thus can has the possibility of W different values. Therefore, there are

alternative representations for an OFDM symbol. The search complexity

increases exponentially with the number of sub-blocks M To reduce the search

complexity and overhead (by reducing/avoiding the usage of side

information)[166].These methods achieve significant reduction in search

complexity with marginal PAPR performance degradation. In 2007, R. J. Baxley

et.al [167] gave a useful comparison between PTS and SLM techniques. It has been

shown that the PTS outperforms SLM in terms of PAPR reduction at the cost of

increase side.


59

Figure 4.8 Block diagram of partial transmit sequence (PTS) technique for PAPR

Reduction

4.10 Tone Reservation : In TR subcarriers, called Peak Reduction Tones (PRT‟s) [168], are set aside for PAPR

reduction as shown in the transceiver block diagram in Figure 4.9.

Tone reservation implemented a projection onto convex sets (POCS) method. Later,

Tellado and Cioffi [169] discussed the idea of tone reservation as a linear

programming problem that has an exact solution (the POCS method is suboptimal).

The linear programming solution can be reached with complexity O [N log N].

The idea behind tone reservation is to isolate energy used to cancel large peaks to a

predefined set of tones. These tones do not bear any useful information and are

orthogonal to the data bearing tones. This orthogonality makes recovering the data

trivial [100].

The advantages of TR technique include:

1. No need for side information

2. Fewer complex-multiplications as only one time IFFT operation is needed. But

multiple iteration operations are needed after IFFT operation.

3. No special receiver operation is needed

While promising, tone reservation has several shortcomings. First the data rate is

necessarily decreased because some tones are used strictly for PAR reduction. The

second problem is the difficulty of selecting which tones to reserve. A random search

over all the possible sets, B, would greatly increase the complexity of tone

reservation.

Often the tones have to be chosen contiguously because fades often affect contiguous

sets of sub carriers. These contiguous sets of tones are known to have bad PAR

reduction abilities. The third issue is a tradeoff between the quantities of reserved

tones and the rise in average power due to tone reservation. More the tones that are

reserved, lesser the power needs to be allocated for PAPR reduction. On other hand,

more reserved tones mean more unused bandwidth that could be data bearing [100].


60

Figure 4.9 Block diagram of a Tone Reservation (TR) OFDM transceiver.

4.11 Tone Injection: Motivated by the data rate loss of tone reservation, Tellado introduced a new

technique named tone injection [170] as shown in figure 4.10. It reduces the PAPR

without compromising the data rate. In this method the size of the basic constellation

is increased. Hence mapping of original constellation points into numerous

corresponding points in the new stretched out constellation becomes possible. The

distance between these duplicate points can be calculated as d√ , where M=

constellation size, and .

There is no effect on BER and all we have to do is add a modulo-D subsequent to FFT

in the receiver side. Since mapping of each information unit into numerous

corresponding constellation points is done, it gives a margin of free will which can be

used reduction of PAPR [100]

Figure 4.10 Block diagram of a Tone Injection (TI) OFDM transceiver


61

4.12 Interleaving [171][172]:

This method is also termed as Adaptive Symbol Selection Method .Multiple OFDM

symbols are created by bit interleaving of input sequences .The basic Idea is to use W

interleaving ways and selecting one with the lowest PAPR.

Figure 4.11 shows an interleaver, PAPR Reduction capability depends on the number

of interleaver used .To recover the signals the receiver need to know the information

about which interleaver is used.

Figure 4.11: Interleaving

4.13 Active Constellation Extension (ACE) [173][174]: This technique deals with extending the constellation points outside the signal

constellation which is then used to cancel the time domain peaks .Figure 4.12 shows

the points where these constellation points can be extended. Is technique has several

advantages like no loss of data, no degradation in system performance, lower BER as

compared to other techniques and bears no side information like SLM. Some

variations of this method like clipping-based ACE and Adaptive ACE in which

repeated CAF an in later an adaptive control has been used to optimize the

performance.

The drawback is that the technique is useful for larger constellation size modulations

only.

Figure 4.12 Active Constellation Extension (a) for QPSK (b) for 16 QAM


62

4.14 Dummy Sequence Insertion (DSI)[100]: In Dummy sequence insertion (DSI) [175], before IFFT stage in input data a dummy

sequence is added. The sequences which are used may be complementary, correlation

or any other sequence. Since dummy sequence is not used as side information hence

any transmission error does not increase BER. DSI technique is united with PAPR

threshold method. After IFFT, if PAPR is below specific threshold then signal is

transmitted but if it is more than this specific level then insertion of dummy sequence

is done to achieve the required results. The block diagram of DSI system is shown in

figure

Figure 4.13 Block diagram of DSI system

The main advantage of this technique is that BER is not degraded due to transmission

errors in the dummy sequence. So far amongst different sequences, use of

complementary sequence produces better results.

Chapter Five Simulation Results and Analysis

63

Chapter Five

Simulation Results and Analysis

One of the major drawbacks of OFDM system is high PAPR of transmitting signals,

which causes an earnest degradation in performance when a non-linear HPA is

utilized. Therefore, it is compulsory to utilize a congruous PAPR reduction scheme at

the transmitter. In this chapter, the different methods of PAPR reduction are given

with results and new types of PAPR proposed.

5.1 OFDM System model: The system model used in the work is shown in figure 5.1. The OFDM parameter

used in the test is the LTE parameters as shown in table 5.1. The system was tested

under Rayleigh selective fading channel with parameter given in table 5.2 [176]

Figure 5.1 OFDM system model.

Table 5.1 LTE parameter

FFT size 128

Spacing frequency 15 KHz

BW 1.25MHz

CP 32

No symbol 1000

Sampling frequency 192MHz

Modulated type QPSK

Table 5.2 Average Power and Relative Delays with 6 delay taps [176]

Tap no. Relative delay (ns) Average Power (dB)

1 0 0.189

2 0.2 0379

3 0.5 0.239

4 1.6 .095

5 2.3 .061

6 5 .037

Signal

Mapper

Signal

demapper

Equalizer

And

P/S

DFT

OR

FFT

S

/

P

D

/

A

Add

CP

IDFT

OR

IFFT

P

/

S

S

/

P

Multipath Fading Ch.

& noise

A

/

D

Remove

CP

I/P

O/P

P


64

The PAPR was evaluated statistically by using the complementary cumulative

distribution function (CCDF). The CCDF of PAPR, for the proposed PAPR reduction

techniques OFDMA downlink signal, is used to express the probability of exceeding a

given threshold PAPR0 (i.e., CCDF ( )). A simulation result was

compared with each other. PAPR was measured for the transmitted OFDM signal

using the equation:

| |

| | (5.1)

In each case, the BER was measured.

Initially, it is necessary to know the performance of OFDM system without any PAPR

reduction techniques in order to compare it with the PAPR reduction techniques to

find out the amount of improvement in PAPR in each case of PAPR reduction

techniques and their impact on the BER. Fig (5.2) shows the CCDF of PAPR and

SNR at BER for OFDM system without any PAPR reduction techniques which

is equal to (10.84 dB) with PAPR equal to (25.6015 dB) while shows the BER for

OFDM system without any PAPR reduction techniques and SNR at BER is

equal to (11.4314 dB).

Figure (5.2.a)

Figure (5.2.b)

a) is CCDF of PAPR for OFDM system without any PAPR reduction techniques b)

is BER for OFDM system without any PAPR reduction techniques

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CCDF

(Pr[P

APR>

PAPR

0])

Orignal

0 5 10 1510

-4

10-3

10-2

10-1

100

SNR

BER

Bit error probability curve for qpsk using OFDM

simulated


65

5.2 PAPR techniques used:

5.2.1 Repeated clipping and frequency domain filtering (RCF): In the clipping technique hard limiting is applied to the amplitude of the complex

values of the IFFT output. The filtering technique is designed to alleviate or cancel

OOB distortion dependent on oversampling value but cannot correct in-band

distortion. [98]

Figure 5.3 shows the block diagram of the new PAPR reduction scheme [177].

The input vector is first converted from the frequency to the time domain

wing an oversize IFFT. N is the number of subcarriers in each OFDM symbol. For an

oversampling factor of, the input vector is extended by adding ( ) zeros; in the

middle of the vector. This results in the trigonometric interpolation of the time domain

signal [178].

Trigonometric interpolation gives perfect interpolation when the original signal

consists of integral frequencies over the FFT window. This is the case for OFDM. The

input of the Nyquist frequency, has been omitted, as the interpolation technique

does not work for this value [178]. This is not a practical limitation as all applications

of OFDM null this input and most do not use a number of adjacent subcarriers. The

interpolated signal is then clipped.

In this Technique hard-limiting is applied to the amplitude of the complex values of

the IFFT output [12]

After an IFFT, the original signal is clipped in the time domain. The clipping can be

described as shown below:

*

√ ,| | -

| | +| |

| | (5.2)

Where represents the output of the time domain signal,

,| | - (5.3)

, Is the threshold clipping level, | | ; ,| | - Is the mean

power.

N*𝐼

Point

inverse

DFT

over

sampling

rate 𝐼

Nonlinear

Processing

Clipping

Ratio =

CR

N*𝐼

Point

DFT over

sampling

rate 𝐼

N*𝐼

Point

inverse

DFT

over

sampling

rate 𝐼

Add

cp

𝑁 × (𝐼 )

0

0 Zeroes

Input data

zero padded Interpolated

baseband signal

Clipped

Interpolated

baseband signal Frequency domain

filtering

𝑁

× (𝐼 )

0

0

Zeroes

Iterative clipping and filtering fft/iffft

𝑎

𝑎𝑁

𝑐𝑁

𝑐


66

The clipping ratio is defined as the ratio of the clipping level to the mean power of

the unclipped baseband signal.

As shown in the equation (5.2), the discrete time domain signal is clipped in the

amplitude. At every point where the complex time domain signal exceeded the

clipping level, the amplitude was reduced to the clipping level while the phase of the

complex signal was unchanged [179].

The clipping is followed by frequency domain filtering to reduce OOB power caused

by clipping. The filter consists of two FFT operations [12].

The clipped time domain signal c is then converted back into the discrete frequency

domain using an FFT ,The inband discrete frequency components of the clipped

signal

are passed unchanged to the inputs of the second

IFFT while the OOB components,

are nulled [13 and 180] this

technique is repeated, depending on iteration number, usually choose between one

and four.

In this work has been selected four.

Although frequency domain filtering is a common signal processing technique the

form shown in figure 5.3 is unusual. In most filtering applications the filter is

designed to meet particular specifications in the continuous frequency domain. In this

application, the wanted signal is an OFDM signal, which is the sum of discrete

frequency components in each symbol period. The filter must therefore have as little

effect as possible on the in-band discrete frequency domain while attenuating as much

as possible any OOB components. This is precisely what is achieved by the simple

filter structure in Figure 5.3 because the filter operates on a symbol by symbol basis;

there is no filtering across symbol boundaries and so no resultant ISI. The filtering

does cause some peak regrowth. However, this is much less than for clipping before

interpolation [12, and 18]

The clipping noise is added at the transmitter rather than the receiver. In fading

channels this means that in general the clipping noise will cause less degradation in

bit error rate than noise added in the channel since the clipping noise fades along with

the signal.

However the second oversize IFFT could be replaced by any of the transform, up

sampling and filtering arrangements commonly used in OFDM systems. So the

technique can be implemented by replacing the IFFT block in an existing OFDM

system with the new configuration [12].

The FFT/IFFT transform filter can be replaced by DCT/IDCT transform and this

technique has been described in [28].

In paper [18] present a new PAPR reduction technique which exploits the use of

unused carriers as well as the phase information of pilot signals in OFDM systems to

reduce the PAPR while not degrading channel estimation or frequency offset.

Compared to conventional techniques such as clipping and windowing, this technique

introduces much less OOB distortions and provides a lower BER since there is no

interference to the original data signals. There is also no requirement for side

information to be transmitted to the receiver.

To reduce PAPR at LTE downlink, the RCF is applied to OFDM signal for different

CR and oversampling filter and notes their impact on PAPR and BER.

The reason to choose this method is because the filter improves the BER if the

oversampling is high and clipping improves PAPR (it's possible to improve the BER

& PAPR together and this way we have explained previously).


67

The OFDM system model with RCF as shown in figure 5.4.

For this simulation I = (1, pilot, 1.125, 1.25, 1.5, 2, 3, 4) and CR = (4, 3, 2, 1.75, 1.5),

in order to see the impact of CR on the (BER) and (PAPR), this technique is repeated,

depending on iteration number ( four is used in this simulation)

The transmitted signals pass through Rayleigh fading channel.

Figure 5.4 the OFDM system model with RCF.

Figure 5.5 illustrate the effect of repetition clipping and filtering on PAPR where CR

=3, I =2, where CCDF of PAPR for, one RCF = 7.7581, two RCF = 6.5462, three

RCF = 5.8319, and four RCF = 5.401,

Note that there is an improvement in CCDF of PAPR for one RCF (2.8935 dB), two

RCF (4.1054 dB), three RCF (4.8197 dB), and four RCF (5.2506 dB). But the

proportion of improvement, between (N) RCF and (N-1) RCF decrease as N increase.

Whenever a CR reduces the PAPR is improving and contrast SNR at BER is

increased, The best value of PAPR is for CR =1.5, but the SNR at BER for this

case is the worst, as shown in table CR have a positive relationship with PAPR and

negative relationship with SNR at BER

Whenever oversampling increased the SNR at BER is improving and contrast

PAPR is increased and vice versa. The best value of PAPR is for I =1 this mean there

is no filter, but the SNR at BER for this case is the worst, while The best value

of SNR at BER is for I =4, but the PAPR for this case is the worst, as shown in

table (A.1) I have a positive relationship with SNR at BER and negative

relationship with PAPR

+p

ilot sy

mb

ol

Rem

ove

+p

ilot sy

mbol

Sig

nal m

app

er S

ignal

dem

apper

One T

ap

Equalizer

And P

/S

Rem

ove C

P

Ad

d C

P

S

/

P

P

/

S

S

/

P

RC

F

IDF

T O

R IF

FT

DF

T O

R F

FT

Multipath

Fading Ch. &

noise

O

/

P

I

/

P P

𝑃

D

/

A

A

/

D


68

Figure 5.5 CCDF of PAPR for OFDM system with repeated clipping and frequency

domain filtering where CR =3, I =2

Figure 5.6 shows the following:

There is a clear improvement in the CR3 CCDF of PAPR reduction in rate

SNR at BER is relatively small compared with the CR4. Briefly, that‟s

mean the percentage of improvement in CCDF of PAPR More than the

degradation in BER

For the CR2 the PAPR improved more than CR3 and CR4 but SNR at BER

gets worse

the CR1.75 had a little improvement in CCDF of PAPR in comparison with

the CR2) but SNR at BER degradation more than The amount of

improvement

For the CR1.5 the PAPR improved PAPR in comparison with the CR1.75 only

in a small proportion, while SNR at BER Substantially worse.

Figure 5.7 shows the impact of the oversampling (CCDF of PAPR) and (BER),

is conclusion through drawing and table following:

whenever increase the PAPR will increase too only in small percentages, for

this figure PAPR for I4 Worsened by (1.7978 dB) compared with I1

whenever increase the CCDF of PAPR will increase too only in small

percentages, for this figure CCDF of PAPR for I4 Worsened by (.9939 dB)

compared with I1

whenever increase the SNR at BER will improved , for this for I4 the

SNR at BER improved by(5.5382) compared with I1

for I2 the SNR at BER Improved by (2.8466) , CCDF PAPR Worsened

by (.5831) and PAPR Worsened by (.6559 dB) compared with I1

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

Orignal

One clip and filter

Two clip and filter

Three clip and filter

Four clip and filter


69

The Conclusion from table (A.1) Summarizes as follows:

CR 4 have the best SNR at BER and the worst PAPR compared with the

rest of the CR

PAPR at CR 3 better than CR 4 by (2-3 dB improvement in PAPR) but SNR at

BER at CR3 worse than CR 4 only by small percentage (less than 1 dB

in all cases) PAPR at CR 2 better than CR3 by (2.5 - 3.4 dB improvement in

PAPR) but SNR at BER at CR2 worse than CR = 3 by (2- 3.7 dB

degradation in SNR at BER )

PAPR at CR 1.75 better than CR2 by (Maximum improvement is 1.0059) but

SNR at BER at CR1.75 worse than CR2by (2.2 – 3.4 dB degradation

in SNR at BER )

CR 1.5 have the best PAPR and the worst SNR at BER compared with

the rest of the CR, the SNR should higher than 30 dB have the desired SNR at

BER that‟s mean SNR at BER is deteriorating by a large margin

The following conclusion when comparing the proposed method with an OFDM

system without PAPR reduction method:

As (CR =4, 3 and I =4, 3,2, pilot, 1.5,1.25) and when (CR =2and I =4) and finally

(CR =4 and I = 1.125), The best one improvement in PAPR and CCDF of PAPR

is at I =3 and CR =2. The improvement in PAPR by = (14.9490 dB), CCDF of

PAPR = (6.2850 dB), and the SNR at BER by = (1.0134 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = Pilot and CR

=2. The improvement in PAPR by = (16.1583 dB), and CCDF of PAPR = (6.9604

dB), while the SNR at BER deteriorated by = (-1.5686 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1 and CR =2.

The improvement in PAPR by = (17.3529 dB), and CCDF of PAPR = (7.7214

dB), while the SNR at BER deteriorated by = (-2.9442 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1 and CR

=1.75. The improvement in PAPR by = (18.2213 dB), and CCDF of PAPR =

(8.2460 dB), while the SNR at BER deteriorated by = ( -5.2886 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1 and . The improvement in PAPR by = (19.2177 dB), and CCDF of PAPR =

(7.9400 dB), while the SNR at BER deteriorated by = ( -18.0686 dB).


70

Figure 5.6.a

Figure 5.6.b

(a)CCDF of PAPR for OFDM system with RCF where I =2 (b) BER for OFDM

system with RCF where I =2

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

CR =4

CR =3

CR =2

CR =1.75

CR =1.5

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


CR =4

CR =3

CR =2

CR =1.75

CR =1.5


71

Figure 5.7.a

Figure 5.7.b

Figure 5.7 (a) CCDF of PAPR for OFDM system with RCF where CR =3

(b) BER for OFDM system with RCF where CR =3.

0 1 2 3 4 5 610

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

I =1

I =pilot

1.125

I =1.25

I =1.5

I= 2

I =3

I =4

0 5 10 1510

-4

10-3

10-2

10-1

100

SNR

BE

R


I =1

I =pilot

1.125

I =1.25

I =1.5

I= 2

I =3

I =4


72

5.2.2 Repeated frequency domain filtering and clipping RFC: The proposed method is the same as previous method RCF, but with almost a simple

change and it changes the location of the filter becomes before the clipping as shown

in figure 5.8, the frequency domain filtering that depends on the interpolation As

noted by previous results that improve the BER As noted by previous results but

increases the PAPR. The basic idea of this method is proposed that this filter will

improve the performance of the OFDM to improve the BER and then the clipping will

improves PAPR method is the almost same as RCF, where have the same receiver and

channel But there is a difference in One block in the transmitter. This block is RFC as

shown in figure 5.9. Interpolated baseband signal followed by frequency domain

filtering, the same filter which are explained in the case of RCF. The filtering signal is

clipped in the time domain. The clipping block is described previously in the case of

RCF.

Figure 5.8 shows the block of the OFDM system model for this proposed

Figure 5.9 the OFDM system model with RFC.

N*𝐼

Point

inverse

DFT

over sampling

rate 𝐼

N*𝐼

Point

DFT

over sampling

rate 𝐼

N*𝐼

Point

inverse

DFT

over samplin

g rate 𝐼

Add

cp

𝑁 × (𝐼 )

0

0 Zeroe

Input data

zero padded

Interpolated

baseband signal Clipped

the filtering signal

Frequency

domain filtering

𝑁× (𝐼 )

0

0

Zeroes

Iterative filtering fft/iffft and clipping

𝑎

𝑎𝑁

𝑐𝑁

𝑐 Nonlinear

Processin

g

Clipping

Ratio =

CR

+p

ilot sy

mbo

l

Rem

ov

e

+p

ilot sy

mb

ol

Sig

nal m

apper

Sig

nal

dem

app

er

On

e Tap

Equ

alizer

An

d P

/S

Rem

ove C

P

Add

CP

S

/

P

P

/

S

S

/

P

RF

C

IDF

T O

R IF

FT

DF

T O

R F

FT

Multipath

Fading Ch. & noise

O

/

P

I

/

P P

𝑃

D

/

A

A

/

D


73


system without PAPR reduction method as shown in table A.2:

At (CR =4,3 and I =4,3,2, pilot,1.5,1.25) , (CR =2and I =4,3) , ( CR =1.75 and I =

4) and finally (CR =4 and I = 1.125), The best one improvement in PAPR and

CCDF of PAPR is at I =4 and CR =1.75. The improvement in PAPR by = (18.2789

dB), CCDF of PAPR = (8.0187 dB), and the SNR at BER( ) by = (0.6101 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 3 and CR


(8.0088 dB), while the SNR at BER( ) deteriorated by = (-0.2686 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I =2 and CR



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1.5 and CR



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1.125 and CR



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 4 and CR =1.5.


dB), while the SNR at BER( ) deteriorated by = ( -16.7886 dB).

The following conclusion when comparing the proposed method with RCF:

1. CCDF of PAPR was improved in all cases except when (I = pilot and CR=4

by (-0.3570)). The improvement ratio was increased with the decrease of CR

and the increase of I. The biggest improvement is in the case (I = 3 and CR =4

(2.9062))

2. PAPR was improved in all cases except when (I = 1.125 and CR=2 by (-

0.0866)) and (I = 1.5 and CR=4 by (-0.0015)) the improvement ratio was

increased with the decrease of CR and the increase of I. The

biggest improvement is in the case( I = 3 and CR =4 (1.5600))

3. SNR at BER( )

a) SNR at BER( ) was improved for (I =3 and I =4 in all cases of CR )

b) For I =2 SNR at BER( ) was improved in all cases except when (CR=2)

deteriorated by (-0.1548)

c) For I =1.5 SNR at BER( )was improved except when (CR=4) deteriorated

by (-0.1236)

d) For I =1.25 SNR at BER( ) was improved except when (CR=4)

deteriorated by (-0.5700) and (CR=2) deteriorated by (-0.0375)

e) For I =1.125 SNR at BER( ) was improved except when (CR=3)

deteriorated by (-0.1700) and (CR=1.75) deteriorated by (-0.2390)

f) For I =pilot SNR at BER( ) was improved except when (CR=4)

deteriorated by (-0.2400) and (CR=3) deteriorated by (-0.1585)


74

g) The best value of the improvement is where the (I =4 and CR =1. 75 by

(2.1787))

4. RFC is better than RCF because when I increase the SNR at BER( )

improve and PAPR almost preserves its value

That was the conclusion from a comparison of figure 5.6 and figure 5.10 the

following:

1. There is an obvious improvement in the CCDF of PAPR of the RFC In

comparison with the RCF

2. There is an improvement in the SNR at BER( ) of the RFC In comparison

with the RCF

3. The CCDF of PAPR of the RFC at CR=2 is better than the CCDF of PAPR of

the RCF at CR=1.5, in addition to that the SNR at BER( ) of the RFC at

CR=2 is better than the SNR at BER( ) of the RCF at CR=1.5 by ((17.71721

dB)

4. The amounts of improvement, are described in the table A.2

Figure 5.11 shows the impact of the oversampling (CCDF of PAPR) and (BER), is the

conclusion through drawing and table following:

The PAPR for I(N) was increased only by A small amount compared with I1,

for this figure PAPR was declined amount (0.0251 - 0.3086 dB).

The CCDF of PAPR for I(N) was increased only Avery small amount could

be neglected in comparison with I1, for this figure PAPR was declined

amount ( 0.0102 - 0.0837 dB).

SNR at BER( ) degraded whenever I increased between (0.8374 -

6.2451dB)

RFC have the same complexity and cost RCF because RFC has not added a new

function for RCF but the only change filter location.


75

Figure 5.10.a

Figure 5.10.b

Figure 5.10 (a)CCDF of PAPR for OFDM system with RFC where I =2 (b) BER for

OFDM system with RCF where I =2

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

CR =4

CR =3

CR =2

CR =1.75

CR =1.5

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

snr

BE

R


CR =4

CR =3

CR =2

CR =1.75

CR =1.5


76

Figure 5.11.a

Figure 5.11.b

Figure 5.11 (a)CCDF of PAPR for OFDM system with RCF where CR =3 (b) BER

for OFDM system with RCF where CR =3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

I =pilot

I =1.125

I =1.25

I =1.5

I= 2

I =3

I =4

0 5 10 1510

-4

10-3

10-2

10-1

100

SNR

BE

R


I =pilot

1.125

I =1.25

I =1.5

I= 2

I =3

I =4


77

5.2.3 The OFDM System with discrete time companding: Compresses the signal at the input and expands the signal at output in order to keep

the signal level above the noise level during processing. At the output, the original

input signal is then restored by a simple attenuation. Companding increases the SNR

when the input signal is low and therefore reduces the effect of a system‟s noise

source.

5.2.3.1 A-law companding: In this companding method, the compressor characteristic is piecewise, made up of a

linear segment for low level inputs and a logarithmic segment for high level inputs.

Figure 5.12 shows the A-law compressor characteristics for different values of A.

Corresponding to A=1, we observe that the characteristic is linear (no compression)

which corresponds to a uniform quantization. A-law has mid riser at the origin. Hence

it contains non-zero value. The practically used value of “A” is 87.6. The A-law

companding is used for PCM telephone systems. The linear segment of the

characteristic is for low level inputs whereas the logarithmic segment is for high level

inputs. This technique can be used to reduce the PAPR which is the main

disadvantage of OFDM [181, and 182].

( )

{

| |

( ) ( )

| |

[ 0

| |

1]

( ) ( )

| |

}

(5.4)

Where

x=input signal.

y=output signal.

( ) =sign of the input (+ or -).

|x|=absolute value (magnitude of x).

A=87.6 (defined by CCITT (Consultative Committee for International

Telephony and Telegraphy) ).

This A-law companding technique is used in Europe, Asia, Russia, Africa, China, etc

[183].

Initially, A companding as discussed used with OFDM.

Figure 5.13 illustrates the effect of A parameter on the PAPR, CCDF of PAPR, and

SNR at BER( ). When increasing the values of A parameter, the CCDF of PAPR

improves. The relationship between A parameter and CCDF of PAPR is the inverse

relationship.

CCDF of PAPR (A =20) - CCDF of PAPR (A =120) = (1.15 dB)

A is not linear companding , A possible divided into three areas. The first area is that

when A increases lead to improvement in the CCDF of PAPR is relatively large

compared with the second and third region (the example for this area is A

(CCDF of PAPR (A =5) - CCDF of PAPR (A =20) = (2.955 dB) amount of

improvement in the CCDF of PAPR )

In the second area, when A was increased the CCDF of PAPR was improved but a

small quantity less than the first region example of this area when A (CCDF of

PAPR (A =20) - CCDF of PAPR (A =120) = (1.15 dB) as is evident A increased by


78

(100) and the improvement in CCDF of PAPR is (1.15 dB) while in the first area A

increased by (15) but the improvement in CCDF of PAPR (2.955 dB))

In the third area, when A was increased the CCDF of PAPR was not affected even if

improved but very small.

Figure 5.12. A-law Compressor Characteristics [99].

Figure 5.13 the relationship between A parameter and (PAPR, CCDF of PAPR and

BER)

0 20 40 60 80 100 1202

4

6

8

10

12

14

16

18

20

22

[dB

]

A

SNR at (BER =10-4)

CCDF of PAPR

PAPR

Normalized

Output

Normalized input


79

When A parameter was increased the PAPR improved while the SNR at BER( )

deteriorated. The A parameter has a positive relationship BER with but an inverse

relationship with the PAPR.

PAPR (A =5) – PAPR (A = 20) = (4.466 dB)

SNR at BER( ) (A= 5) – SNR at BER( ) (A=20) = (-4.4 dB)

PAPR (A =20) – PAPR (A = 120) = (3.03 dB)

SNR at BER( ) (A= 20) – SNR at BER( ) (A=120) = (-3.386 dB)

In the first area A increased by (15) but the improvement in PAPR (4.466 dB) and the

degradation in SNR at BER( ) (4.4 dB) while in the second area A increased by

(100) and the improvement in PAPR (3.03 dB) and the degradation in SNR at

BER( ) (-3.386 dB) . The first and the second area evident in the Figure 5.14



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at A =5. The

improvement in PAPR by = (6.6954 dB), and CCDF of PAPR = (4.200 dB), while the

SNR at BER( ) deteriorated by = (-2.1686 dB).

For SNR at BER( )


improvement in PAPR by = (10.9098 dB), and CCDF of PAPR = (6.1100 dB),

while the SNR at BER( ) deteriorated by = (-4.6886 dB).

For SNR at BER( )




For SNR at BER( )




Figure 5.14 shows the CCDF of PAPR and the BER of A companding for various A

parameter. For more details see table A.4


80

5.14.a

5.14.b

Figure 5.14 (a) CCDF of PAPR OFDM system A companding for various A

parameter. (b) BER for OFDM system A companding for various A parameter.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

A =5

A =10

A =30

A =50

A =87.6

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


A =5

A =10

A =30

A =50

A =87.6


81

5.2.3.2 μ-law companding technique: In the μ-law companding, the compressor characteristic is piecewise, made up of a

linear segment for low level inputs and a logarithmic segment for high level inputs.

Figure 5.15 shows the μ-law compressor characteristics for different values of μ.

Higher the value of μ more is the compression.

Corresponding to μ=0, we observe that the characteristic is linear (no compression)

which corresponds to a uniform quantization. μ-law has mid tread at the origin. Hence

it contains a zero value. The practically used value of “μ” is 255 [183].

The signal by utilized μ -Law compression characteristic is defined as:

( ) .

| |

/

( ) ( ) (5.5)

Where V is the peak amplitude of the signal, and x is the instantaneous amplitude of

the input signal. Decompression is simply the inverse of (5.5). Compression improves

the quantization resolution of small amplitude signals at the cost of lowering the

resolution of large signals. This also introduces quantization noise; however, the

effect of the quantization noise due to reduction in resolution of the peaks is relatively

small as the peaks occur less frequently. The compression algorithm as described by

amplifying the signals of lower amplitude with the peaks remaining unchanged. [184],

[185].

Figure. 5.15 μ-law Compressor Characteristics [186]

A-law and law coefficients are responsible for the compression ratio. Compression

increases with increasing value of the coefficients. Originally A-law and -law

companders were used for voice compression, as it can be seen, A-law and -law

companders have logarithmic compressing profile. In fact they work as follows,

instead of compressing the high peaks; companding schemes increase the value of

small signals in a way, to bring them in the same level with the high peaks [186].

Normalized

Output

Normalized input


82

Thus original Gaussian distributed OFDM signal will be transformed to a signal with

quasi uniform distribution. However, because of increased level of the small signals,

average power of the signal will be increased. That means noise will be increased as

well. This is disadvantage of A-law and -law companding schemes as compared

with exponential companding, which is claimed to adjust both small and large signals

without changing the average power of the signal [187].

Figure 5.16 illustrates the effect of parameter on the PAPR, CCDF of PAPR and

SNR at BER( ). In General, when parameter was increased, the CCDF of PAPR

was decreased except for some cases are as follows:

At the CCDF of PAPR Larger than the by (0.06 dB)





At the CCDF of PAPR Larger than the by ( 0.11 dB )

Even in exceptional cases, the amount of the decline is a few and not exceed

(0.205 dB)

The max CCDF of PAPR at ( ) = 6.416 dB while the min CCDF of PAPR at

( ) =2.17 dB

The parameter has a positive relationship SNR at BER( ) with but an inverse

relationship with the PAPR.

When parameter was increased SNR at BER( ) deteriorated except at be

better than the by (0.132 dB).

In General, when parameter was increased, the PAPR was decreased except for

some cases are as follows:

At the PAPR Larger than the by (0.7103 dB)







The max PAPR at ( ) = 17.4332 dB while the min PAPR at ( ) =10.8218

dB

The max SNR at BER( ) at ( ) = 23.764 dB while the min SNR at

BER( ) at ( ) =13.3363 dB



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at MU =10. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at MU =20 . The




83

For SNR at BER( )




For SNR at BER( )


improvement in PAPR by = ( 14.7797 dB), and CCDF of PAPR = (8.6700 dB),


Fig (5.17) shows the CCDF of PAPR of companding and the BER of

companding for various parameter and illustrates the former explanation.

PAPR improved by ( 8.1683 -14.7797 dB)

CCDF of PAPR improved by (4.4240 - 8.6700 dB)

The amount of SNR at BER( ) degradation is (1.9049 - 12.3326dB )

Figure 5.16 the relationship between parameter and (PAPR, CCDF of PAPR and

BER)

0 50 100 150 200 250 3002

4

6

8

10

12

14

16

18

20

22

[dB

]

MU

SNR at (BER =10-4)

CCDF of PAPR

PAPR


84

Figure 5.17.a

5.17.b

Figure 5.17 (a) CCDF of PAPR OFDM system companding for various

parameters. (b) The BER of companding for various parameters.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

orginal

MU =5

MU =50

MU =100

MU =160

MU =200

MU = 255

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


MU =5

MU =50

MU =100

MU =160

MU =200

MU = 255


85

5.2.3.3 Rooting Companding Technique (RCT): The proposed Rooting companding has the same principle of SQRT.

Rooting companding equation is given by:

( ) | | ( ) (5.6)

Where ( )= sign(x)

sign(x) was used in RCT to maintain the phases of the OFDM signal Where the

phases of the OFDM output signals are kept unchanged while only the amplitudes

are treated and changed . The amount of change in amplitude depends on the value of

R

Rooting decompanding equation is given by:

( ) | |

( ) (5.7)

The following can be observed from table A.6 and figure 5.18

When y parameter decreases the PAPR and CCDF of PAPR also decrease

while SNR at BER( ) increase

The best value for the PAPR is (2.8726) when R =0.1 while the worst value is

(21.8631) when R=0.9

The best value for the CCDF of PAPR is (1.268) when R =0.1 while the worst

value is (9.55) when R =0.9

The best value for the SNR at BER( ) is (11.6765) when R =0.9 while the

worst value is (28.3) when R=0.1

PAPR improved by (3.7384 - 22.7289 dB )

CCDF of PAPR improved by (1.2900 -9.5720 dB )

The amount of SNR at BER( ) degradation is (0.2451 - 16.8686 dB )



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R =0.7. The

improvement in PAPR by = ( 7.4724 dB), and CCDF of PAPR = ( 2.7820 dB),


For SNR at BER( )




For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R =0.4 . The



For SNR at BER( )




For SNR at BER( )


86




For SNR at BER( )




5.18.a

5.18.b

Figure 5.18 (a)CCDF of PAPR OFDM system RCT for various parameter. (b) The

BER of RCT for various parameter

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

R = .9

R = .8

R =.7

R =.6

R = .5

R = .4

R =.3

R =.2

R = .1

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


R = .9

R = .8

R =.7

R =.6

R = .5

R = .4

R =.3

R =.2

R = .1


87

Figure 5.19 illustrates the effect of R parameter on the PAPR, CCDF of PAPR and

SNR at BER( ). The y parameter has a positive relationship with PAPR and

CCDF of PAPR but an inverse relationship with the SNR at BER( ).

Figure 5.19 the relationship between parameter and (PAPR, CCDF of PAPR, and

SNR at BER( ))

5.2.3.4 New error function Companding (NERF) : The new type of companding was proposed depends on erf. The NERF companding

equation is:

( ) (| |

√ ) ( ) (5.8)

NERF De_companding:

( ) |√ .| |

/| ( ) (5.9)

When used this type of companding the PAPR was improved by (15.422 dB) and the

CCDF of PAPR also was improved by (6.4045 dB) while the SNR at BER( )

was deteriorated by (2.2466 dB).

The rate of improvement in the PAPR and CCDF of PAPR is greater than the rate of

the decline in SNR at BER( ) as shown in figure 5.20 and table 4.3.

Table 5.3 NERF performance

NERF PAPR CCDF of PAPR SNR at BER( )

10.1795 4.4355 13.678

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

5

10

15

20

25

30

[dB

]

R

SNR at (BER =10-4)

CCDF of PAPR

PAPR


88

5.20.a

5.20.b

Figure 5.20 (a) CCDF of PAPR OFDM system NERF companding (b) the BER

of NERF companding.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

Orignal

NERF

0 2 4 6 8 10 12 1410

-4

10-3

10-2

10-1

100

SNR

BE

R


NERF


89

5.2.3.5 Absolute Exponential companding (AEXP) : The proposed AEXP equation is derived based on EXP companding and Trapezoidal

power companding:

( ) ( ) √ 0 . | |

/1

(5.10 )

EXP companding since received signal with EXP companding is so distorted that the

square root part in (5.10) would be an imaginary or complex number, we then take the

absolute value of the square root parts to eliminate any further potential phase

distortion.

Where ( ) is sign function? The positive constant determines the average power

output signals. In order to keep the input and output signals at the same average power

level

(

[| | ]

* √[ ( | |

)] +

)

(5.11 )

At the receiver side, the inverse function ( ) of is used in the De_companding

operation

( ) ( ) |√ ( | |

) | (5.12)

The following can be observed from the table (A.7) and figure 5.21

When d parameter decreases the PAPR and CCDF of PAPR also decrease

while SNR at BER( ) increase

Values less than d =.8 the deterioration in the SNR at BER( ) becomes

large

For

The best value for the CCDF of PAPR is (2.92) when d =0.8 while the worst

value is (5.1533) when d =2

The best value for the PAPR is (6.0806) when d =0.8 while the worst value is

(13.0811) when d =2

The best value for the SNR at BER( ) is (14.73) when d =2 while the

worst value is (24.833) when d =0.8

PAPR improved by (12.5205 - 19.5209 dB )

CCDF of PAPR improved by (5.6867 - 7.9136 dB )

The amount of SNR at BER( ) degradation is (3.2986 - 13.4016 dB )


90

5.21.a

5.21.b

Figure 5.21 (a) CCDF of PAPR OFDM system AEXP companding for various

parameters. (b) The BER of AEXP companding for various parameters.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

orginal

d = 2

d = 1.8

d =1.6

d = 1.4

d =1.2

d = 1

d = .8

d =.6

d = .4

d =.2

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


d = 2

d = 1.8

d =1.6

d = 1.4

d =1.2

d = 1

d = .8

d =.6

d = .4

d =.2


91



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at d= 1.1. The



For SNR at BER( )




For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at d =0.8.



Figure 5.22 illustrates the effect of d parameter on the PAPR, CCDF of PAPR and

SNR at BER( ). The relationship between the d parameter and SNR at

BER( ) is a direct correlation, while the relationship between the d parameter and

PAPR, and CCDF of PAPR is the inverse relationship.

Figure 5.22 the relationship between parameter and (PAPR, CCDF of PAPR, and

SNR at BER( ))

5.2.3.6 Cos companding: The new type of companding was proposed depends on cos . The proposed cos

companding eauation is:

( ) ( )√ 0 . | |

/1

(5.13 )

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

[dB

]

d

SNR at (BER =10-4)

CCDF of PAPR

PAPR


92

At the receiver side, the inverse function ( ) of is used in the de-companding

operation,

( ) ( ) | ( | |

)| (5.14)

The positive constant determines the average power of output signals. In order to

keep the input and output signals at the same average power level.

( [| | ]

[ √0 . | |

/1

]

)

(5.15 )

The following can be observed from table A.8 and the following figures

At y =2 the PAPR deteriorate incremented by (.7122 dB) as well as the CCDF

of PAPR, deteriorate incremented by (.515 dB) until the SNR at BER( )

deteriorates by (3.9886 dB)

For , in this region whenever d decreased lead to improvement in

(PAPR, CCDF of PAPR and SNR at BER( )) compared with values when y

= 2.

Figure 5.23 shows the best CCDF of PAPR and PAPR at y =0.1. The amount

of improvement in CCDF of PAPR by (9.9192 dB) and in PAPR (23.6085 dB)

compared with OFDM system without companding . Whereas the BER

deteriorates considerably in this case.

Figure 5.23 show the best value for the SNR at BER( ) in cos companding

is when y = 1. Where it has less value deterioration in the SNR at BER( )

by (0.2717 dB) while PAPR improved by (9.9547 dB), as well as it CCDF of

PAPR improved by (3.8892 dB),compared with OFDM system without

companding.

For this area is better than region in terms of PAPR and

CCDF of PAPR and almost have the same SNR at BER( ) as shown in

figure 5.24. So were selected d values less or equal to one.

The relationship between the y parameter in cos companding and PAPR is a direct

correlation, as shown in figure 5.25. Whenever y increased the PAPR and CCDF of

PAPR also increased. But it's different from SNR at BER( ). y =1 is the point of

separation and switching between two contradictory in relation to the SNR at

BER( ). For whenever y decreased the SNR at BER( ) degradation

increases simply means the relationship is an inverse relationship between y and SNR

at BER( ) when . While for is quite unlike the previous case.

Whenever y decreased the SNR at BER( ) degradation also decreased.


93

5.23.a

5.23.b

Figure 5.23 (a) CCDF of PAPR OFDM system cos companding for various

parameter. (b) The BER of cos companding for various parameter

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

y = 1

y = .9

y = .8

y = .7

y = .6

y = .5

y = .4

y = .3

y = .2

y = .1

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


y = 1

y = .9

y = .8

y = .7

y = .6

y = .5

y = .4

y = .3

y = .2

y = .1


94



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at y =0.8. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at y=0.6. The



For SNR at BER( )




For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at y =0.4. The



For SNR at BER( )




Figure 5.24 the effect of y parameter of cos companding on PAPR and SNR at (BER

= )

10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

2.1> y >1

1.1>y>0


95

Figure 5.25 illustrates the effect of y parameter in cos companding on the PAPR,

CCDF of PAPR and SNR at BER( )

5.2.3.7 tangent Rooting Companding (tanhR):

The proposed companding depend on tanh and the companding equation will be as

follows:

( ) ((| | × ) ) ( ) (5.16 )

Where k is positive numbers controlling the companding level applied to the envelope

x, | | and sign(x) was used to maintain the phases of the OFDM signal.

Decompanding equation will be as follows:

( ) |( (| |

))

| ( ) (5.17)

Figure 5.26 shows the CCDF of PAPR and BER for OFDM system with tanhR

companding at k =10 and y change from 0.1 to 1 by 0.1 every time. The CCDF of

PAPR was improved by (6.6795 - 23.9603 dB) and the PAPR was improved by

(6.6999 - 10.1412 dB) while the SNR at BER( ) was deteriorated by (3.0726 -

18.5686 dB) compared with an OFDM system without companding.

Whenever y decreased the PAPR and CCDF of PAPR was improved while increasing

the SNR at BER( ) values

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

[dB

]

y

SNR at (BER =10-4)

CCDF of PAPR

PAPR


96

5.26.a

5.26.b

Figure 5.26 (a) CCDF of PAPR OFDM system tanhR companding for various

parameter. (b) The BER of tanhR companding for various parameters

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

y =1

y =.9

y =.8

y =.7

y =.6

y =.5

y =.4

y =.3

y =.2

y =.1

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


y =1

y =.9

y =.8

y =.7

y =.6

y =.5

y =.4

y =.3

y =.2

y =.1


97

Figure 5.27 and table A.10 shows the CCDF of PAPR and BER for OFDM system

with tanhR companding at y=1 and with different k,

k = ( 5,10) The CCDF of PAPR was improved by ( 3.6235 - 6.6310 dB) and the

PAPR was improved by (9.1388 - 16.6703 dB) while the SNR at BER ( ) was

deteriorated by (0.4931 - 3.2172 dB) compared with an OFDM system without

companding.

k = ( 15 ,20) . The SNR at BER ( ) was deteriorated considerably up. where at

k =15 the SNR at BER( ) reach to 30 dB And more than 30 dB at k =20.

While the CCDF of PAPR was improved by (8.2413 - 9.0172 dB) and the PAPR

was improved by ( 20.3701 - 22.2234 dB) compared with an OFDM system

without companding.

Whenever k was increased the PAPR and CCDF of PAPR was improved while

increasing the BER values

Fig (5.28) and table(A.10) shows the CCDF of PAPR and SNR at BER( ) for

OFDM system with tanh companding at y=.8 and with different k .

k = ( 5,10) The CCDF of PAPR was improved by (1.3349- 0.4511 dB) and

the PAPR was improved by (3.0266 - 0.7832 dB) While the SNR at BER

( ) was deteriorated by (0.8925 - 0.2835 dB) compared with an OFDM

system with tanh companding at y=1

k = ( 5,10) The CCDF of PAPR was improved by (4.9584 - 7.0821 dB) and

the PAPR was improved by (12.1654 - 17.4535 dB) While the SNR at BER

( ) was deteriorated by (1.3856 - 3.5007 dB) compared with an OFDM

system without companding.

k =15. The SNR at BER ( ) was improved by (7.5344) and the CCDF of

PAPR was improved by (0.0722 dB) While the PAPR was deteriorated by (-

0.0115 dB) but the deterioration ratio is less than the proportion of

improvement.

At k =20, The SNR at BER ( ) was improved ,the CCDF of PAPR was

improved by (0.0184 dB) and the PAPR was improved by ( 0.2169 dB).

k = ( 15,20). where at k =15 the The SNR at BER ( ) was deteriorated

by (11.0342 dB )And more than 30 dB at k =20. While the CCDF of PAPR

was improved by (8.3135 - 8.9988 dB) and the PAPR was improved by

(20.3586 - 22.0065 dB) compared with an OFDM system without

companding.

Figure 5.29 and table A.9 shows the following:

For k =5, 10, whenever y was decreased the PAPR and CCDF of PAPR improved

while deteriorating the SNR at BER( ).

For k =15, whenever y was decreased the PAPR and CCDF of PAPR improved,

but the rate of improvement is less than at k = 5, 10. The best value of SNR at

BER( ) at y =.5 while the worst at y =1, where up to 30 dB.

For k =20, The influence of y on PAPR and CCDF of PAPR very little, except

when y = .2 which improve the SNR at BER( ), and CCDF PAPR of PAPR

with a small amount


98

Figure 5.27.a

Figure 5.27.b


parameters at y=1. (b) The BER of tanhR companding for various parameters at

y=1.

0 1 2 3 4 5 6 7 810

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

k= 5

k= 10

k= 15

k= 20

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

snr

BE

R


k= 5

k= 10

k= 15

k= 20


99

Figure 5.28.a

Figure 5.28.b


parameters at y=.8. (b) The BER of tanhR companding for various parameters at

y=.8.

0 1 2 3 4 5 610

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

k= 5

k= 10

k= 15

k= 20

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


k= 5

k= 10

k= 15

k= 20


100



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =5 ,y =0.8 . The


while the BER deteriorated by = (-1.2398 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =10 ,y =0.9 .

The improvement in PAPR by = ( 17.0445 dB), and CCDF of PAPR = (6.8431


For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=10 ,y =0.6 . The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =10 , y=0.5. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k = 5, y=0.2. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =20 ,y =0.2. The



Figure 5.29 illustrates the effect of y and k parameter in tanhR companding on the

PAPR and SNR at BER( )

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

y

[dB

]

BER k = 5

BER k = 10

BER k = 15

BER k = 20

PAPR k = 5

PAPR k = 10

PAPR k = 15

PAPR k = 20


101

5.2.3.8 Logarithmic Rooting Companding (logR): The logarithm ( ) Rooting companding equation will be as follows:

( ) ((| | × ) ) ( ) (5.18)

Decompanding equation will be as follows:

( ) |( .| |

/ )

| ( ) (5.19)

Where is positive number controlling the amount of companding.

We used to control the companding level applied to the envelope x, | | and sign(x)

was used to maintain the phases of the OFDM signal.

Figure 5.30 shows the CCDF of PAPR and BER for OFDM system with log

companding at y =1 and k change from 10 to 100 by 10 every time.

The CCDF of PAPR was improved by (15.5911 - 6.6150 dB) and the PAPR was

improved by (8.8595 -15.5911 dB) while the SNR at BER( ) was deteriorated by

(1.0686 -18.5686 dB) compared with an OFDM system without companding.

Whenever k was increased the PAPR and CCDF of PAPR was decreased while


Figure 5.31 shows the CCDF of PAPR and BER for OFDM system with tanh

companding at k =10 and y change from 0.1 to 1 by 0.1 every time. The CCDF of

PAPR was improved by (3.5255 - 9.9080 dB) and the PAPR was improved by

(8.8595 -23.4987 dB) while the SNR at BER( ) was deteriorated by (1.0686 -

18.1686 dB) compared with an OFDM system without companding.

Whenever y decreased the PAPR and CCDF of PAPR was improved while


The following conclusion when comparing the proposed method (from the table

(A.12)) with an OFDM system without PAPR reduction method:

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =10, y =0.9. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=10 ,y =0.6. The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =20 ,y =0.5 .

The improvement in PAPR by = ( 16.6873 dB), and CCDF of PAPR = (6.9120


For SNR at BER( )





102

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at . The



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at . The



Figure 5.30.a

Figure 5.30.b

Figure 5.30 (a)CCDF of PAPR OFDM system logR companding for various

parameter. (b) the BER of logR companding for various parameter.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

k = 10

k = 20

k = 30

k = 40

k = 50

k =60

k = 70

k = 80

k = 90

k = 100

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


k = 10

k = 20

k = 30

k = 40

k = 50

k =60

k = 70

k = 80

k = 90

k = 100


103

Figure 5.31.a

Figure 5.31.b

Figure 5.31 (a)CCDF of PAPR OFDM system logR companding for various

parameter. (b) The BER of logR companding for various parameter.

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

original

y = 1

y = .9

y = .8

y = .7

y = .6

y = .5

y = .4

y = .3

y = .2

y = .1

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


y = 1

y = .9

y = .8

y = .7

y = .6

y = .5

y = .4

y = .3

y = .2

y = .1


104

5.2.4 Pre-distortion methods: The pre-distortion methods are based on the reorientation or spreading the energy of

data symbol before taking IFFT. The pre-distortion schemes include DFT spreading,

pulse shaping or precoding and constellation shaping. The methods like Tone

Reservation (TR) and Tone Injection (TI) are the example of constellation shaping

schemes [188]

5.2.4.1 Pulse Shaping or Pre-coding: The pulse shaping or pre-coding technique is an efficient and flexible way for

reducing the PAPR of OFDM signals. In this method, each data block is multiplied by

a pre-coding matrix prior to OFDM modulation and transmission. This method is

data-independent and, thus, avoids block based optimization. It also works with an

arbitrary number of subcarriers and any type of baseband modulation used. In terms

of BER performance, it takes advantage of the frequency variation of the fading

multipath channel and improves the BER of OFDM signals in comparison to

conventional OFDM (no pre-coding). The implementation complexity of the proposed

technique is acceptable, since a predefined pre-coding matrix is used and thus, no

handshake is needed between transmitter and receiver. Having the same pre-coding

matrix for all OFDM blocks avoids all the processing needed in block-based

optimization methods [189].

Precoded OFDMA consists of using a precoding matrix P that spreads the energy of

symbols over the subcarriers allocated to the user. Uniform energy distribution is

favored in practice. [190]

The OFDM system with an orthogonal precoder is considered. In precoded OFDM

system instead of sending uncoded symbols (one per subcarrier), the idea is to send

different linear combinations of the information symbols on the subcarriers. This

corresponds to signal space diversity. [191]

Precoding based techniques are simple linear techniques. These techniques can reduce

the PAPR up to the PAPR of single carrier systems (Slimane, 2007). WHT precoding

based techniques, DCT precoding based techniques, DHT precoding based techniques

are common examples of precoding based PAPR reduction techniques (Slimane,

2007; Min & Jeoti, 2007; Baig & Jeoti, 2010a, 2010b, 2010c) [14]

Figure. 5.32 shows the block diagram of Precoding Based OFDM System. We

implemented the Precoding matrix P of dimension N × N before the IFFT to reduce

the PAPR.

The Precoding matrix P can be written as:

[

( )

( )

( ) ( ) ( )( ) ]

(5.20)

Where P is a Precoding Matrix of size N ×N is shown in equation (5.20). The complex

baseband OFDM signal with N subcarriers can be written as:

( )

√ ∑

0 t NT (5.21)


105

We can express modulated OFDM vector signal with N subcarriers as:

* + (5.22)

[192], [193]

Figure 5.32 Block diagram of Precoding based OFDM system

5.2.4.2 Discrete Hartley transform (DHT) : The DHT is a linear transform. In DHT, N real numbers are

transformed into N real numbers a. According to [91], the N-point

DHT can be defined as follows:

∑ 0

1 ∑ ( )

(5.23)

Where and

P is precoding matrix of size N×N shown, m and n are integers from . The

DHT is also invertible transform which allows us to recover the from and

inverse can be obtained by simply multiplying DHT of by

[194].

5.2.4.3 Walsh-Hadamard Transform (WHT): The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–

Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an

example of a generalized class of Fourier transforms. It performs an orthogonal,

symmetric, involutional, linear operation on real numbers (or complex numbers,

although the Hadamard matrices themselves are purely real). The Hadamard

transform can be regarded as being built out of size × × × × DFTs, and is

in fact equivalent to a multidimensional DFT of size. It decomposes an arbitrary input

vector into a superposition of Walsh functions [195].

Sig

nal m

app

er

Sig

nal

dem

apper

One T

ap

Equalizer

And P

/S

Rem

ove C

P

Ad

d C

P

S

/

P

P

/

S

S

/

P

IDF

T O

R IF

FT

D

FT

OR

FF

T

Multipath

Fading Ch. &

noise

O

/

P

I

/

P P

𝑃


106

The WHT is a non-sinusoidal and it is an orthogonal technique which decomposes a

signal into set of basic functions. These functions are called Walsh functions, the

hadamard transform scheme reduce the occurrence of the high peaks comparing the

conventional OFDM system. The hadamard transform used because it reduce the

autocorrelation of the input sequence to reduce the PAPR of OFDM signal. It also not

requires to send side information to the receiver [36]

WHT can be implemented by a butterfly structure as in FFT. This means that

applying WHT does not require the extensive increase of system complexity. The

kernel of WHT can be written as follows:-

, - (5.24)

√ 0

1 (5.25)

√ [

] (5.26)

Where denotes the binary complement of [14]

Note that Hadamard transform is an orthogonal linear transform and can be

implemented by a butterfly structure as in FFT. This means that applying Hadamard

transform does not require the extensive increase of system complexity. The received

vector signal corrupted by noise vector n can be recovered to as [11]:

* + (5.27)

*

* * + + (5.28)

The FWHT for a signal x of length N is defined as:

∑

( ) (5.29)

Where i = 0,1,..., N-1 and ( ) are Walsh functions

5.2.4.4 Discrete Cosine Transform (DCT): DCT is a technique to transform a signal into frequency domain. DCT denotes a row

of data in terms of the sum of cosine functions that oscillate at different frequencies.

DCT is similar to the DFT, but the DCT only uses real number without imaginary

component. The idea of using DCT in this study is to reduce the autocorrelation of the

input row to reduce PAPR and it does not require the information transmitted to the

receiver. The idea of using DCT is for reduce auto-correlation from input data to

reduce PAPR problem [117].

DCT matrix P of size N-by-N can be created by using equation


107

{

√

√

( )

}

(5.30)

and DCT can be defined as:-

∑ ,

.

/ - k=0,1 … N-1 (5.31)

5.2.4.5 Discrete Sine Transform (DST) Precoding Technique:

For an input signal , the discrete sine transform can be defined as:

∑ ,

( )( )- k=1, 2 … N-1 (5.32)

DST precoding matrix D can be generated as follows:

{

√

√

( )

}

(5.33)

The DST matrix must satisfy the following criteria:

1. Same magnitude for all the elements of the precoding matrix.

2. The magnitude must be equal to

√ .

3. The DST precoding matrix must be non-singular matrix.

These criteria ensure that every output symbol has the same amount of information of

every input data; it preserves the power at the output and also ensures the recovery of

the original data at the receiver. When DST precoding is applied to the complex input

vector of size M, this input vector is transformed to a new vector of size L that can be

written as follows:

Y = D. X = , - (5.34)

Where D is a DST precoder matrix of size N = L × L, generated by Eq. (5.33) and

∑ , (5.35)

correspond to row and column of DST precoder matrix [40].

5.2.4.6 The Discrete Fourier Transform (DFT) Precoding: The only difference between the DFT-spread OFDM and the conventional OFDM is

the presence of a DFT and an IDFT block in the transmitter and receiver, respectively


108

In the DFT-spread OFDM, the PAPR of the signal is fairly low as compared with the

conventional OFDM because the DFT operation spreads data into subcarriers [197].

The DFT of a sequence of length N can be defined as

( ) ∑ ( ) – (5.36)

The sinusoids of the DFT (or Inverse Discrete Fourier Transform (IDFT)) form an

orthogonal basis set and a signal in vector space of the DFT (or IDFT) can be

represented as a linear combination of the orthogonal sinusoids. Thus the IDFT at the

transmitter maps an input signal into a set of orthogonal subcarriers. Similarly the

transform DFT is used at the receiver to reverse the mapping of IDFT and signal from

the subcarriers are combined to form an estimate of the source signal from the

transmitter. Since the basis function of DFT is uncorrelated, the correlation performed

in DFT for the given subcarrier only sees energy for that corresponding subcarrier.

The energy from other subcarrier does not contribute because they are uncorrelated.

This separation of the signal energy is the reason that OFDM subcarriers spectrum

can overlap without causing interference. [198]

5.2.4.7 Simulation results and analysis of OFDM system with pre-

coding: 5 types of Pre-coding are used in this section and then compare them with each other.

The best type of reduced PAPR and BER is the DFT pre-coder. The best type of

reduced PAPR and BER is the DFT pre-coder as shown in figure 5.34 and table (A.3)

but suffer from link performance loss in a frequency-selective channel when high-

order modulation techniques are used. The presence of carrier frequency offsets

(CFOs) between the transmitter and the receiver results in a loss of orthogonality

among subcarriers and an intercarrier interference (ICI). CFOs also introduce multiple

access interference (MAI) and degrade the bit error rate (BER) performance in the

DFT pre-coder system. [92]

The following is the conclusion from the table (A.3) and figure 5.33

WHT pecoder was improved each of the PAPR by (2.7941 dB), CCDF of

PAPR by (0.8684 dB) and SNR at BER( ) (0.01dB). But the amount of

improvement in WHT pre-coding is the least in comparison with the rest

kinds of pre-coding

DCT pecoder was improved each of the PAPR by (7.5208 dB), CCDF of

PAPR by (3.109 dB) and SNR at BER( ) (0.012dB). DCT pre-coding

results better than WHT pre-coding but worse than the rest

DST pecoder was improved each of the PAPR by (8.1669 dB), CCDF of

PAPR by (3.25 dB) and SNR at BER( ) (0.012 dB).

DST and DCT have the same SNR at BER( ) but DST better than DCT in

PAPR and CCDF of PAPR

DHT pecoder was improved each of the PAPR by (18.6731 dB), CCDF of

PAPR by (7.423 dB) and SNR at BER( ) (0.058dB). DHT pecoder results

are better than other types of pre-coding except DFT pre-coder.

DFT pecoder was improved each of the PAPR by (25.6118 dB), CCDF of

PAPR by (10.773 dB) and SNR at BER( ) (0.171dB). DFT pecoder

results are the best compared with other types of pre-coding


109

5.33.a

5.33.b

Figure 5.33 (a)CCDF of PAPR for OFDM system with different type of pre-coding

(b) BER for OFDM system with different type of pre-coding

0 2 4 6 8 10 1210

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

orignal

WHT

DCT

DST

DHT

DFT

0 5 10 1510

-4

10-3

10-2

10-1

100

SNR

BE

R


orignal

WHT

DCT

DST

DHT

DFT

Simulation Results and Analysis of Hybrid PAPR techniques

110

Chapter six


6.1 Hybrid pre-coding with RCF: Proposed a method based on the integration of all of precoding with RCF as shown in

figure 6.1. The results of this method better than the results of the RCF and pre-

coding each alone, except in the case of DHT with RCF (I = 2, pilot) ,where the

results of the DHT itself better than the pre-coding with RCF hybrid (DHT with

RCF).The best result for the PAPR is when RCF (I =1) with (DHT).

WHT, DCT, DST, and DHT pre-coders are used with RCF is used with the following

specifications (I =1, pilot, and 2, CR =4, 3, 2) The OFDM system model with the

proposed technique as shown in figure 6.1.

Figure 6.1 the OFDM system model with precoding + RCF.

The following conclusion from table A.29 when comparing the proposed method with

an OFDM system without PAPR reduction method:

There are improved in PAPR, CCDF of PAPR and BER in many points that have

been tested, but The best one improvement in PAPR and CCDF of PAPR is at I =

pilot, CR =2,and DHT. The improvement in PAPR by = (17.2780 dB), CCDF of

PAPR = (7.2062 dB), and the SNR at BER( ) by = (0.1105 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at I = 1, CR =1.5,

and DHT. The improvement in PAPR by = (20.4339 dB), and CCDF of PAPR =


For SNR at BER( )

+p

ilot sy

mbol

Rem

ove

+pilo

t sym

bo

l

Sig

nal m

apper

Sig

nal

dem

apper

One T

ap

Equalizer

And P

/S

Rem

ove C

P

Add C

P

S

/

P

P

/

S

S

/

P

RC

F

IDF

T O

R IF

FT

DF

T O

R F

FT

Multipath

Fading Ch. &

noise

O

/

P

I

/

P P

𝑃

D

/

A

A

/

D


111

The best one improvement in PAPR and CCDF of PAPR is at I = 1, CR =1.3,

and DHT. The improvement in PAPR by = (21.0373 dB), and CCDF of PAPR =



an OFDM system with RCF method:

The PAPR and the CCDF of PAPR were improved and the least amount of

improvement was when I=1, CR = 4 and WHT .The PAPR improvement is equal

to (0.1148 dB) and the CCDF of PAPR improvement is equal to (0.0031 dB),

while the vast amount of improvement is where I=1, CR = 4 and DHT the PAPR

improvement is equal to (7.1348 dB) and the CCDF of PAPR improvement is

equal to (3.0141 dB )

The SNR at BER( ) at I =2, pilot, there is an improvement in some of the

points and the degradation the other .The largest amount of improvement is when

I=2, CR =1.5 and DHT is equal to (17.1418 dB).The largest amount of

degradation is when I=2, CR =1.5 and WHT is equal to (-0.4 dB).

The SNR at BER( ) was improved at I =1.The least amount of improvement in

SNR at BER( ) when DST and CR = 4 and is equal to (0.063 dB). The largest

amount of improvement is when DHT and CR =1.5 is equal to (16.905 dB).


an OFDM system with pre-coding method:

The PAPR was improved, except when (DHT, I =2, pilot, and CR = 4,3 ,2 ,1.5)

PAPR was degraded and the maximum degradation is (-7.269 dB).the least

amount of improvement was at (DHT, I =1, and CR = 2) and is equal to (0.0008

dB), while the vast amount of improvement is where (WHT, I =1, and CR = 1.1)

and is equal to (18.3732 dB).

The CCDF of PAPR was improved, except when (DHT, I =2, pilot, and CR = 4,3

,2 ,1.5) and (DHT, I =2, and CR = 1.5) CCDF of PAPR were degraded and the

maximum degradation is (-2.9015 dB).the least amount of improvement was at

(DHT, I = Pilot, and CR = 1.5) and is equal to (0.1727 dB), while the vast amount

of improvement is where (WHT, I =1, and CR = 1.1) and is equal to (8.6977 dB).

The SNR at BER( ) was degraded, except when (DHT, I =2, pilot, and CR =

4,3 ,2 ) SNR at BER( ) were degraded and the maximum improvement is

(2.7701 dB).The least amount of degradation in SNR at BER( ) when d=1.1

and DHT and is equal to (dB). The largest amount of degradation is when I =1,

CR =1.1, 1.3 and WHT is equal to (-18.37 dB).

Figures (6.2, 6.3, 6.4, and 6.5) shows the performance of the hybrid pre-coding with

RCF (at I =1 for different CR).

Hybrid pre-coding with RCF is the same as the RCF where whenever CR decreased

the (PAPR and CCDF of PAPR) improved and the SNR at BER( ) degraded but

the proposed better than RCF because the improvement (CCDF and PAPR of PAPR)

is greater than the amount of degradation in the SNR at BER( ) if we compared

both with the original signal and RCF to gather.

The best species is (`DHT + RCF) comes after ((DCT + RCF and RCF + DST)

together) and in the end come (WHT + RCF)


112

Figure 6.2.a

Figure 6.2.b

Figure 6.2 (a)CCDF of PAPR for OFDM system with WHT +RCF where I =1 for

different CR (b) BER for OFDM system with WHT +RCF where I =1 for different

CR

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

WHT+RCF (CR = 4)

WHT+RCF(CR = 3)

WHT+RCF(CR = 2)

WHT+RCF(CR = 1.5)

WHT+RCF(CR = 1.3)

WHT+RCF(CR = 1.1)

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


WHT+RCF (CR = 4)

WHT+RCF(CR = 3)

WHT+RCF(CR = 2)

WHT+RCF(CR = 1.5)

WHT+RCF(CR = 1.3)

WHT+RCF(CR = 1.1)


113

Figure 6.3.a

Figure 6.3.b

Figure 6.3 (a)CCDF of PAPR for OFDM system with DCT +RCF where I =1 for

different CR (b) BER for OFDM system with DCT +RCF where I =1 for different CR

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

DCT+RCF (CR = 4)

DCT+RCF(CR = 3)

DCT+RCF(CR = 2)

DCT+RCF(CR = 1.5)

DCT+RCF(CR = 1.3)

DCT+RCF(CR = 1.1)

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


DCT+RCF (CR = 4)

DCT+RCF(CR = 3)

DCT+RCF(CR = 2)

DCT+RCF(CR = 1.5)

DCT+RCF(CR = 1.3)

DCT+RCF(CR = 1.1)


114

Figure 6.4.a

Figure 6.4.b

Figure 6.4 (a)CCDF of PAPR for OFDM system with DST +RCF where I =1 for

different CR (b) BER for OFDM system with DST +RCF where I =1 for different CR

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

DST+RCF (CR = 4)

DST+RCF(CR = 3)

DST+RCF(CR = 2)

DST+RCF(CR = 1.5)

DST+RCF(CR = 1.3)

DST+RCF(CR = 1.1)

0 5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR

BE

R


DST+RCF (CR = 4)

DST+RCF(CR = 3)

DST+RCF(CR = 2 )

DST+RCF(CR = 1.5)

DST+RCF(CR = 1.3)

DST+RCF(CR = 1.1)


115

Figure 6.5.a

Figure 6.5.b

Figure 6.5 (a)CCDF of PAPR for OFDM system with DHT +RCF where I =1 for

different CR (b) BER for OFDM system with DHT +RCF where I =1 for different CR

Figure 6.6 shows the proposed method with I =2 and CR =4, Can note the following

form the figure 6.6, primarily the proposed method on despite of the different type of

pre-coding used but it has almost the same PAPR (0.1690 dB), CCDF of PAPR

(0.1493 dB) and SNR at BER( ) (0.1111 dB). This means that it is not based on

the type of pre-coding.

0 0.5 1 1.5 2 2.5 3 3.510

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

DHT+RCF(CR =4)

DHT+RCF(CR =3)

DHT+RCF(CR =2)

DHT+RCF(CR =1.5)

DHT+RCF(CR =1.3)

DHT+RCF(CR =1.1)

0 5 10 1510

-4

10-3

10-2

10-1

100

SNR

BE

R


DHT+RCF (CR = 4)

DHT+RCF (CR = 3)

DHT+RCF (CR = 2)

DHT+RCF (CR = 1.5)

DHT+RCF (CR = 1.3)

DHT+RCF (CR = 1.1)


116

Figure 6.6.a

Figure 6.6.b


+RCF where I =2, CR =4 (b) BER for OFDM with different type of pre-coding +RCF

where I =2, CR =4

Figure 6.7 shows the proposed method with I =pilot and CR =4 , Can note the

following form the figure 6.7 ,primarily the proposed method on despite of the

different type of pre-coding used but it has almost the same PAPR (0.0378 dB),

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

WHT + RCF

DCT + RCF

DST + RCF

DHT+ RCF

0 1 2 3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

100

SNR

BE

R


WHT + RCF

DCT + RCF

DST + RCF

DHT + RCF


117

CCDF of PAPR(0.0219 dB) and SNR at BER( ) 0.3759 dB). This means that it

is not based on the type of pre-coding.

Figure 6.7.a

Figure 6.7.b


+RCF where I =pilot, CR =4 (b) BER for OFDM with different type of pre-coding

+RCF where I =pilot, CR =4

0 1 2 3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

100

snr

BE

R


DCT +RCF

DST +RCF

DHT +RCF

0 1 2 3 4 5 6 710

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

DCT+RCF

DST+RCF

DHT +RCF


118

Figure 6.8 shows the proposed method with I =1 and CR =1.5. Here depends on the

type of the pre-coding, where different values of (PAPR, CCDF of PAPR and BER).

For each type of pre-coding, and best as is evident is the DHT of figure 6.8.

Figure 6.8.a

Figure 6.8.b

Figure 6.8(a) CCDF of PAPR for OFDM system with different type of pre-coding

+RCF where I =pilot, CR =4 (b) BER for OFDM with different type of pre-coding

+RCF where I =1, CR =1.5

0 0.5 1 1.5 2 2.510

-3

10-2

10-1

100

PAPR0 [dB]

CC

DF

(P

r[P

AP

R>

PA

PR

0])

WHT + RCF

DCT + RCF

DST + RCF

DHT + RCF

0 5 10 15 20 2510

-4

10-3

10-2

10-1

100

SNR

BE

R


WHT + RCF

DCT + RCF

DST + RCF

DHT + RCF


119

6.2 Hybrids RCF with companding: The clipping is the easiest technique to reduce the power by setting a maximum level

for the transmitted signal. In addition to these benefits in clipping, the use of

frequency domain filtering, this improves the BER. On the other hand, the

companding has also been considered a good technique, because it has the good

PAPR reduction capability with no bandwidth expansion and low computational

complexity. The other advantage of companding is that the signal can be recovered at

the receiver through inverse companding transform [10]

With the understanding on RCF and companding techniques, an idea emerged to

combine the philosophy of companding and RCF.

This hybrid technique shows good results because of first RCF reduce the PAPR and

improves the BER constant and then companding more reduces the amount of the

PAPR.

The OFDM system model with the proposed technique is as shown in figure 6.9. RCF

is used with the following specifications (I =2, CR =4, 3, 2) as for the companding has

been using all kinds of previous companding.

Figure 6.9 the OFDM system model with RCF with companding .

6.2.1 RCF + A companding: The following conclusion from table A.13 and figure 6.10 when

comparing the proposed method with an OFDM system without PAPR reduction

method:

At A =5 and CR =4 , There is an improvement in PAPR by = 17.1305 dB ,CCDF

of PAPR = 6.9730 dB, and the BER by =0.3409 dB. This point was chosen

because all the variables improved.

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at A =10 and CR =4

. The improvement in PAPR by = (18.8751 dB), and CCDF of PAPR = (7.7750


For SNR at BER( )

+p

ilot sy

mb

ol

Rem

ov

e

+p

ilot sy

mb

ol

Co

mp

and

ing

Sig

nal m

app

er

Sig

nal

dem

app

er

De-C

om

pan

din

g

On

e Tap

Eq

ualizer

An

d P

/S

S

/

P

P

/

S

S

/

P

RC

F

IDF

T O

R IF

FT

DF

T O

R F

FT

O

/

P

I

/

P

Rem

ov

e CP

A

dd

CP

Multipath

Fading Ch. &

noise

D

/

A

A

/

D


120

The best one improvement in PAPR and CCDF of PAPR is at A = 20 and CR =4



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at A =4 and CR =4.



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at A=200 and CR



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at A =90 and CR =2.



The following conclusion from table A.13 and figure 6.10 when

comparing the proposed method with an OFDM system with a companding PAPR

reduction method:

The PAPR was improved and the least amount of improvement was when A = 90

and CR = 4 and is equal to (6.9071 dB), While the vast amount of improvement is

where A = 5 and CR = 1.5 and is equal to (13.4875 dB).

Less the amount of improvement in CCDF of PAPR when A = 100 and CR = 4

and is equal to (0.815 dB), While the vast amount of improvement is where A = 5

and CR = 1.5 and is equal to (4.64 dB).

The SNR at BER( )was improved when CR =4,3 . The vast amount of

improvement is where A = 10 and CR = 4 and is equal to (3.1882 dB), while Less

the amount of improvement in SNR at BER( )when A = 30 and CR = 3 and is

equal to (0.88 dB)

The SNR at BER( ) was degraded when CR = 2. The least amount of

degradation in SNR at BER( ) when A =80 and CR = 2 and is equal to (-

1.2408 dB). The largest amount of degradation is when A= 90 and is equal to (-

3.0453 dB).

The SNR at BER( ) was degraded when CR = 1.5, when A =5 the amount of

degradation is equal to (-16.4 dB).

The following conclusion from table A.13 and figure 6.10 when comparing the

proposed method with an OFDM system with RCF method:


improvement was when A =5 and CR = 1.5 .The PAPR improvement is equal to

(2.9643 dB) and the CCDF of PAPR improvement is equal to (1.6257 dB), while

the vast amount of improvement is where A = 200 and CR = 4 and the PAPR


equal to (4.64 dB )

The SNR at BER( ) was degraded, except when A =5 and CR =1.5 the BER

maintains its value. The least amount of degradation in SNR at BER( ) when

A =5 and CR = 4 and is equal to (-2.3226 dB). The largest amount of degradation

is when A= 90 and CR =2 is equal to (-12.072 dB).


121

Figure 6.10.a

Figure 6.10.b

Figure 6.10(a) Shows the values of the PAPR and SNR at BER = for each of the

RCF, Acompanding , and Hybird (RCF+A ) b) Shows the values of the CCDF of

PAPR and SNR at BER = for each of the RCF, Acompanding , and Hybird

(RCF+A )

6.2.2 RCF + : The following conclusion when from table A.14 and figure 6.11 comparing the

proposed method with an OFDM system without PAPR reduction method:

At =5 and CR = 4, There is an improvement in PAPR by = (16.5081 dB), CCDF

of PAPR = (6.7470 dB), and the SNR at BER( ) by = (0.8014 dB). This point

was chosen because all the variables improved.

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at =10 and CR



8 10 12 14 16 18 20 22 24 260

5

10

15

20

25

30

SNR at (BER =10-4)

PAPR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

A

RCF (CR=4) + A

RCF (CR=3) + A

RCF (CR=2) + A

8 10 12 14 16 18 20 22 24 261

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

A

RCF (CR=4) + A

RCF (CR=3) + A

RCF (CR=2) + A


122

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at =30 and CR =3.


dB), while the BER deteriorated by = (-3.5686 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at =70 and CR =3 .



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at = 220 and CR




comparing the proposed method with an OFDM system with companding PAPR

reduction method:

The PAPR was improved and the least amount of improvement was when = 220


where = 80 and CR = 2 and is equal to (12.6705 dB).

Less the amount of improvement in CCDF of PAPR when = 220 and CR = 4 and

is equal to (0.8325 dB), while the vast amount of improvement is where = 5 and

CR = 1.5 and is equal to (4.1715 dB).

The SNR at BER( ) was improved when CR =4,3 . The vast amount of

improvement is where = 50 and CR = 4 and is equal to (3.0554 dB), while Less

the amount of improvement in SNR at BER( )when = 255 and CR = 3 and is

equal to (1.468 dB)


degradation in SNR at BER( ) when =240 and is equal to (-0.985 dB). The

largest amount of degradation is when = 20 and is equal to (-2.625 dB).

The SNR at BER( ) was degraded when CR = 1.5, when =5 the amount of

degradation is equal to (-16.6637dB).


comparing the proposed method with an OFDM system with RCF method:


and CR = 1.5 and is equal to (2.8281 dB), while the vast amount of improvement is


Less the amount of improvement in CCDF of PAPR when = 5 and CR = 2 and is

equal to (1.4656 dB), while the vast amount of improvement is where = 255 and

CR = 4 and is equal to (4.6677 dB).

The SNR at BER( ) was degraded, except when =5 and CR =1.5 the SNR at

BER( ) maintains its value. The least amount of degradation in SNR at

BER( ) when MU =5 and CR = 4 and is equal to (-1.8621 dB). The largest

amount of degradation is when MU= 220 and CR =2 is equal to (-10.872 dB).


123

Figure 6.11.a

Figure 6.11.b

Figure 6.11 ( a) Shows the values of the PAPR and SNR at BER = for each of

the RCF, companding , and Hybird (RCF+ ). b) Shows the values of the CCDF of

PAPR and SNR at BER = for each of the RCF, companding , and Hybird

(RCF+ ).

6.2.3 RCF + RCT: The following conclusion from table A.15 and figure 6.12 when


method:

At these values (R =0.9, 0.8, 0.7, 0.6 and CR = 4, 3) There are improved in PAPR,

CCDF of PAPR and the BER dB). Point was chosen because all the variables

improved. The best one improvement in PAPR and CCDF of PAPR is at R =0.6

8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RCF (CR =4)

RCF (CR =3)

RCF (CR=2)

MU

RCF (CR =4) + MU

RCF (CR =3) + MU

RCF (CR=2) + MU

8 10 12 14 16 18 20 22 241

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

RCF (CR =4)

RCF (CR =3)

RCF (CR=2)

MU

RCF (CR =4) + MU

RCF (CR =3) + MU

RCF (CR=2) + MU


124

and CR =3. The improvement in PAPR by = (17.2514 dB), CCDF of PAPR =

(7.0968 dB), and the SNR at BER( ) by = (0.5079 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R =0.5 and CR =3.



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R =0.4 and CR =3



For SNR at BER( )




For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R= 0.2 and CR =3.



For SNR at BER( )





proposed method with an OFDM system with RCT PAPR reduction method:


improvement was when R =0.1 and CR = 4 .The PAPR improvement is equal to

(0.9227 dB) and the CCDF of PAPR improvement is equal to(0.363 dB), While

the vast amount of improvement is where R =0.9 and CR = 1.5 and the PAPR


equal to (6.201dB )

The SNR at BER( ) was improved when CR =4,3 . The vast amount of

improvement is where R =0.7 and CR = 4 and is equal to (3.0902 dB), while Less

the amount of improvement in CCDF of PAPR when R =0.1 and CR = 3 and is

equal to (1.8866 dB)


degradation in SNR at BER( ) when R =0.5 and is equal to (-0.7285 dB). The

largest amount of degradation is when R =0.3 and is equal to (-1.475 dB).

The SNR at BER( ) was degraded when CR = 1.5, when R =0.9 the amount of

degradation is equal to (-18.3235dB).



The PAPR was improved and the least amount of improvement was when R =0.9


where R =0.1and CR = 4 and is equal to (12.5433 dB).


125

Less the amount of improvement in CCDF of PAPR when R = 0.9 and CR =2 and

is equal to (0.1506 dB), while the vast amount of improvement is where R= 0.1

and CR = 4 and is equal to (5.5127 dB).

The SNR at BER( ) was degraded, except when R =0.1 and CR =1.5 the SNR

at BER( ) maintains its value. The least amount of degradation in CCDF of

PAPR when R =0.9 and CR = 4 and is equal to (-0.1291 dB). The largest amount

of degradation is when R= 0.1 and CR =2 is equal to (-17.181 dB).

figure 6.12.a

Figure 6.12. b

figure 6.12 (a) Shows the values of the PAPR and SNR at BER = for each of the

RCF, RCT , and Hybird (RCF+RCT) b) Shows the values of the CCDF of PAPR and

SNR at BER = for each of the RCF, RCT, and Hybird (RCF+ RCT).

10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

Roots

RCF (CR=4) + Rooting



10 15 20 25 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

Roots





126

6.2.4 RCF + AEXP: The following conclusion from table A.16 and figure 6.13 when


method:

At these values ((d =2-1by .1 every time and CR = 4) and when (d =1.8, 1.5 and

CR =3) There are improved in PAPR, CCDF of PAPR and SNR at BER( )).

The best one improvement in PAPR and CCDF of PAPR is at d = 1 and CR =4.

The improvement in PAPR by = (18.7316 dB) ,CCDF of PAPR = (7.7135 dB),

and the SNR at BER( ) by = ( 0.2467 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at d =0.9 and CR =3.

The improvement in PAPR by = (19.6985 dB), and CCDF of PAPR = (8.1400 dB),


For SNR at BER( )




For SNR at BER( )




For SNR at BER( )





comparing the proposed method with an OFDM system with AEXP companding

PAPR reduction method:

The PAPR was improved except when d =1.6 and CR =3 the PAPR was degraded

by (-1.6855dB) .the least amount of improvement was when d =0.4 and CR = 4

and is equal to (0.1356 dB), while the vast amount of improvement is where d =2


Less the amount of improvement in CCDF of PAPR when d =0.4 and CR = 4 and

is equal to (0.14 dB), while the vast amount of improvement is where d =2 and CR

= 1.5 and is equal to (1.7799 dB).

The SNR at BER( ) was improved when CR =4,3 except when (d =0.4 and CR

=4,3) and when (d =0.5 and CR =3) the SNR at BER( ) maintains its value.

The vast amount of improvement is where d =0.7 and CR = 4 and is equal to

(15.5893 dB), while Less the amount of improvement in SNR at BER( ) when

d=0.1 and CR = 4 and is equal to (0.79 dB)

The SNR at BER( ) was degraded when CR = 2 except when (d =0.6, 0.5, 0.4)

the SNR at BER( ) maintains its value. The least amount of degradation in

SNR at BER( ) when d =0.7 and is equal to (-0.9 dB). The largest amount of

degradation is when R =1.1 and is equal to (-7.65 dB).

The SNR at BER( ) was degraded when CR = 1.5, when d =2. the amount of



127



The PAPR was improved and the least amount of improvement was when d =2 and

CR = 1.5 and is equal to (0.0555 dB), while the vast amount of improvement is

where d =0.4 and CR = 4 and is equal to (11.2249 dB).

Less the amount of improvement in CCDF of PAPR when d =2 and CR =2 and is

equal to (0.3051 dB), while the vast amount of improvement is where d =0.4 and


The SNR at BER( ) was degraded, except when d =2 and CR =1.5 the SNR at

BER( ) maintains its value. The least amount of degradation in SNR at

BER( ) when d =1.3 and CR = 4 and is equal to (-1.6321 dB). The largest amount

of degradation is when d =0.4 and CR =4 is equal to (-21.2321 dB).

Figure 6.13.a

Figure 6.13.b

Figure 6.13 (a) Shows the values of the PAPR and SNR at BER = for each of the

RCF, AEXP companding , and Hybird (RCF+AEXP)

b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the

RCF, AEXP companding , and Hybird (RCF+ AEXP)

5 10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

AEXP

RCF (CR=4) + AEXP

RCF(CR=3) +AEXP

RCF(CR=2) +AEXP

5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

AEXP

RCF (CR=4) + AEXP

RCF (CR=3) + AEXP

RCF(CR=2) + AEXP


128

6.2.5 RCF + cos : The following conclusion from table A.17 and figure 6.14 when


method:

At these values ((y =1,0.9, 0.8,0.7 and CR = 4,3) and when ( y =0.6 and CR =4)

There are improved in PAPR, CCDF of PAPR and BER ). The best one

improvement in PAPR and CCDF of PAPR is at y =.7 and CR =3. The

improvement in PAPR by = (17.1463 dB),CCDF of PAPR = (7.0651 dB), and the

BER by = ( 0.5250 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at y=0.5 and CR =4.



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at y = 0.6 and CR



For SNR at BER( )




For SNR at BER( )




For SNR at BER( )





proposed method with an OFDM system with cos companding PAPR reduction

method:

The PAPR was improved and the least amount of improvement was when y =0.3


where y =1 and CR = 1.5 and is equal to (8.2456 dB).

The CCDF of PAPR was improved and the least amount of improvement is where

y =0.3 and CR = 4 and is equal to (0.4597 dB).

, while the vast amount of improvement is where y =1 and CR = 1.5 and is equal

to (3.4937 dB).

The SNR at BER( ) was improved when CR =4, 3. The vast amount of

improvement when y=0.3 and CR = 4 and is equal to (4.4287 dB), while Less the

amount of improvement in BER when y = 1 and CR = 3 and is equal to (2.061dB)


degradation in SNR at BER( ) when y =0.6 and is equal to (-0.3705 dB). The

largest amount of degradation is when y =0.3 and is equal to (-8.3 dB).

The SNR at BER( ) was degraded when CR = 1.5, when y =1 the amount of



129



The PAPR was improved and the least amount of improvement was when y =1 and

CR = 1.5 and is equal to (1.0679 dB), While the vast amount of improvement is

where y =0.1 and CR = 4 and is equal to (12.7916 dB).

Less the amount of improvement in CCDF of PAPR when y=1 and CR = 1.5 and is

equal to (0.2694 dB), while the vast amount of improvement is where y=0.1 and


The SNR at BER( ) was degraded, except when y =1 and CR =1.5 the BER

maintains its value. The least amount of degradation in SNR at BER( ) when y

=1 and CR = 4 and is equal to (-0.3076 dB). The largest amount of degradation is

when y =0.2, 0.1 and CR =4 is equal to (21.2321 dB).

Figure 6.14.a

Figure 6.14.b


RCF, cos companding , and Hybird (RCF+cos) (b)Shows the values of the CCDF of

PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird

(RCF+ cos)

5 10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

cos

RCF (CR=4) + cos

RCF (CR=3) +cos

RCF (CR=2) + cos

5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CCDF

of P

APR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

cos

RCF (CR=4) + cos

RCF (CR=3) + cos

RCF (CR=2) + cos


130

6.2.6 RCF + NERF :



method:

At these values (CR = 4, 3) and There are improved in PAPR, CCDF of PAPR and

BER). The best one improvement in PAPR and CCDF of PAPR is at CR =3. The

improvement in PAPR by = (17.0615 dB), CCDF of PAPR = (7.2730 dB), and the

SNR at BER( ) by = (0.5314 dB).

Figure 6.15.a

Figure 6.15.b


RCF, cos companding , and Hybird (RCF+NERF) b) Shows the values of the CCDF

of PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird

(RCF+ NERF)

5 10 15 20 25 306

8

10

12

14

16

18

20

22

24

26

SNR at (BER =10-4)

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

RCF (CR=1.5)

NERF

RCF (CR=4) + NERF

RCF (CR=3) + NERF

RCF (CR=2) + NERF

RCF (CR=1.5) + NERF

5 10 15 20 25 302

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RCF (CR=4)

RCF (CR=3)

RCF (CR=2)

RCF (CR=1.5)

NERF

RCF (CR=4) + NERF

RCF (CR=3) + NERF

RCF (CR=2) + NERF

RCF (CR=1.5) + NERF


131

6.2.7 RCF + tanhR : The following conclusion from table A.19 when comparing the proposed method with


At these values (k =5,10,15 and y =1 ,0.8 for CR = 4,3) and when (k =20 and y

=1 ,0.8 for CR =3) There are improved in PAPR, CCDF of PAPR and BER ). The

best one improvement in PAPR and CCDF of PAPR is at k=20 y = .8 for CR =4 .

The improvement in PAPR by = (18.6958 dB), CCDF of PAPR = (7.5530 dB), and

the SNR at BER( ) by = ( 0.1527dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=10, y=.5 and

CR =4 . The improvement in PAPR by = (19.3352 dB), and CCDF of PAPR =


The best one improvement in CCDF of PAPR is at k=5 , y=.5 and CR =3 . The


while the SNR at BER( ) deteriorated by = (--1.4569dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =20, y=0.5 and

CR =3. The improvement in PAPR by = (21.1850 dB), and CCDF of PAPR =


For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k =30, y=0.5 and



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=40, y =0.5and



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k= 40, y =0.2 and

CR =3.The improvement in PAPR by = (23.9630 dB), and CCDF of PAPR =


For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k = 20, y=.2 and




an OFDM system with tanhR companding PAPR reduction method:

The PAPR was degraded at (k =15, 20, y =1,0.8 and CR = 4, 3, 2), (k= 10, y

=1,0.8 and CR=4,3), (k =15 ,y =0.5 ,and CR =4) , and when (k =20 ,y =0.5 ,and

CR =4,3). The least amount of degradation in PAPR when k=10, y=0.8 and CR = 3

and is equal to (-0.5458 dB). The largest amount of degradation is when k =15, y =

1 and CR =4 is equal to (-4.717 dB).

The CCDF of PAPR was degraded at [for CR = 4, 3, 2 at (k =20, y =1,0.8,0.5 )

and (k =15, y =1,0.8) ], [for CR =4,3 at (k =15 ,y =0.5 ),and((k= 10, y =1,0.8 and

CR=4,3)], and when [k =10 ,y =0.5 ,and CR =4] The least amount of degradation

in CCDF of PAPR when k=20, y=0.5 and CR = 2 and is equal to (-0.0085 dB).

The largest amount of degradation is when k =20, y = 1 and CR =4 is equal to (-

2.0457 dB).


132

Except the points already mentioned, the PAPR was improved and the least

amount of improvement was when k =20, y =0.5 and CR = 2 and is equal to

(0.0043 dB), while the vast amount of improvement is where k =5, y =1 and CR =

1.5 and is equal to (8.0365dB).

Except the points already mentioned, the CCDF of PAPR was improved and Less

the amount of improvement in CCDF of PAPR when k =10, y =1 and CR = 2 and

is equal to (0.3585 dB), While the vast amount of improvement is where k =5, y =1


The SNR at BER( ) was degraded when CR =2 at (k =5, y =1,0.8,0.5,0.2 ), ( y

=0.2, k =10,15,20), and when (k =10 ,y =0.5). The least amount of degradation in

SNR at BER( ) when k=5, y=0.8 and is equal to (-0.11 dB). The largest

amount of degradation is when k =10, y = 0.2 is equal to (-2.4641 dB).

The SNR at BER( ) was degraded when CR = 1.5, when k =5, y =1 .The

amount of degradation is equal to (-18.0900 dB).

Except the points already mentioned, the SNR at BER( ) was improved and

the vast amount of improvement is where k =15, y =1 and CR = 4 and is equal to

(19.803 dB), while Less the amount of improvement in SNR at BER( ) when k

=5, y=0.8 and CR = 2 and is equal to (1.1217 dB).



The PAPR was improved and the least amount of improvement was when k=5 ,y

=1 and CR = 1.5 and is equal to (0.0685 dB), While the vast amount of

improvement is where k =40, y =0.2 and CR = 4 and is equal to (12.6433 dB).

The CCDF of PAPR was improved, except when k =5 ,y =1 and CR =2 the CCDF

of PAPR was degraded by (-0.0264 dB). Less the amount of improvement in

CCDF of PAPR when k =5, y= 1 and CR = 3 and is equal to (0.027 dB), While the

vast amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to

(5.5377 dB).

The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5 the

SNR at BER( ) maintains its value. The least amount of degradation in SNR at

BER( ) when k =5, y =1 and CR = 3 and is equal to (-0.1386 dB). The largest

amount of degradation is when k =50, y =1, 0.8, 0.2 and CR =3 is equal to (-

20.5860 dB).

6.2.8 RCF +logR : The following conclusion from table A.20 when comparing the proposed method with


At these values (k =5,10,20 and y =1 ,0.8 for CR = 4,3) and when (k =30,40,50

,70 and y =1 ,0.8 for CR =4) and finally, when (k =30,40 and y =1 for CR =3)

There are improved in PAPR, CCDF of PAPR and BER ). The best one

improvement in PAPR and CCDF of PAPR is at k=70, y = .8 for CR =4 . The

improvement in PAPR by = (18.0511 dB) ,CCDF of PAPR = (7.3815 dB), and the

SNR at BER( ) by = (0.2025 dB).

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=5, y =0.5 and




133

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=90, y=0.5 and



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=50 ,y =0.5and



For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=70 ,y =0.2and

CR =3. The improvement in PAPR by = (23.3205dB), and CCDF of PAPR =


For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=.9, y =0.2 and




an OFDM system with logR companding PAPR reduction method:

The PAPR was improved and the least amount of improvement was when k =5, y

=1 and CR = 1.5 and is equal to (0.4088 dB), While the vast amount of


The CCDF of PAPR was improved and the least amount of improvement was

when k =5, y =1 and CR = 2 and is equal to (0.2136 dB), while the vast amount of


The SNR at BER( ) was degraded when k =5 ,y =1 and CR =1.5 the BER

degraded by (-17.9300 dB).

The SNR at BER( ) was improved when CR =4, 3. The least amount of

improvement in BER when k =5, y =1 and CR = 3 and is equal to (2.0783 dB). The

largest amount of improvement is when k =90, y =1 and CR =4 is equal to (18.215

dB).

The SNR at BER( ) was degraded when CR = 2, except when (k =30,90 and y

=1) and CR =1.5 the SNR at BER( ) was improved by (0.5122 - 0.825 dB).

The least amount of degradation in SNR at BER( ) when k=10, y =1 and is

equal to (-0.1639 dB). The largest amount of degradation is when k=90, y =0.8 and

is equal to (-4.0838 dB).



The PAPR was improved and the least amount of improvement was when k=5, y

=1 and CR = 1.5 and is equal to (0.4088 dB), while the vast amount of


Less the amount of improvement in CCDF of PAPR when k =5, y= 1 and CR = 2

and is equal to (0.2136 dB), While the vast amount of improvement is where k

=90, y =0.2 and CR = 4 and is equal to (5.2067 dB).

The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5 the

SNR at BER( ) maintains its value. The least amount of degradation in SNR at

BER( ) when k =5, y =1 and CR = 3 and is equal to (-0.1171 dB). The largest

amount of degradation is when k =90, y =1 and CR =2 is equal to (-17.047 dB).


134

6.3 Hybrid RFC with companding: The process of companding enlarges the amplitudes of the small signals, while the

peaks remain unchanged. Therefore, the average power is increased and thus the

Peak-to Average Power Ratio (PAPR) of the OFDM systems can be reduced, which

in turn helps in increasing the efficiency of the power amplifiers and also reduces the

complexity of the Analog-to-Digital Converter (ADC) and Digital-to-Analog

Converter (DAC) [53].

As we demonstrated earlier RFC batter than RCF, because it reduces the impact of the

filter on the PAPR and also when increasing I the BER improves.

The previous method hybrid technique depend on RCF and companding give a good

result and as we demonstrated earlier RFC batter than RCF, because it reduces the

impact of the filter on the PAPR and also the BER improves. When increasing I. So

we proposed a new hybrid method based on RFC and companding. The OFDM

system model with RFC and companding was shown in the Figure 6.16.

This hybrid technique shows good results better than the previous method, because of

first RFC improve the PAPR and the BER more than RCF and then companding more

reduces the amount of the PAPR.

RFC is used with the following specifications (I =4, CR =4, 3, 2) as for the

companding has been using all kinds of previous companding.

Figure 6.16 the OFDM system model with RFC + companding .

6.3.1 RFC + A companding: The following conclusion from table A.21 and figure 6.17 when comparing the


At these values (A =5, 10 for CR = 4, 3) and There are improved in PAPR, CCDF

of PAPR and BER). The best improvement in PAPR and CCDF of PAPR is at

A=10 for CR =3. The improvement in PAPR by = (19.8043 dB), CCDF of PAPR =

(8.4633 dB), and the SNR at BER( ) by = ( 0.5300 dB).

For SNR at BER( )

+p

ilot sy

mb

ol

Rem

ov

e

+p

ilot sy

mbo

l

Co

mp

and

ing

Sig

nal m

app

er

Sig

nal

dem

app

er

De-C

om

pan

din

g

On

e Tap

Eq

ualizer

An

d P

/S

S/

P

P/

S

S/

P

RF

C

IDF

T O

R IF

FT

DF

T O

R F

FT

O

/

P

I

/

P

Rem

ov

e CP

A

dd

CP

Multipath

Fading Ch. &

noise

D

/

A

A

/

D


135

The best improvement in PAPR and CCDF of PAPR is at A =20 and CR =3. The



The best improvement in CCDF of PAPR is at A =20 and CR =2. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A = 80 and CR =4 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A = 100 and CR =2.



The best improvement in CCDF of PAPR is at A = 120 and CR =3. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A=80 and CR =2. The

improvement in PAPR by = (22.2111 dB), and CCDF of PAPR = (9.8161dB),


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A =90 and CR =2. The




comparing the proposed method with an OFDM system with A companding PAPR

reduction method:

The PAPR was improved and the least amount of improvement was when A = 40


where A = 5 and CR = 1.5 and is equal to (14.3373 dB).

The least amount of improvement in CCDF of PAPR when A = 100 and CR = 4

and is equal to (0.8784 dB), while the vast amount of improvement is where A = 5


The SNR at BER( ) was improved when CR =4,3,2 . The vast amount of

improvement is where A = 70 and CR = 4 and is equal to (6.3199 dB), while The

least amount of improvement in SNR at BER( ) when A = 5 and CR = 2 and

is equal to (0.5206 dB)

The SNR at BER( ) was degraded when CR = 1.5, when A =5. The amount of

degradation is equal to (-16 dB).




improvement was when A =5 and CR = 1.5 .The PAPR improvement is equal to

(1.8221 dB) and the CCDF of PAPR improvement is equal to(0.9794 dB), while

the vast amount of improvement is where A = 200 and CR = 4 and the PAPR




136

The SNR at BER( ) was degraded. The least amount of degradation in SNR at

BER( ) when A =5 and CR = 1.5 and is equal to (-1.38 dB). The largest

amount of degradation is when A= 80 and CR =2 is equal to (-11.2484 dB).

Figure 6.17.a

Figure 6.17.b


RFC, companding , and Hybird (RFC+ ). b) Shows the values of the CCDF of

PAPR and SNR at BER = for each of the RCF, companding , and Hybird

(RFC+ ).

4 6 8 10 12 14 16 18 20 220

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

A

RFC (CR=4) + A

RFC (CR=3) + A

RFC (CR=2) + A

4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

A

RFC (CR=4) + A

RFC (CR=3) + A

RFC (CR=2) + A


137

6.3.2 RFC + companding: The following conclusion from table A.22 and figure 6.18 when


method:

At these values ( =5,10,20 for CR = 4,3) and when ( =30 ,40 for CR = 4)

There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best

improvement in PAPR is at =40 for CR =4. The improvement in PAPR by =

(20.0157 dB) ,CCDF of PAPR = (8.3564 dB), and the SNR at BER( ) by = (

0.0116dB).

The best improvement in CCDF of PAPR is at =20 for CR =3 . The

improvement in PAPR by = (19.9252 dB) ,CCDF of PAPR = (8.5044 dB), and

the SNR at BER( ) by = ( 0.2052 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at = 40 and CR =3. The



For SNR at BER( )

The best improvement in PAPR is at =180 and CR =4. The improvement in

PAPR by = (21.4145 dB) , and CCDF of PAPR = (9.0247 dB), while the SNR at

BER( ) deteriorated by = (-3.4087 dB).

The best improvement in CCDF of PAPR is at =80 and CR =3. The

improvement in PAPR by = (21.1482 dB) , and CCDF of PAPR = (9.0856 dB),


For SNR at BER( )

The best improvement in PAPR is at =240 and CR =4. The improvement in

PAPR by = (21.605 dB), and CCDF of PAPR = (9.0964 dB), while the SNR at


The best improvement in CCDF of PAPR is at =255and CR =4. The

improvement in PAPR by = (21.1370 dB),and CCDF of PAPR = (9.1356 dB),


A.22 and figure 6.18 when comparing the proposed method with an OFDM system

with companding PAPR reduction method:




The least amount of improvement in CCDF of PAPR when = 220 and CR = 4

and is equal to (0.9042 dB), while the vast amount of improvement is where = 5


The SNR at BER( ) was improved when CR =4,3. The vast amount of

improvement is where = 50 and CR = 4 and is equal to (6.4429 dB), while The

least amount of improvement in SNR at BER( ) when = 5 and CR = 3 and is


The SNR at BER( ) was improved when CR =2. The vast amount of

improvement is where = 220 and is equal to (2.4273 dB), while The least

amount of improvement in CCDF of PAPR when = 30 and is equal to (1.2065

dB)

The SNR at BER( ) was degraded when CR = 1.5, when =5 .the amount of



138


comparing the proposed method with an OFDM system with RFC method:


and CR = 1.5 and is equal to (1.4566 dB), While the vast amount of improvement

is where = 240 and CR = 4 and is equal to (10.1873 dB).

The least amount of improvement in CCDF of PAPR when = 5 and CR = 1.5 and

is equal to (0.8031 dB), while the vast amount of improvement is where = 255


The SNR at BER( ) was degraded the least amount of degradation in SNR at

BER( ) when =5 and CR = 1.5 and is equal to (-1.78 dB). The largest

amount of degradation is when = 255 and CR =2 is equal to (-10.882 dB).

Figure 6.18.a

Figure 6.18.b


RFC, companding , and Hybird (RFC+ ) b) Shows the values of the CCDF of

4 6 8 10 12 14 16 18 20 220

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

MU

RFC (CR=4) + MU

RFC (CR=3) + MU

RFC (CR=2) + MU

4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

12

SNR at (BER =10-4)

CCDF

of P

APR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

MU

RFC (CR=4) + MU

RFC (CR=3) + MU

RFC (CR=2) + MU


139

PAPR and SNR at BER = for each of the RFC, companding , and Hybird

(RFC+ )

6.3.3 RFC + RCT:



method:

At these values (R =0.9 ,0.8 ,0.7,0.6,0.5 and CR = 4,3,2) and when (R =0.4 and

CR = 4,3) There are improved in PAPR, CCDF of PAPR and the SNR at

BER( ) dB). These points were chosen because all the variables improved. The

best improvement in PAPR and CCDF of PAPR is at R =0.5 and CR =2. The

improvement in PAPR by = (20.5192 dB) ,CCDF of PAPR = (8.7312 dB), and the

SNR at BER( ) by = ( 0.0156 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R = 0.4 and CR = 2 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R = 0.2 and CR =3 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R= 0.2 and CR =2.



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R= 0.1 and CR =3.



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R = 0.1and CR =2 .




comparing the proposed method with an OFDM system with RCT companding PAPR

reduction method:


improvement was when R =0.1 and CR = 4 .The PAPR improvement is equal to

(0.9577 dB) and the CCDF of PAPR improvement is equal to(0.3934 dB), While

the vast amount of improvement is where R =0.9 and CR = 1.5 and the PAPR



The SNR at BER( ) was improved when CR =4,3,2 . The vast amount of

improvement is where R =0.6 and CR = 4 and is equal to (6.298 dB), while The

least amount of improvement in SNR at BER( ) when R =0.4 and CR = 2 and

is equal to (2.3582 dB)


140

The SNR at BER( ) was degraded when CR = 1.5, when R =0.9 the amount of


Figure 6.19.a

Figure 6.19.b

figure 6.19 (a) Shows the values of the PAPR and SNR at BER = for each of the

RFC, RCT, and Hybird (RFC+ RCT) b) Shows the values of the CCDF of PAPR and

SNR at BER = for each of the RFC, RCT, and Hybird (RFC+ RCT).



The PAPR was improved and the least amount of improvement was when R =0.9


where R =0.1and CR = 4 and is equal to (12.2689 dB).

The least amount of improvement in CCDF of PAPR when R = 0.9 and CR =2

and is equal to (0.1621 dB), while the vast amount of improvement is where R=

0.1 and CR = 4 and is equal to (5.1904 dB).


BER( ) when R =0.9 and CR = 4 and is equal to (-0.2943 dB). The largest

amount of degradation is when R= 0.1 and CR =3 is equal to (-17.4447 dB).

5 10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PAP

R

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

Roots

RFC (CR=4) + Rooting



5 10 15 20 25 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

Roots





141

6.3.4 RFC + AEXP:



method:

At these values ((d =2-0.7 by .1 every time and CR = 4,3) and when (d =2 and

CR =2) and finally when(d =0.6 and CR =2)There are improved in PAPR, CCDF

of PAPR and SNR at BER( )). The best improvement in PAPR and CCDF of

PAPR is at d = 0.6 and CR =4. The improvement in PAPR by = (21.0509dB)

,CCDF of PAPR = (8.7178 dB), and the SNR at BER( ) by = ( 0.0116 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d =0.6 and CR = 3.

The improvement in PAPR by = (21.3545 dB) , and CCDF of PAPR = (8.8589


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d= 0.5 and CR =4 .


dB), while the SNR at BER( ) deteriorated by= (-2.9912 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d = 0.7 and CR =2 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d =0.6 and CR =2. The






The PAPR was improved the least amount of improvement was when d =0.4 and

CR = 4 and is equal to (0.1528 dB), while the vast amount of improvement is

where d =2 and CR = 1.5 and is equal to (5.8486 dB).

The least amount of improvement in CCDF of PAPR when d =1.2 and CR = 4 and

is equal to (0.0659 dB), while the vast amount of improvement is where d =2 and

CR = 1.5 and is equal to (2.6752 dB).

The SNR at BER( ) was improved when CR =4,3,2 except when (d =0.4 and

CR =4, 3, 2) and when (d =0.5 and CR =2) the SNR at BER( ) maintains its

value. The vast amount of improvement is where d =0.7 and CR = 4 and is equal

to (19.5598 dB), while The least amount of improvement in CCDF of PAPR when

d=1 and CR = 2 and is equal to (0.217 dB)

The SNR at BER( ) was degraded when CR = 1.5, when d =2 The amount of




The PAPR and the CCDF of PAPR were improved except when d =2 and CR

=1.5,2 they were degraded. The least amount of improvement was when d =1.9 and

CR = 2 and the PAPR improvement is equal to (0.2144 dB) and the CCDF of

PAPR improvement is equal to (0.0784 dB), while the vast amount of


142

improvement is where d =0.4 and CR = 4 and the PAPR improvement is equal to

(10.9327 dB) and the CCDF of PAPR improvement is equal to (4.5281 dB).


BER( ) when d =2 and CR = 4 and is equal to (-1.5141 dB). The largest

amount of degradation is when d =0.4 and CR =4 is equal to (-24.3287 dB).

Figure (6.20.a)

Figure (6.20.b)


RFC, AEXP companding , and Hybird (RFC+AEXP)


RFC, AEXP companding , and Hybird (RFC+ AEXP).

5 10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

AEXP

RFC (CR=4) + AEXP

RFC (CR=3) + AEXP

RFC (CR=2) + AEXP

5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

AEXP

RFC (CR=4) + AEXP

RFC (CR=3) + AEXP

RFC (CR=2) + AEXP


143

6.3.5 RFC + cos :



At these values ((y =1,0.9, 0.8,0.7 and CR = 4,3,2) and when ( y =0.6,0.5 and CR

=4,3 ) and finally at (y =.4 and CR =4) There are improved in PAPR, CCDF of

PAPR and SNR at BER( )). The best improvement in PAPR and CCDF of

PAPR is at y =0.7 and CR =2. The improvement in PAPR by = (19.9896 dB),

CCDF of PAPR = (8.3417 dB), and the SNR at BER( ) by = (1.1469 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y = 0.5 and CR =2 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y =0.4 and CR =2 .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y =0.3 and CR =2.



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y=0.2 and CR =3. The




comparing the proposed method with an OFDM system with cos companding PAPR

reduction method:


improvement was when y =0.3 and CR = 4.The PAPR improvement is equal to


the vast amount of improvement is where y =1 and CR = 1.5 and the PAPR



The SNR at BER( ) was improved when CR =4, 3,2. The vast amount of

improvement when y=0.3 and CR = 4 and is equal to (7.6706 dB), while The least

amount of improvement in SNR at BER( ) when y = 0.8 and CR = 2 and is


The SNR at BER( ) was degraded when CR = 1.5, when y =1 the amount of



proposed method with an OFDM system with RFC method:


improvement was when y =1 and CR = 1.5 .The PAPR improvement is equal to


the vast amount of improvement is where y =0.1 and CR = 4 and the PAPR




144

The SNR at BER( ) was degraded, the least amount of degradation in SNR at

BER( ) when y =1 and CR = 2 and is equal to (-0.0444 dB). The largest

amount of degradation is when (y =0.2, 0.1 and CR =4)(y =0.1 and CR =3) is equal

to (-23.9287 dB).

Figure 6.21.a

Figure 6.21.b


RCF, cos companding , and Hybird (RCF+cos)


RCF, cos companding , and Hybird (RCF+ cos)

5 10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

cos

RFC (CR=4) + cos

RFC (CR=3) + cos

RFC (CR=2) + cos

5 10 15 20 25 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

cos

RFC (CR=4) + cos

RFC (CR=3) + cos

RFC (CR=2) + cos


145

6.3.6 RFC + NERF : The following conclusion from table A.26 and figure 6.22 when


method:

At these values ( CR = 4,3) and There are improved in PAPR, CCDF of PAPR and

SNR at BER( )). The best improvement in PAPR and CCDF of PAPR is at CR

=3. The improvement in PAPR by = (17.3493 dB) ,CCDF of PAPR = (7.3822 dB),

and the SNR at BER( ) by = ( 3.6500 dB).

Figure 6.22.a

Figure 6.22.b

Figure (a) Shows the values of the PAPR and SNR at BER = for each of the

RCF, cos companding , and Hybird (RCF+NERF) (b)Shows 6.22 the values of the

CCDF of PAPR and SNR at BER = for each of the RCF, cos companding , and

Hybird (RCF+ NERF)

5 10 15 20 25 305

10

15

20

25

30

SNR at (BER =10-4)

PAPR

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

RFC (CR=1.5)

NERF

RFC (CR=4) + NERF

RFC (CR=3) + NERF

RFC (CR=2) + NERF

RFC (CR=1.5) + NERF

5 10 15 20 25 301

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

RFC (CR=4)

RFC (CR=3)

RFC (CR=2)

RFC (CR=1.5)

NERF

RFC (CR=4) + NERF

RFC (CR=3) +NERF

RFC (CR=2) + NERF

RFC (CR=1.5) + NERF


146

6.3.7 RFC + tanhR : The following conclusion from table A.27 when comparing the proposed method with


There are improved in PAPR, CCDF of PAPR and SNR at BER( ) in many

points that have been tested, but The best improvement in PAPR and CCDF of

PAPR is at k=40, y = .5 for CR =4 . The improvement in PAPR by = (20.7866

dB),CCDF of PAPR = (8.5636 dB), and the SNR at BER( ) by = ( 0.1129

dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k =20, y =0.5 and CR

=2 . The improvement in PAPR by = (21.4382 dB) , and CCDF of PAPR =


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k =30, y=0.5 and CR

=2 . The improvement in PAPR by = (21.7998 dB), and CCDF of PAPR =


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k =40, y=0.5 and CR



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k=40, y=0.2 and CR



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is k=40, y=0.2 and CR =2.




an OFDM system with tanhR companding PAPR reduction method:

The PAPR was degraded at (k =20, y =1,0.8,0.5 and CR = 4, 3, 2), (k= 10, y

=1,0.8 and CR=4,3), (k =15 ,y =0.5 ,and CR =4,3) , and when (k =10 ,y =0.5 ,and

CR =4). The least amount of degradation in SNR at BER( ) when k=20, y=0.5

and CR = 2 and is equal to (-0.3872 dB). The largest amount of degradation is

when k =20, y = 1 and CR =4 is equal to (-8.4594 dB).

The CCDF of PAPR was degraded at (k =15, 20, y =1,0.8,0.5 and CR = 4, 3), (k=

20, y =1,0.8 and CR=2), (k =10 ,y =1,0.8 ,and CR =4,3) , and when (k =10 ,y =0.5

,and CR =4). The least amount of degradation in SNR at BER( ) when k=15,

y=0.8 and CR = 2 and is equal to (-0.2065 dB). The largest amount of degradation

is when k =20, y = 1 and CR =4 is equal to (-3.426 dB).

Except the points already mentioned, the PAPR was improved and the least

amount of improvement was when k =20, y =0.2 and CR = 4 and is equal to

(0.1722 dB), while the vast amount of improvement is where k =5, y =1 and CR =

1.5 and is equal to (10.4523 dB).

Except the points already mentioned, the CCDF of PAPR was improved and the

least amount of improvement in CCDF of PAPR when k =10, y =1 and CR = 2 and

is equal to (0.0055 dB), While the vast amount of improvement is where k =5, y =1



147

The SNR at BER( ) was degraded when CR = 1.5, when k =5, y =1 the

amount of degradation is equal to (-16.346 dB).

Except the points already mentioned, the SNR at BER( ) was improved and

The vast amount of improvement is where k =20, y =1 and CR = 4 and is equal to

(23.5747 dB), while the least amount of improvement in SNR at BER( ) when

k =5, y=0.2 and CR = 2 and is equal to (3.0185 dB).


an OFDM system with RFC method:

The PAPR was improved except when k =5, y =1 and CR =2 the SNR at

BER( ) degraded by (-0.1918 dB), the least amount of improvement was

when k=5, y =1 and CR = 1.5 and is equal to (0.1303 dB), While the vast amount

of improvement is where k =40, y =0.2 and CR = 4 and is equal to (12.0608 dB).

The CCDF of PAPR was improved , except when k =5,y =1 and CR =1.5,2,3,4 the

CCDF of PAPR was degraded. The least amount of improvement in CCDF of

PAPR when k =10, y= 1 and CR = 2 and is equal to (0.0255 dB), while the vast

amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to (5.0792

dB).

The SNR at BER( ) was degraded, except when k =10, y =1 and CR =3 the

SNR at BER( ) was improved by (0.018 dB). The least amount of degradation

in SNR at BER( ) when k =10, y =1 and CR = 4 and is equal to (-0.1085 dB).

The largest amount of degradation is when k =30, y =0.2 and CR =2 is equal to (-

16.7147 dB).

6.3.8 RFC +logR : The following conclusion from table A.28 when comparing the proposed method with


There are improved in PAPR, CCDF of PAPR and SNR at BER( ) in many

points that have been tested, but the best improvement in PAPR and CCDF of

PAPR is at k=90, y = .5 for CR =3 . The improvement in PAPR by = (20.6844 dB)

,CCDF of PAPR = (8.6603 dB), and the SNR at BER( ) by = ( 0.2112 dB).

For SNR at BER( )


=2 . The improvement in PAPR by = (20.9759 dB) , and CCDF of PAPR =


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k=50 ,y =0.5 and CR

=2 . The improvement in PAPR by = (21.6271 dB), and CCDF of PAPR =


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k =90 ,y= 0.5 and CR



For SNR at BER( )


=. The improvement in PAPR by = (23.3566 dB), and CCDF of PAPR = (9.8323


For SNR at BER( )


148

The best improvement in PAPR and CCDF of PAPR is at = and CR =. The




an OFDM system with logR companding PAPR reduction method:

The PAPR was improved and the least amount of improvement was when k =90 ,y

=1 and CR = 4 and is equal to (0.4691 dB), while the vast amount of improvement

is where k =10, y =1 and CR = 2 and is equal to (8.8239 dB).

The CCDF of PAPR was improved and the least amount of improvement was

when k =70 , y =1 and CR = 4 and is equal to (0.2321 dB), while the vast amount

of improvement is where k =10, y =1 and CR = 2 and is equal to (4.1088 dB).

The SNR at BER( ) was improved . The least amount of improvement in

SNR at BER( ) when k =90 ,y =0.2 and CR = 2 and is equal to (1.437 dB).

The largest amount of improvement is when k =90, y =1 and CR =4 is equal to

(22.5168 dB).


an OFDM system with RFC method:

The PAPR was improved and the least amount of improvement was when k=5 ,y

=1 and CR = 1.5 and is equal to (0.0422 dB), while the vast amount of


The CCDF of PAPR was improved, except when k =5 ,y =1 and CR =2 the CCDF

of PAPR was degraded by (-0.0018 dB). The least amount of improvement in

CCDF of PAPR when k =5, y= 1 and CR = 3 and is equal to (0.0727 dB), while the

vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to

(4.8674 dB).

The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5, 2. the

BER was improved by (8.7803 - 0.1415 dB). The least amount of degradation in

SNR at BER( ) when k =5, y =1 and CR = 4 and is equal to (-0.0429 dB). The

largest amount of degradation is when k =90, y =0.2 and CR =2 is equal to (-

14.496 dB).

6.4 Pre-coding + companding: The companding scheme can be implemented with low complexity, without any

iterative computations, comparing with coding, partial transmit and selective mapping

schemes, in which either delay due to coding or extra overheads to synchronize

transmitter and receiver are required. On the other hand, the pre-coding has also been

considered as a best among all these techniques, because it improves the PAPR

without increasing much complexity and destroying the orthogonality between

subcarriers. The pre-coding also improves the BER in comparison to the normal

OFDM system because of diversity gain obtained due to the spreading of the data

symbol on more than one subcarrier.

The OFDM system model with the proposed technique is as shown in figure 6.23.

WHT, DCT, DST, and DHT pre-coders are used as for the companding has been

using all kinds of previous companding. The results of the proposed method are good

and the best result for the PAPR is when (DHT + tanhR).


149

Figure 6.23 the OFDM system model with precoding + companding .

6.4.1 Pre-coding + A companding: The following conclusion from table A.30 and figure 6.24 when comparing the


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A =5 and DHT . The



For SNR at BER( )

The best improvement in PAPR is at A = 20 and DHT. The improvement in



The best improvement in CCDF of PAPR is at A = 15 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at A =120 and DHT. The



For SNR at BER( )

The best improvement in CCDF of PAPR is at A =120 and DST. The




proposed method with an OFDM system with A companding PAPR reduction

method:

The PAPR was improved except when (WHT and A =30, 87.6 ,100,120) the

PAPR was degraded and maximum degraded at A =30 by (-0.9647). The least

amount of improvement was when A = 40 and WHT and is equal to (0.0499 dB),

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On

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Eq

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An

d P

/S

S

/

P

P

/

S

S

/

P

IDF

T O

R IF

FT

DF

T O

R F

FT

O

/

P

I

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P P

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150

While the vast amount of improvement is where A = 5 and DHT and is equal to

(14.2226 dB).

The least amount of improvement in CCDF of PAPR when A = 30 and WHT and

is equal to (0.1178 dB), while the vast amount of improvement is where A = 5 and

DHT and is equal to (4.2193 dB).

The SNR at BER( ) was degraded at DST and WHT. The least amount of

degradation in The SNR at BER( ) when A =70 and WHT and is equal to (-

0.036 dB). The largest amount of degradation is when A =50 and WHT and is

equal to (-1.5167 dB).

The SNR at BER( ) was improved at DHT. The least amount of improvement

in The SNR at BER( ) when A =5 and is equal to (1.1517 dB). The largest

amount of improvement is when A= 120 is equal to (5.377 dB).

The SNR at BER( ) was improved at DCT and A= 30, 40, 70, 87.6, 100,120.

The least amount of improvement in The SNR at BER( ) when A =40 and is

equal to (0.0804 dB). The largest amount of improvement is when A= 70 is equal

to (0.4065 dB).

The SNR at BER( ) was degraded at DCT and A= 5, 10, 15, 20, 35, 50. The

least amount of degradation in the SNR at BER( ) when A =15 and DHT and

is equal to (-0.0769 dB). The largest amount of degradation is when A= 5 and

WHT is equal to (-0.5053 dB).


comparing the proposed method with an OFDM system with pre-coding method:


improvement was when A =5 and DHT.The PAPR improvement is equal to

(2.2752 dB) and the CCDF of PAPR improvement is equal to ( 0.9293 dB), while

the vast amount of improvement is where A = 120 and WHT and the PAPR



The SNR at BER( ) was degraded. The least amount of degradation in the

SNR at BER( ) when A =5 and DHT and is equal to (-0.8663 dB). The largest

amount of degradation is when A= 90 and WHT is equal to (-10.9249 dB).


151

Figure 6.24.a

Figure 6.24.b


precodings, companding , and Hybird (precodings + ). b) Shows the values of the

CCDF of PAPR and SNR at BER = for each of the precodings, companding ,

and Hybird (precodings + ).

10 12 14 16 18 20 22 240

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

A

WHT + A

DCT+ A

DST + A

DHT + A

10 12 14 16 18 20 22 241

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

WHT

DCT

DST

DHT

A

WHT + A

DCT+ A

DST + A

DHT + A


152

6.4.2 Pre-coding + : The following conclusion from table A.31 and figure 6.25 when comparing the


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at MU =10 and DHT.



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at MU =50 and DHT. The



For SNR at BER( )

The best improvement in PAPR is at MU =320 and DHT. The improvement in



The best improvement in CCDF of PAPR is at MU =320 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at MU = 1000 and DHT.



For SNR at BER( )

The best improvement in CCDF of PAPR is at MU = 1000 and DCT. The




comparing the proposed method with an OFDM system with A companding PAPR

reduction method:

The PAPR was improved except when (WHT and MU =60, 160,180) the PAPR

was degraded and the maximum degraded at MU=160 by (-0.7606 dB). The least

amount of improvement was when MU = 220 and WHT and is equal to (0.0305

dB), while the vast amount of improvement is where MU = 20 and DHT and is


The least amount of improvement in CCDF of PAPR when MU = 160 and WHT

and is equal to (0.0919 dB), while the vast amount of improvement is where MU =

5 and DHT and is equal to (3.8402 dB).

The SNR at BER( ) was degraded at DST and WHT except at (MU =700 and

WHT the SNR at BER( ) was improved by (0.1826 dB) . The least amount of

degradation in The SNR at BER( ) when MU =100 and WHT and is equal to

(-0.071dB). The largest amount of degradation is when MU =30 and DST and is

equal to (-0.8137 dB).


in The SNR at BER( ) when MU =5 and is equal to (1.0094 dB). The largest

amount of improvement is when MU= 700 is equal to (6.2032 dB).

The SNR at BER( ) at DCT there is an improvement in some of the points

and the degradation the other .The largest amount of improvement is when MU

=120 is equal to (0.4501 dB).The largest amount of degradation is when MU =10


153

is equal to (-0.3151 dB). Is clearly the amount of improvement and degradation

less than 0.5 in all cases.

Figure 6.25.a

Figure 6.25.b


precodings, companding , and Hybird (precodings + ). b) Shows the values of the

CCDF of PAPR and SNR at BER = for each of the precodings, companding ,

and Hybird (precodings + ).

10 12 14 16 18 20 22 240

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

MU

WHT + MU

DCT+ MU

DST + A

DHT + MU

10 12 14 16 18 20 22 241

2

3

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

WHT

DCT

DST

DHT

MU

WHT + MU

DCT+ MU

DST + MU

DHT + MU


154


proposed method with an OFDM system with pre-coding method:


improvement was when MU =5 and DHT .The PAPR improvement is equal to

(1.9728dB) and the CCDF of PAPR improvement is equal to (0.7742 dB), While

the vast amount of improvement is where MU = 1000 and WHT and the PAPR


equal to (8.0035 dB ).


BER( ) when MU =5 and DHT and is equal to (-0.7449 dB). The largest

amount of degradation is when MU =1000 and WHT is equal to (-12.5842 dB).

6.4.3 Pre-coding + RCT: The following conclusion from table A.32 and figure 6.26 when


method:

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R =0.6 and DHT . The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R =0.4 and DHT . The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R =0.3 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R= 0.2 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R = 0.1 and DHT. The



The best improvement in PAPR and CCDF of PAPR is at R = 0.2 and DST. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at R =0.1 and DST. The




155

Figure 6.26.a

Figure 6.26.b

Figure 6.26 (a) Shows the values of the PAPR and SNR at BER = for each of

the precodings, RCT, and Hybird (precodings +RCT). b) Shows the values of the

CCDF of PAPR and SNR at BER = for each of the precodings, RCT, and Hybird

(precodings +RCT ).

10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

Rooting

WHT + Rooting

DCT+ Rooting

DST + Rooting

DHT + Rooting

10 15 20 25 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

WHT

DCT

DST

DHT

Rooting

WHT + Rooting

DCT+ Rooting

DST + Rooting

DHT + Rooting


156

The following conclusion from table A.32 and figure 6.26 when comparing

the proposed method with an OFDM system with RCT PAPR reduction method:

The PAPR and the CCDF of PAPR were improved .The least amount of

improvement was when R =0.1 and WHT.The PAPR improvement is equal to

(0.218 dB) and the CCDF of PAPR improvement is equal to (0.0754 dB), while

the vast amount of improvement is where R =0.9 and DHT and the PAPR




in The SNR at BER( ) when R = 0.8 and is equal to (0.1258 dB). The largest

amount of improvement is when R=0.1 is equal to (7.1402 dB).

The SNR at BER( ) at DCT, DST and WHT there is an improvement in some

of the points and the degradation the other .The largest amount of improvement is

when R = 0.2 and DCT is equal to (0.4058 dB).The largest amount of degradation

is when R=0.4 and WHT is equal to (-0.4257 dB). Is clearly the amount of

improvement and degradation less than 0.5 in all cases.




improvement was when R =0.9 and DHT.The PAPR improvement is equal to

(0.356 dB) and the CCDF of PAPR improvement is equal to (0.1052 dB), While

the vast amount of improvement is where R =0.1 and WHT and the PAPR



The SNR at BER( ) was degraded, except when R=0.9 and DHT the SNR at

BER( ) was improved by (0.0718 dB).The least amount of degradation in

SNR at BER( ) when R=0.9 and DCT and is equal to (-0.0147 dB). The

largest amount of degradation is when R =0.1 and WHT is equal to (-17.0412 dB).

6.4.4 Pre-coding + AEXP: The following conclusion from table A.33 and figure 6.27 when comparing the


For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d =0.8 and DHT . The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d =0.4 and DHT. The


while the SNR at BER( ) deteriorated by (-3.1563 dB).

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at d =0.3 and DHT. The


while the SNR at BER( ) deteriorated by (-4.2985dB).

For SNR at BER( )


157

The best improvement in PAPR and CCDF of PAPR is at d =0.2 and DHT . The






The PAPR was improved, except when (DHT and d =0.1, 0.2, 0.3, 0.4) and (WHT

and d =0.1) PAPR was degraded and the maximum degradation is(-2.1186 dB).the

least amount of improvement was when d = 0.2 and WHT and is equal to (0.0148

dB), while the vast amount of improvement is where d = 1.9 and DHT and is


The CCDF of PAPR was improved, except when (DHT and d =0.1, 0.2, 0.3, 0.4,

0.5) and (WHT and d =0.1, 0.2, 0.3) PAPR were degraded and the maximum

degradation is (-1.5086 dB). The least amount of improvement in CCDF of PAPR

when d =0.7 and WHT and is equal to (0.017 dB), while the vast amount of

improvement is where d =0.9 and DHT and is equal to (2.085 dB).


in The SNR at BER( ) when d =2 and is equal to (1.5749 dB). The largest

amount of improvement is when d =0.3 is equal to (14.2701 dB).



when d = 0.7 and WHT is equal to (11.8003 dB).The largest amount of

degradation is when d = 1 and DST is equal to (-14.3dB).



The PAPR was improved, except when DHT and d =2, 1.9, 1.8, 1.7 PAPR was

degraded and the maximum degradation is(-0.6937 dB).the least amount of

improvement was when d = 1.6 and DHT and is equal to (0.0648 dB), while the

vast amount of improvement is where d = 0.1 and WHT and is equal to (21.8494

dB).

The CCDF of PAPR was improved. The least amount of improvement in CCDF

of PAPR when d =2 and DHT and is equal to (0.1988 dB), while the vast amount

of improvement is where d =0.1 and WHT and is equal to (9.3313 dB).


BER( ) when d=1.1 and DHT and is equal to (-0.3896 dB). The largest

amount of degradation is when d =0.2 and DHT is equal to (-18.418 dB).


158

Figure 6.27.a

figure 6.27.b


precodings, AEXP companding , and Hybird (precodings +AEXP ). b) Shows the

values of the CCDF of PAPR and SNR at BER = for each of the

precodings,AEXP companding , and Hybird (precodings +AEXP ).

10 15 20 25 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

AEXP

WHT + AEXP

DCT+ AEXP

DST +AEXP

DHT +AEXP

10 15 20 25 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of

PA

PR

original

WHT

DCT

DST

DHT

AEXP

WHT + AEXP

DCT+ AEXP

DST +AEXP

DHT +AEXP


159

6.4.5 Pre-coding + cos : The following conclusion from table A.34 and figure 6.28 when


method:

For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y=0.6 and DHT . The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y = 0.4 and DHT .



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y = 0.3 and DHT. The



For SNR at BER( )

The best improvement in CCDF of PAPR is at y = 0.3 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at y = 0.2 and DST. The




comparing the proposed method with an OFDM system with cos companding PAPR

reduction method:

The PAPR was improved .The least amount of improvement was when y = 0.3

and WHT and is equal to (0.0641 dB), while the vast amount of improvement is

where y = 1 and DHT and is equal to (7.7327 dB).

The CCDF of PAPR was improved except at y =1 and WHT the CCDF of PAPR

was degraded by (-0.0179 dB). The least amount of improvement in CCDF of

PAPR when y =0.4 and WHT is equal to (0.0715 dB), while the vast amount of

improvement is where y =0.1 and DHT and is equal to (3.7092 dB).


in The SNR at BER( ) when y = 1 and is equal to (0.0513 dB). The largest

amount of improvement is when y =0.3 is equal to (5.7052 dB).



when y = 0.3 and DST is equal to (1.0307 dB).The largest amount of degradation

is when y =0.3 and DCT is equal to (-1.6335 dB).



The PAPR was improved, except when DHT and y =1, 9, the PAPR was degraded

and the maximum degradation is (-0.8692 dB). the least amount of improvement

was when y = 0.8 and DHT and is equal to (0.0275 dB), while the vast amount of

improvement is where y = 0.1 and WHT and is equal to (20.8281 dB).


160


of PAPR when y =1 and DHT is equal to (0.2092 dB), while the vast amount of

improvement is where y =0.1 and WHT and is equal to (8.9201 dB).


BER( ) when y =1 and DCT and is equal to (-0.0147 dB). The largest amount

of degradation is when y=0.2 and DHT is equal to (-18.418 dB).

Figure 6.28.a

Figure 6.28.b


precodings, cos companding , and Hybird (precodings +cos ). b) Shows the values of

the CCDF of PAPR and SNR at BER = for each of the precodings,cos

companding , and Hybird (precodings +cos ).

10 12 14 16 18 20 22 24 26 28 300

5

10

15

20

25

30

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

cos

WHT + cos

DCT+ cos

DST +cos

DHT +cos

10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

WHT

DCT

DST

DHT

cos

WHT + cos

DCT+ cos

DST +cos

DHT +cos


161

6.4.6 Pre-coding + tanhR : The following conclusion from table A.36 when comparing the proposed method with


For SNR at BER( )

The best improvement in PAPR is at k=15, y=.8 and DHT. The improvement in



The best improvement in CCDF of PAPR is at k=20, y=1 and DHT. The



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k= 5, y =0.2 and DCT.



For SNR at BER( )

The best improvement in PAPR and CCDF of PAPR is at k = 30, y=.2 and DCT.




an OFDM system with a tanhR companding method:

The PAPR, there is an improvement in some of the points and the degradation the

other .The largest amount of improvement is when k =5, y = 1 and DST is equal to

(15.7833 dB).The largest amount of degradation is when k =20, y =1 and WHT is

equal to (-19.3807dB).

The CCDF of PAPR, there is an improvement in some of the points and the

degradation the other .The largest amount of improvement is when k =5, y = 1 and

DST is equal to (6.7541dB).The largest amount of degradation is when k =20, y

=1 and WHT is equal to (-7.703 dB).

The SNR at BER( ) at DHT, WHT, and DCT, there is an improvement in

some of the points and the degradation the other .The largest amount of

improvement is when k=15, y = 1 and WHT is equal to (17.8776 dB).The largest

amount of degradation is when k=5, y =0.3 and DHT is equal to (-0.856 dB).

The SNR at BER( ) at DST was degraded more than 30 dB in all cases

The following conclusion from table A.36 when comparing the

proposed method with an OFDM system with pre-coding method:

The PAPR was improved, except when WHT and y =1, k =5 the PAPR was

degraded by (-0.1807 dB).the least amount of improvement was when k=10, y = 1


where k =40, y = 0.1 and WHT and is equal to (19.3685 dB).


of PAPR when k=5, y =1 and WHT is equal to (0.2938 dB), while the vast amount

of improvement is where k =40, y =0.2 and WHT and is equal to (8.3837 dB).


BER( ) when k=30, y =1 and WHT and is equal to (-0.1416 dB). The largest

amount of degradation is when k=30, y =0.2 and DCT is equal to (-17.5814 dB).


162

6.4.7 Pre-coding + logR : The following conclusion from table A.37 when comparing the proposed method with


For SNR at BER( )

The best improvement in PAPR is at k=90, y =0.5 and DHT. The improvement

in PAPR by = (21.9089 dB) , and CCDF of PAPR = (8.6957 dB), while the SNR

at BER( ) deteriorated by = (-1.4569 dB).

The best improvement in CCDF of PAPR is at k=70, y =0.5 and DHT. The



For SNR at BER( )

The best improvement in PAPR is at k=90, y=0.2 and DHT. The improvement

in PAPR by = (22.2456 dB), and CCDF of PAPR = (8.8018 dB), while the SNR at


The best improvement in CCDF of PAPR is at k=20, y=0.2 and DHT. The



For SNR at BER( )

The best improvement in PAPR is at k=90 ,y =0.2and DCT. The improvement

in PAPR by = (22.8075 dB), and CCDF of PAPR = (9.5916 dB), while the SNR

at BER( ) deteriorated by = (-12.8773 dB).

The best improvement in CCDF of PAPR is at k=10 ,y =0.2 and DST. The



For SNR at BER( )

The best improvement in PAPR is at k=90, y =0.2 and DST. The improvement in



The best improvement in CCDF of PAPR is at k=70 , y =0.2 and DST. The



The following conclusion from table A.37 when comparing the

proposed method with an OFDM system with a logR companding method:

The PAPR, there is an improvement in some of the points and the degradation the

other .The largest amount of improvement is when k =10, y = 1 and DHT is equal

to (12.2879 dB).The largest amount of degradation is when k =70, y =1 and WHT

is equal to (-7.3198 dB).

The CCDF of PAPR, there is an improvement in some of the points and the

degradation the other .The largest amount of improvement is when k =10, y = 1

and DHT is equal to (4.9342 dB).The largest amount of degradation is when k

=90, y =1 and WHT is equal to (-2.7575 dB).

The SNR at BER( ) at DHT, WHT, and DCT, there is an improvement in

some of the points and the degradation the other .The largest amount of

improvement is when k=90 , y = 1 and WHT is equal to (17.7596 dB).The largest

amount of degradation is when k=70, y =0.2 and DHT is equal to (-0.7874 dB).

The SNR at BER( ) at DST was degraded, except when k =90 ,y =1 the SNR

at BER( ) maintains its value .The least amount of degradation in SNR at


163

BER( ) when k =50, y =1 and is equal to (-1.012 dB). The largest amount of

degradation is when k=40, y=0.2 is equal to (-14.3135dB).


an OFDM system with pre-coding method:

The PAPR was improved, the least amount of improvement was when k=5, y = 1


where k =40, y = 0.2 and WHT and is equal to (18.9242 dB).


of PAPR when k=5, y =1 and WHT is equal to (0.5366 dB), while the vast amount

of improvement is where k =90, y =0.2 and WHT and is equal to (8.2281 dB).

The SNR at BER( ) was degraded, except at WHT, y =1 and k=5, 10, 40 the

SNR at BER( ) was improved and the maximum improvement is (0.0852

dB). The least amount of degradation in SNR at BER( ) when k=50, y =1 and

WHT and is equal to (-0.1544 dB). The largest amount of degradation is when

k=50, y =0.8 and DST is equal to (-17.2375 dB).

6.4.8 Pre-coding + NERF: This method did not work with DST and DHT, whereas the BER gives us an error.



method:

When using DCT and WHT with ERF, there are improved in PAPR, CCDF of PAPR.

The best improvement in PAPR and CCDF of PAPR is at DCT. The improvement in

PAPR by = (16.5312 dB),CCDF of PAPR = (6.8931 dB), while the SNR at

BER( ) degraded by = (-1.9789 dB).

Figure 6.29.a

11.5 12 12.5 13 13.5 146

8

10

12

14

16

18

20

22

24

26

SNR at (BER =10-4)

PA

PR

original

WHT

DCT

DST

DHT

NERF

WHT + NERF

DCT+NERF


164

Figure 6.29.b


precodings, cos companding , and Hybird (precodings +cos ). b) Shows the values of

the CCDF of PAPR and SNR at BER = for each of the precodings,cos

companding , and Hybird (precodings +cos ).

11.5 12 12.5 13 13.5 143

4

5

6

7

8

9

10

11

SNR at (BER =10-4)

CC

DF

of P

AP

R

original

WHT

DCT

DST

DHT

NERF

WHT + NERF

DCT+ NERF

Chapter seven Conclusions and future work

A. 165

Chapter seven

Conclusions and future work

7.1Conclusions: 1. The RFC and RCF can improve the PAPR and BER at the same time.

2. RFC is better than RCF in performance especially when I ≥ 2 while maintaining

the complexity and price of RCF.

3. The performance of all kinds of proposed companding is better than the

performance μ-law and A-law compandings

4. AEXP can be considered the best types of companding that we used in terms of

BER and performance, followed by tanhR.

5. TanhR has better results when used in the hybrid technique and also the y, k

parameters give it a kind of flexibility

6. TanhR and logR and NERF is better than tanh, log, and erf that Mohit was used in

his paper, because the performance of the proposed techniques better than μ-law

and A-law compandings and also the μ-law and A-law compandings better than

Mohit methods.

7. The performance of logR companding asymptotic to tanhR but the tanhR have

better results in most cases.

8. The performance of cos companding asymptotic to AEXP but the AEXP have

better results in most cases.

9. The best type precoding in term of reduced PAPR and BER is the DFT

10. DST and DCT precodings give almost the same performance, the DST improves

the PAPR more than DCT even a few percent.

11. The worst type of precoding in term of reducing the PAPR and BER is the WHT.

12. As it is clear from the results that the hybrid methods have better results but at the

expense of complexity.

13. The results of hybrid pre-coding with RCF is better than the results of the RCF

and pre-coding each alone, except in the case of DHT with RCF (I = 2, pilot)

where the results of the DHT is better

14. For the hybrid pre-coding with RCF, the PAPR value is better when RCF (I = 1)

because in this case the effect of the filter on the PAPR cancels.

15. The hybrid RCF with companding shows good results better than the results of the

RCF and pre-coding each alone, because of RCF reduces the PAPR and improves

the BER constant and then companding more reduces the amount of the PAPR.

16. The hybrid RCF with companding can improve the PAPR and BER at the same

time with amount greater than the RCF and the best one improvement in PAPR is

at (RCF + AEXP).

17. The hybrid RFC with companding shows good results better than the results of

the hybrid RCF with companding, because as we demonstrated earlier RFC batter

than RCF.

18. The hybrid RFC with companding can also improve the PAPR and BER at the

same time, and the best one improvement in PAPR is at (RFC + AEXP).

19. The results of the hybrid precoding with companding are provides good results

and the best result for the PAPR is when (DHT with tanhR) except at (DST with

tanhR, DST with NERF, DHT with NERF). At DST with tanhR the BER

performance significantly degraded

20. The best results are obtained at these techniques:

RFC:


A. 166

The best one improvement in PAPR and CCDF of PAPR is at I =4 and CR =1.75. The

improvement in PAPR by = (18.2789 dB), CCDF of PAPR = (8.0187 dB), and the

SNR at BER ( ) by = (0.6101 dB).

AEXP companding:

For SNR at BER( )


improvement in PAPR by = (18.8515 dB), and CCDF of PAPR = (7.6480 dB), while

the SNR at BER ( ) deteriorated by = (-4.8686 dB).

LogR companding:

For SNR at BER( )




TanhR companding:

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k = 5, y=0.2. The



LogR companding:

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at . The improvement in

PAPR by = (23.5788 dB), and CCDF of PAPR = (9.9600 dB), while the SNR at BER

( ) deteriorated by = (-18.1686 dB).

Hybird

RFC + AEXP:

The best one improvement in PAPR and CCDF of PAPR is at d = 0.6 and CR =4. The

improvement in PAPR by = (21.0509dB), CCDF of PAPR = (8.7178 dB), and the

SNR at BER ( ) by = (0.0116 dB).

Pre-coding + tanhR :

For SNR at BER( )

The best one improvement in PAPR is at k=15, y=.8 and DHT. The improvement in

PAPR by = (22.7711 dB), and CCDF of PAPR = (8.9691 dB), while the SNR at BER

( ) deteriorated by = (-1.1828 dB).

RFC + tanhR

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at k=40, y=0.2 and CR


dB), while the SNR at BER ( ) deteriorated by = (-8.0074 dB).

RFC + tanhR

For SNR at BER( )


A. 167

The best one improvement in PAPR and CCDF of PAPR is k=40, y=0.2 and CR =2.


while the SNR at BER ( ) deteriorated by = (-13.0440 dB).

RFC + RCT

For SNR at BER( )

The best one improvement in PAPR and CCDF of PAPR is at R = 0.1and CR =2. The



7.2 Future work: 1. Use another type of filter that does not effect on the PAPR or have little

impact with clipping.

2. Find a new type of companding to recduce the PAPR with maintaining the

BER performance.

3. The proposed companding PAPR reduction methods can be combined with

different PAPR reduction techniques such as PTS, SLM, TR and etc.

4. The proposed RFC can be combined with different PAPR reduction

techniques such as coding, interleaving, TI and DSI etc.

5. The RCF, proposed RFC can be combined with different existing companding

techniques such as airy companding, linear companding, Trapezoidal power

companding and etc.

6. The proposed companding PAPR reduction methods can be combined with

Zadoff-Chu matrix Transform precoding.

7. Analysis of the proposed techniques and find out its impact on the PAPR

mathematically.

8. proposed new hybrid techniques by using the proposed method

9. Study the impact of these proposed techniques on bandwidth, noise , distortion

and the ratio of power saving.

10. Study the impact of these proposed techniques on statistical distribution.

11. The proposed PAPR reduction methods can be used with MIMO OFDM

system.

12. The proposed PAPR reduction methods can be used with other multicarrier

system

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Appendix A: Tables of Results

A. 1

Appendix A

Tables of Results

A.1 RCF Results

Table A.1 RCF Results

Oversampling(I) CR PAPR CCDF of

PAPR

SNR

(BER= )

1 4 14.0866 6.0244 11.82

3 11.5702 4.785 12.3711

2 8.2486 3.1186 14.3756

1.75 7.3802 2.594 16.72

1.5 6.3838 2.9 29.5

pilot 4 14.1973 6.248 9.36

3 11.9169 5.225 9.8415

2 9.4432 3.8796 13

1.75 8.5889 3.56 16.063

1.5 8.0108 3.3568 29.6( )

1.125 4 14.1010 6.0337 11.3068

3 11.6977 4.8436 11.65

2 8.4887 3.2413 14.4

1.75 7.4828 2.7737 16.875

1.5 6.6241 2.3785 29.6( )

1.25 4 14.1392 6.06 10.572

3 11.7044 4.8772 11.4

2 8.5374 3.43 13.72

1.75 7.7741 3.023 16

1.5 6.8767 2.618 29.6( )

1.5 4 14.1728 6.145 10

3 11.8777 5 10.4765

2 8.9992 3.6 13.7445

1.75 8.0760 3.315 16.4373

1.5 7.8858 2.976 29.6( )

2 4 14.4932 6.4177 8.7679

3 12.2742 5.355 9.414

2 9.6073 4.1216 12.128

1.75 9.0189 3.785 15

1.5 8.3829 3.6257 29.6( )

3 4 15.3850 6.7374 7.1838

3 13.2843 5.79 7.665

2 10.6525 4.555 10.418

1.75 9.8847 4.2435 13.1725

1.5 9.4795 3.9 29.6( )

4 4 15.5738 6.8674 5.8315

3 13.4298 5.8 6.0725

2 10.7538 4.4432 9.6955

1.75 10.0109 4.0712 13

1.5 9.2193 3.725 29.6( )


A. 2

A.2 RFC Results

If (A, B, and C) positive values that's mean there is an improvement,

while if the negative values this mean there is a deterioration in values

The (PAPR, CCDF OF PAPR and BER) were calculated with different

value of (CR (4, 3, 2, 1.75, 1.5) and I (1, pilot, 1.125, 1.25, 1.5, 2, 3, 4)

These values have been placed on the table. A, B, and C also added to the

table for comparison with the RCF

Where A = PAPR (RCF) – PAPR (RFC)

B =CCDF of PAPR (RCF) – CCDF of PAPR (RFC)

C = SNR at BER ( ) (RCF) – SNR at BER ( ) (RFC)

Table A.2 RFC Results

I CR A PAPR B CCDF of

PAPR

C SNR

(BER= )

Pilot 4 0.0232 14.1741 -0.3570 6.605 -0.2400 9.6

3 .0906 11.8263 0.3490 4.876 -0.1585 10

2 .6703 8.7729 0.5706 3.309 0.5000 12.5

1.75 .334 8.2549 0.7329 2.8271 1.0630 15

1.5 1.1464 6.8644 0.9448 2.412 0 29.6( )

1.125 4 0.0042 14.0968 0.0069 6.0268 0.0708 11.236

3 0.0159 11.6818 0.0436 4.8 -0.1700 11.82

2 -0.0866 8.5753 0.0513 3.19 0.7455 13.6545

1.75 0.1119 7.3709 0.0837 2.69 -0.2390 17.114

1.5 0.0771 6.5470 0.2372 2.1413 0.9000 28.7

1.25 4 0.0215 14.1177 0.0279 6.0321 -0.5700 11.142

3 0.0583 11.6461 0.0732 4.804 0.1900 11.21

2 0.2114 8.3260 0.1750 3.255 -0.0375 13.7575

1.75 0.2031 7.5710 0.3118 2.7112 0.3286 15.6714

1.5 0.2754 6.6013 0.3556 2.2624 0 29.6( )

1.5 4 -0.0015 14.1743 0.1018 6.0432 -0.1236 10.1236

3 0.0533 11.8244 0.1655 4.8345 0.0510 10.4255

2 0.4553 8.5439 0.3289 3.2711 0.5733 13.1712

1.75 0.6558 7.4202 0.2343 3.0807 0.1286 16.3087

1.5 1.3184 6.5674 0.5919 2.3841 0 29.6( )

2 4 0.3083 14.1849 0.3577 6.06 0.1089 8.659

3 0.5039 11.7703 0.5010 4.854 0.0440 9.37

2 1.0271 8.5802 0.7472 3.3744 -0.1548 12.2828

1.75 1.4327 7.5862 0.9370 2.848 0.3875 14.6125

1.5 1.7478 6.6351 1.1757 2.45 0 29.6(

3 4 1.1714 14.2136 0.6830 6.0544 0.4163 6.7675

3 1.5283 11.7560 0.9373 4.8527 0.2842 7.3808


A. 3

2 2.1453 8.5072 1.2550 3.3 0.4380 9.98

1.75 2.2903 7.5944 1.4123 2.8312 1.4725 11.7

1.5 2.9062 6.5733 1.5600 2.34 0 29.6( )

4 4 1.3900 14.1838 0.8024 6.065 0.1602 5.6713

3 1.6108 11.8190 0.9360 4.864 0.0340 6.0385

2 2.2738 8.4800 1.1382 3.305 1.2685 8.427

1.75 2.6883 7.3226 1.2499 2.8213 2.1787 10.8213

1.5 2.8284 6.3909 1.3092 2.4158 1.3800 28.22

Table A.3 Precoding Results

Precoding PAPR CCDF of PAPR SNR (BER= )

25.6318 10.773 11.64

WHT 22.8377 9.9046 11.63

DCT 18.1110 7.664 11.628

DST 17.4649 7.523 11.628

DHT 6.9587 3.35 11.582

DFT 0.0200 0 11.469

Table A.4 Companding Results

PAPR CCDF of PAPR SNR (BER= )

5 18.9061 6.64 13.6

10 14.6917 4.73 16.12

15 14.3265 4.126 17.576

20 14.4401 3.685 18

30 11.8545 3.32 19.2

35 12.9100 3.413 19.2

40 12.6028 3.26 19.757

50 12.5906 3.1 20.116

70 12.1503 2.837 21.13

80 11.8738 2.7542 21.2592

87.6 11.7819 2.723 21.372

90 11.7517 2.739 21.1547

100 11.3543 2.58 21.62

120 11.4101 2.535 21.863

Table A.5 Companding Results

PAPR CCDF of

PAPR SNR (BER= )

5 17.4332 6.416 13.3363

10 16.5470 5.77 14.64

20 17.2573 5.078 16.25

30 14.6356 4.4 17.165

40 13.8453 4 17.75

50 13.8622 3.8 18.27

60 13.7327 3.66 18.777

70 14.2237 3.646 19


A. 4

80 16.5501 3.33 19.474

90 12.8584 3.3 19.6

100 12.4142 3.12 20

120 12.1142 3 20.28

140 12.7379 3.06 20.5

160 12.2083 2.866 20.475

180 12.5858 2.88 21.05

200 12.2423 2.76 21

220 11.2722 2.645 21.2385

240 15.9144 2.85 21.6

250 12.4358 2.723 21.6

255 11.7434 2.68 21.468

260 11.9863 2.666 21.7

280 11.5891 2.6 21.9

300 11.9073 2.61 22

320 12.3703 2.7 22.125

500 11.1951 2.3 22.92

700 10.8218 2.17 23.5

1000 12.8953 2.28 23.764

Table A.6 RCT Results

R PAPR CCDF of PAPR SNR (BER= )

.9 21.8631 9.55 11.6765

.8 21.1311 8.6815 11.987

.7 18.1291 8.058 12.4137

.6 15.3547 6.6825 13.4

.5 13.9264 5.835 14.45

.4 11.5292 4.8215 16.145

.3 8.5529 3.71 18.525

.2 5.9888 2.5745 22.25

.1 2.8726 1.268 28.3

Table A.7 AEXP Companding Results

AEXP d PAPR CCDF of PAPR SNR (BER= )

2 13.0811 5.1533 14.73

1.9 13.0240 5.14 14.7

1.8 12.1983 4.9185 14.858

1.7 11.2173 4.77 14.45

1.6 10.6962 4.585 14.2

1.5 10.0664 4.358 14.3138

1.4 9.6145 4.1465 14.5685

1.3 8.9815 3.98 14.3

1.2 8.4500 3.806 15.33

1.1 7.9523 3.5995 14.85

1 7.3774 3.374 15.3

.9 6.7500 3.192 16.3

.8 6.0806 2.9264 24.833

.7 5.5253 2.637 30 ( )


A. 5

.6 4.7892 2.34 30 ( )

.5 4.1344 2.1075 30 ( )

.4 3.4039 1.74 30 ( )

.3 2.6518 1.3272 30 ( )

.2 1.8358 .9425 30 ( )

.1 0.9690 .5116 30 ( )

Table A.8 Cos Companding Results

y PAPR CCDF of PAPR SNR (BER= )

2 26.3137 11.355 15.42

1.9 25.5112 10.858 15

1.8 24.6392 10.47 14.255

1.7 23.3428 10.129 13.75

1.6 22.8762 9.682 13.675

1.5 21.4160 9.256 12.765

1.4 20.1972 8.875 12.25

1.3 19.4063 8.5 11.95

1.2 18.1770 7.919 12

1.1 16.8064 7.48 11.832

1 15.6468 6.9508 11.7031

.9 14.3909 6.3817 12.1428

.8 13.1204 5.796 12.6966

.7 11.7106 5.2196 13.6465

.6 10.2401 4.6249 14.2953

.5 8.7575 3.9743 15.7648

.4 7.2067 3.3453 17.7538

.3 5.5700 2.59 29.2836

.2 3.8470 1.8074 >30

.1 1.9930 .9208 >>30

Table A.9 tanhR Companding Results

k

y PAPR CCDF of

PAPR SNR (BER= )

5 1 16.7129 7.4076 12.1294

5 .8 13.8472 6.0581 12.6712

5 .5 8.9043 3.8861 15.5877

5 .2 3.5446 1.5275 24.7231

10 1 8.9016 4.1605 14.504

10 .9 8.5570 3.9969 14.8376

10 .8 8.2624 3.915 15.1789

10 .7 7.7099 3.5129 15.8577

10 .6 7.0350 3.2427 16.8


A. 6

10 .5 6.5160 2.9176 17.8871

10 .4 5.4230 2.4367 20.2622

10 .3 4.4318 1.9475 22.3763

10 .2 3.0296 1.3133 25.7854

10 .1 1.6412 .6988 >30

15 1 5.2314 2.5987 29.6

15 .8 5.2429 2.5265 22.4656

15 .5 4.9419 2.2321 21.4747

15 .2 2.7437 1.2027 26.5576

20 1 3.3781 1.8228 30( )

20 .8 3.5950 1.8412 30

20 .5 3.7761 1.7895 30

20 .2 2.5710 1.1315 28.9392

Table A.10 tanhR Companding Results at y =1

k y PAPR CCDF of

PAPR SNR at BER

5 1 16.4627 7.2165 11.9245

10 1 8.9312 4.209 14.6486

15 1 5.2314 2.5987 30

20 1 3.3781 1.8228 30( )

Table A.11 tanhR Companding Results at y =0.8

k y PAPR CCDF of

PAPR SNR (BER= )

5 .8 13.4361 5.8816 12.817

10 .8 8.1480 3.7579 14.9321

15 .8 5.2429 2.5265 22.4656

20 .8 3.5950 1.8412

Table A.12 logR Companding Results

y k PAPR CCDF

OF

PAPR

SNR

(BER= )

k PAPR CCDF

OF

PAPR

SNR

(BER= )

1 1 23.9381 10.14 11.65 1 23.9381 10.14 11.65

1 5 19.4187 8.54 12.07 5 19.4187 8.54 12.07

1 10 16.7420 7.3145 12.5 20 14.4171 6.291 13.616

.9 10 15.7785 6.783 12.712 20 13.5847 6.078 13.913

.8 10 14.7339 6.3775 13.2 20 12.8391 5.53 13.838

.7 10 12.3933 5.526 13.8265 20 11.3170 5 14.54

.6 10 11.1271 4.87 14.7521 20 10.1279 4.4434 15.325

.5 10 9.7424 4.24 15.75 20 8.9142 3.928 16.4332

.4 10 8.1924 3.532 17.27 20 7.4148 3.317 18.082

.3 10 5.9023 2.625 20 20 5.8346 2.526 20.237

.2 10 4.0933 1.8 23.65 20 4.1963 1.892 23.9


A. 7

.1 10 2.1028 .932 29.6 20 2.0352 .892 29.4665

1 30 13.7103 5.76 14.7383 40 12.4590 5.417 15.6865

.9 30 12.5344 5.465 14.71 40 11.7370 5.2 15.9

.8 30 11.5140 5.02 14.7 40 10.8267 4.7 16

.7 30 10.4388 4.538 15.37 40 10.1572 4.5 15.68

.6 30 9.6178 4.18 16 40 9.0205 4 16.4

.5 30 8.4340 3.6 17.0828 40 8.3535 3.62 17.612

.4 30 7.2406 3.15 18 40 7.3063 3.05 18.95

.3 30 5.7490 2.5 20.627 40 5.4107 2.4 20.778

.2 30 3.7918 1.686 23.78 40 3.8688 1.6785 23.8868

.1 30 2.2351 .882 30 40 2.1230 .885 30

1 50 11.6535 5.08 16.58 60 11.7483 4.6 18.2856

.9 50 11.1455 4.803 16.82 60 10.3340 4.515 17.52

.8 50 10.2693 4.492 16.26 60 10.2586 4.483 17.08

.7 50 9.9933 4.2685 16.478 60 9.5086 4.0443 17.36

.6 50 9.0562 3.94 16.93 60 8.4069 3.685 17.085

.5 50 7.7808 3.412 17.424 60 7.8869 3.5 18

.4 50 6.5612 2.8765 18.893 60 6.5492 2.6864 19.03

.3 50 5.3972 2.306 21 60 5.2164 2.2685 21.056

.2 50 3.7525 1.668 24.4 60 4.0939 1.6474 23.888

.1 50 2.0379 .908 29.6 60 2.1340 .8863 29.6

1 70 10.3556 4.592 20 80 10.4009 4.6 24.085

.9 70 10.3901 4.5 18.3715 80 10.2191 4.2185 20

.8 70 9.7716 4.3 17.725 80 9.6145 4.288 19.1

.7 70 8.9933 3.935 17.65 80 8.7912 3.84 17.638

.6 70 8.8132 3.64 17.43 80 8.0294 3.56 18.18

.5 70 7.6750 3.366 18.128 80 7.5870 3.3025 17.3734

.4 70 6.3892 2.83 19.15 80 6.6794 3 19.4576

.3 70 5.5562 2.446 21.25 80 5.1697 2.28 21.188

.2 70 3.9753 1.705 23.88 80 3.6822 1.626 24.158

.1 70 2.2114 .9335 29.6 80 2.0227 .88 29.6

1 90 9.6991 4.42 30 100 10.0104 4.225 30

.9 90 9.6399 4.282 21.9 100 9.7016 4.238 24.5

.8 90 9.0726 3.95 20 100 9.2385 4.0666 20.46

.7 90 8.7749 3.707 18.85 100 8.2944 3.65 18.745

.6 90 8.2079 3.488 19.337 100 7.9701 3.504 19.13

.5 90 7.0926 3.124 18.9814 100 7.1942 3.2 19.45

.4 90 6.6478 2.889 19.16 100 6.2888 2.755 19.816

.3 90 6.0873 2.24 21.5175 100 5.0748 2.275 21.6

.2 90 3.6976 1.6 24.36 100 3.6944 1.63 24.4

.1 90 2.1337 .89 29.6 100 2.0774 .9 29.6


A. 8

A.5 Hybrid RCF with companding Results:

X = PAPR (Companding ) – PAPR (Companding +RCF)

Y =CCDF of PAPR (Companding) - CCDF of PAPR (AV+RCF)

Z= SNR (BER= ) (Companding) – SNR (BER= ) (Companding +RCF)

X1 == PAPR (RCF) – PAPR (Companding +RCF)

Y1 =CCDF of PAPR (RCF) - CCDF of PAPR (Companding +RCF)

Z1= SNR (BER= ) (RCF) – SNR (BER= ) (Companding +RCF)

Table A.13 (RCF+A) Results and compared with the results of each of (RCF) and (A companding)

A I CR X X1 PAPR Y Y1 CCDF of

PAPR

Z Z1 SNR

(BER= )

5 2 4 10.4351 6.0222 8.4710 2.7730 2.5507 3.867 2.5095 -2.3226 11.0905

10 2 4 7.9653 7.7668 6.7264 1.6650 3.3527 3.065 3.1882 -4.1639 12.9318

20 2 4 8.7683 8.9350 5.5582 1.6433 3.9350 2.4827 2.5760 -6.2321 15

30 2 4 9.3538 9.4069 5.0863 1.4150 4.1477 2.27 1.8420 -7.3901 16.158

40 2 4 7.0551 9.6938 4.7994 1.1950 4.2927 2.125 2.2000 -8.2321 17

50 4 8.0339 9.9243 4.5689 1.2070 4.3647 2.053 2.3762 -8.6129 17.3808

60 4 8.1579 10.0605 4.4327 1.1400 4.4577 1.96 2.4280 -8.9201 17.688

70 4 7.8844 10.2273 4.2659 0.9340 4.5147 1.903 2.6545 -9.7076 18.4755

80 4 7.7334 10.3528 4.1404 0.9127 4.5762 1.8415 2.8042 -9.6871 18.455

87.6 4 7.6864 10.3977 4.0955 0.8872 4.5819 1.8358 2.7380 -9.8661 18.634

90 4 6.9071 9.6486 4.8446 0.9250 4.6037 1.814 2.6547 -9.7321 18.5

100 4 7.3797 10.5186 3.9746 0.8150 4.6527 1.765 2.9200 -9.9321 18.7

120 4 7.4983 10.5814 3.9118 0.8535 4.7362 1.6815 3.0930 -10.0021 18.77

140 4 10.6902 3.8030 4.7527 1.665 -10.3171 19.085

160 4 10.7995 3.6937 4.7847 1.633 -10.8121 19.58


A. 9

180 4 10.8244 3.6688 4.8297 1.588 -10.9921 19.76

200 4 10.8726 3.6206 4.8657 1.552 -11.1128 19.8807

5 2 3 11.6867 5.0548 7.2194 3.4560 2.1710 3.184 1.6000 -2.5860 12

10 2 3 8.8266 6.4091 5.8651 2.1970 2.8220 2.533 2.1200 -4.5860 14

20 2 3 9.3274 7.2751 4.9991 2.0140 3.2430 2.112 1.4040 -6.7580 16.172

30 9.8178 7.6519 4.6223 1.7750 3.4450 1.91 0.8800 -7.7060 17.12

40 7.5657 7.9854 4.2888 1.5435 3.5785 1.7765 1.4000 -8.3860 17.8

50 8.5241 8.1955 4.0787 1.5333 3.6283 1.7267 1.2927 -9.0503 18.4643

60 8.6321 8.3157 3.9585 1.4440 3.6990 1.656 1.3397 -9.3623 18.7763

70 8.2973 8.4212 3.8530 1.2262 3.7442 1.6108 2.0412 -9.6748 19.0888

80 8.0553 8.4557 3.8185 1.1762 3.7770 1.578 2.0229 -9.8223 19.2363

87.6 8.0755 8.5678 3.7064 1.1790 3.8110 1.544 1.8220 -10.1360 19.55

90 8.0445 8.5670 3.7072 1.1990 3.8150 1.54 1.9285 -9.8122 19.2262

5 2 13.0333 3.7345 5.8728 4.1750 1.6566 2.465 -3.0400 -4.5120 16.64

10 2 9.7802 4.6958 4.9115 2.7300 2.1216 2 -2.6729 -6.6649 18.7929

20 2 10.0069 5.2877 4.3196 2.5260 2.5216 1.6 -2.8240 -8.2720 20.4

30 10.5073 5.6745 3.9328 2.2375 2.6741 1.4475 -2.7420 -8.6140 20.742

40 8.1122 5.8650 3.7423 1.9420 2.7436 1.378 -1.8000 -8.8720 21

50 8.9745 5.9790 3.6283 1.9300 2.7916 1.33 -2.7930 -10.4220 22.55

60 9.0741 6.0908 3.5165 1.8065 2.8281 1.2935 -2.1605 -10.1485 22.2765

70 8.6284 6.0854 3.5219 1.6420 2.9266 1.195 -1.7200 -10.7220 22.85

80 8.5552 6.2887 3.3186 1.5542 2.9216 1.2 -1.2408 -10.3720 22.5

87.6 8.3071 6.1325 3.4748 1.5230 2.9216 1.2 -1.4280 -10.6720 22.8

90 8.4543 6.3099 3.2974 1.5570 2.9396 1.182 -3.0453 -12.0720 24.2

5 1.5 13.4875 2.9643 5.4186 4.64 1.6257 2 -16.4 0 30

Table A.14 (RCF+ ) Results and compared with the results of each of (RCF) and ( companding)


A. 10

I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

5 2 4 8.3398 5.3998 9.0934 2.3230 2.3247 4.093 2.7063 -1.8621 10.63

10 2 4 8.7462 6.6924 7.8008 2.2700 2.9177 3.5 2.8310 -3.0411 11.809

20 2 4 10.6280 7.8639 6.6293 2.1110 3.4507 2.967 2.7100 -4.7721 13.54

30 2 4 8.5826 8.4402 6.0530 1.6640 3.6817 2.736 2.9360 -5.4611 14.229

40 2 8.1186 8.7665 5.7267 1.4470 3.8647 2.553 2.8084 -6.1737 14.9416

50 2 8.3905 9.0215 5.4717 1.3690 3.9867 2.431 3.0554 -6.4467 15.2146

60 2 8.5050 9.2655 5.2277 1.3100 4.0677 2.35 2.8770 -7.1321 15.9

70 2 9.1783 9.4478 5.0454 1.3860 4.1577 2.26 2.7650 -7.4671 16.235

80 2 11.6283 9.5714 4.9218 1.1230 4.2107 2.207 2.9880 -7.7181 16.486

90 2 8.0138 9.6486 4.8446 1.1540 4.2717 2.146 2.7524 -8.0797 16.8476

100 2 7.5968 9.6758 4.8174 1.0110 4.3087 2.109 2.9370 -8.2951 17.063

120 2 7.5493 9.9283 4.5649 0.9738 4.3915 2.0262 2.7300 -8.7821 17.55

140 2 8.2890 10.0443 4.4489 1.0600 4.4177 2 2.8215 -8.9106 17.6785

160 2 7.8826 10.1675 4.3257 0.9080 4.4597 1.958 2.7580 -8.9491 17.717

180 2 8.3477 10.2551 4.2381 1.0160 4.5537 1.864 2.8232 -9.4589 18.2268

200 2 8.0239 10.2748 4.2184 0.9200 4.5777 1.84 2.7820 -9.4501 18.218

220 2 7.1092 10.3302 4.1630 0.8325 4.6052 1.8125 2.8935 -9.5771 18.345

240 2 11.8374 10.4162 4.0770 1.0750 4.6427 1.775 2.9780 -9.8541 18.622

255 2 7.7076 10.4574 4.0358 0.9300 4.6677 1.75 2.7810 -9.9191 18.687

5 2 3 9.6333 4.4743 7.7999 2.9530 1.8920 3.463 1.8508 -2.0715 11.4855

10 2 3 9.7698 5.4970 6.7772 2.8265 2.4115 2.9435 1.7762 -3.4498 12.8638

20 2 3 11.4793 6.4962 5.7780 2.5780 2.8550 2.5 2.1180 -4.7180 14.132

30 2 9.3130 6.9516 5.3226 2.1220 3.0770 2.278 2.1650 -5.5860 15

40 2 8.8229 7.2518 5.0224 1.8440 3.1990 2.156 1.7760 -6.5600 15.974

50 2 9.0029 7.4149 4.8593 1.7330 3.2880 2.067 1.8700 -6.9860 16.4

60 2 9.0355 7.5770 4.6972 1.6600 3.3550 2 1.8585 -7.5045 16.9185

70 2 9.7156 7.7661 4.5081 1.7260 3.4350 1.92 2.1287 -7.4573 16.8713


A. 11

80 2 12.1143 7.8384 4.4358 1.4350 3.4600 1.895 2.2240 -7.8360 17.25

90 2 8.4708 7.8866 4.3876 1.4885 3.5435 1.8115 2.1840 -8.0020 17.416

100 2 8.0750 7.9350 4.3392 1.3600 3.5950 1.76 2.2374 -8.3486 17.7626

120 2 7.9673 8.1273 4.1469 1.3040 3.6590 1.696 2.2800 - 8.5860 18

140 2 8.6065 8.1428 4.1314 1.3750 3.6700 1.685 2.0000 -9.0860 18.5

160 2 8.2211 8.2870 3.9872 1.2240 3.7130 1.642 1.8250 -9.2360 18.65

180 2 8.7098 8.3982 3.8760 1.2734 3.7484 1.6066 2.1320 -9.5040 18.918

200 2 8.4603 8.4922 3.7820 1.2350 3.8300 1.525 1.6372 -9.9488 19.3628

220 2 7.9536 8.9556 3.3186 1.0450 3.7550 1.6 2.0058 -9.8187 19.2327

240 2 12.2074 8.5672 3.7070 1.3300 3.8350 1.52 2.2420 -9.9440 19.358

255 2 8.0323 8.5631 3.7111 1.1800 3.8550 1.5 1.4680 -10.5860 20

5 2 2 10.9878 3.1619 6.4454 3.7600 1.4656 2.656 -1.6637 -2.8720 15

10 2 2 10.9800 4.0403 5.5670 3.5200 1.8716 2.25 -2.6100 -5.1220 17.25

20 2 2 12.1035 4.4535 5.1538 3.1310 2.1746 1.947 -2.6250 -6.7470 18.875

30 2 10.1735 5.1452 4.4621 2.6080 2.3296 1.792 -1.6850 -6.7220 18.85

40 2 9.4721 5.2341 4.3732 2.3440 2.4656 1.656 -1.8810 -7.5030 19.631

50 2 9.5534 5.2985 4.3088 2.1870 2.5086 1.613 -1.6875 -7.8295 19.9575

60 2 9.7161 5.5907 4.0166 2.1380 2.5996 1.522 -1.2230 -7.8720 20

70 2 10.2669 5.6505 3.9568 2.1940 2.6696 1.452 -2.1380 -9.0100 21.138

80 2 12.6705 5.7277 3.8796 1.8640 2.6556 1.466 -2.2047 -9.5507 21.6787

90 2 9.0121 5.7610 3.8463 1.8850 2.7066 1.415 -1.7000 -9.1720 21.3

100 2 2 8.6436 5.8367 3.7706 1.7370 2.7386 1.383 -1.7400 -9.6120 21.74

120 2 2 8.5096 6.0027 3.6046 1.6740 2.7956 1.326 -1.7200 -9.8720 22

140 2 9.1446 6.0140 3.5933 1.7565 2.8181 1.3035 -1.9200 -10.2920 22.42

160 2 8.7169 6.1159 3.4914 1.5840 2.8396 1.282 -2.1100 -10.4570 22.585

180 2 8.9937 6.0152 3.5921 1.6390 2.8806 1.241 -1.1377 -10.0597 22.1877

200 2 8.7313 6.0963 3.5110 1.5600 2.9216 1.2 -1.2575 -10.1295 22.2575

220 2 7.9605 6.2956 3.3117 1.4520 2.9286 1.193 -1.7615 -10.8720 23

240 2 12.5271 6.2200 3.3873 1.6650 2.9366 1.185 -0.9850 -10.4570 22.585


A. 12

255 2 8.3049 6.1688 3.4385 1.5205 2.9621 1.1595 -1.4387 -10.7787 22.9067

5 2 1.5 11.8784 2.8281 5.5548 4.1715 1.4805 2.2445 -16.6637 0 >30

Table A.15 (RCF+ RCT) Results and compared with the results of each of (RCF) and (RCT)

R I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

.9 2 4 8.4451 1.0752 13.4180 3.4485 0.3162 6.1015 2.7795 -0.1291 8.897

.8 2 8.8719 2.2340 12.2592 3.0435 0.7797 5.638 2.9000 -0.3191 9.087

.7 2 7.0561 3.4202 11.0730 3.0450 1.4047 5.013 3.0902 -0.5556 9.3235

.6 2 5.5844 4.7229 9.7703 2.2125 1.9477 4.47 2.8434 -1.7887 10.5566

.5 2 5.5235 6.0903 8.4029 2.0020 2.5847 3.833 2.7734 -2.9087 11.6766

.4 2 4.5640 7.5280 6.9652 1.6643 3.2605 3.1572 2.8350 -4.5421 13.31

.3 2 3.1480 9.0883 5.4049 1.2494 3.9571 2.4606 2.7515 -7.0056 15.7735

.2 2 2.2574 10.7618 3.7314 0.8589 4.7021 1.7156 3.0366 -10.4455 19.2134

.1 2 0.9227 12.5433 1.9499 0.3630 5.5127 0.905 2.3826 -17.1495 25.9174

.9 2 3 10.5065 0.9176 11.3566 4.4625 0.2675 5.0875 2.0289 -0.2336 9.6476

.8 2 3 10.7468 1.8899 10.3843 3.8165 0.4900 4.865 2.2570 -0.3160 9.73

.7 2 8.7402 2.8853 9.3889 3.8460 1.1430 4.212 2.0987 -0.9010 10.315

.6 2 7.0046 3.9241 8.3501 2.9393 1.6118 3.7432 2.4765 -1.5095 10.9235

.5 2 6.7408 5.0886 7.1856 2.5280 2.0480 3.307 2.2000 -2.8360 12.25

.4 2 5.5512 6.2962 5.9780 2.0849 2.6184 2.7366 2.3021 -4.4289 13.8429

.3 2 3.9120 7.6333 4.6409 1.5900 3.2350 2.12 2.0918 -7.0192 16.4332

.2 2 2.7595 9.0449 3.2293 1.0745 3.8550 1.5 2.2500 -10.5860 20

.1 2 1.1778 10.5794 1.6948 0.4965 4.5835 .7715 1.8866 -16.9994 26.4134

.9 2 2 12.9726 0.7168 8.8905 5.5790 0.1506 3.971 -0.7535 -0.3020 12.43

.8 2 2 12.9771 1.4533 8.1540 5.0235 0.4636 3.658 -1.1263 -0.9853 13.1133

.7 2 2 10.6546 2.1328 7.4745 4.8455 0.9091 3.2125 -0.9963 -1.2820 13.41


A. 13

.6 2 8.7222 2.9748 6.6325 3.7330 1.1721 2.9495 -1.3655 -2.6375 14.7655

.5 2 8.2882 3.9691 5.6382 3.3380 1.6246 2.497 -0.7285 -3.0505 15.1785

.4 2 6.8280 4.9061 4.7012 2.7185 2.0186 2.103 -1.2550 -5.2720 17.4

.3 2 4.8319 5.8863 3.7210 2.0413 2.4529 1.6687 -1.4750 -7.8720 20

.2 2 3.4152 7.0337 2.5736 1.4145 2.9616 1.16 -0.9680 -11.0900 23.218

.1 2 2 1.5161 8.2508 1.3565 0.6680 3.5216 0.6 -1.0090 -17.1810 29.309

.9 2 1.5 13.9836 .5034 7.8795 6.201 .2767 3.349 -183235 0 > 30

Table A.16 (RCF+AEXP) Results and compared with the results of each of (RCF) and (AEXP companding)

d I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

2 2 4 1.6576 3.0697 11.4235 0.3643 1.6287 4.789 4.2300 -1.7321 10.5

1.9 4 2.0456 3.5148 10.9784 0.4800 1.7577 4.66 3.9830 -1.9491 10.717

1.8 4 1.6168 3.9117 10.5815 0.4185 1.9177 4.5 4.0281 -2.0620 10.8299

1.7 4 0.9849 4.2608 10.2324 0.4080 2.0557 4.362 3.7540 -1.9281 10.696

1.6 4 0.8895 4.6865 9.8067 0.3930 2.2257 4.192 3.5558 -1.8763 10.6442

1.5 0.8895 5.2963 9.1969 0.3410 2.4007 4..017 3.4196 -2.1263 10.8942

1.4 0.7767 5.6554 8.8378 0.2839 2.5551 3.8626 3.9685 -1.8321 10.6

1.3 0.6955 6.2072 8.2860 0.2534 2.6911 3.7266 3.9000 -1.6321 10.4

1.2 0.5710 6.6142 7.8790 0.2895 2.9012 3.5165 4.6170 -1.9451 10.713

1.1 0.5453 7.0862 7.4070 0.2820 3.1002 3.3175 4.0570 -2.0251 10.793

1 0.5075 7.6233 6.8699 0.2475 3.2912 3.1265 4.1153 -2.4168 11.1847

.9 0.4226 8.1658 6.3274 0.2760 3.5017 2.916 4.0250 -3.5071 12.275

4.8 0.3368 8.7494 5.7438 0.2575 3.7552 2.6625 6.5750 -3.9021 12.67

.7 0.3553 9.3232 5.1700 0.2310 4.0117 2.406 15.5893 -5.6428 14.4107

.6 0.2150 9.9190 4.5742 0.1875 4.2652 2.1525 13.8000 -7.4321 16.2

.5 0.2011 10.5599 3.9333 0.2485 4.5587 1.859 0.7900 -20.4421 29.21

.4 0.1356 11.2249 3.2683 0.1400 4.8177 1.6 0 -21.2321 > 30


A. 14

2 3 2.6002 1.7933 10.4809 0.6733 0.8750 4.48 3.2800 -2.0360 11.45

1.9 3 2.9078 2.1580 10.1162 0.8370 1.0520 4.303 3.2458 -2.0402 11.4542

1.8 3 2.3311 2.4070 9.8672 0.6955 1.1320 4.223 3.6960 -1.7480 11.162

1.7 1.9316 2.9885 9.2857 0.7300 1.3150 4.04 2.9420 -2.0940 11.508

1.6 -1.6855 3.2635 9.0107 0.6150 1.3850 3.97 2.4616 -2.3244 11.7384

1.5 3 1.5155 3.7233 8.5509 0.5820 1.5790 3.776 2.9138 -1.9860 11.4

1.4 1.3912 4.0509 8.2233 0.5390 1.7475 3.6075 3.0907 -2.0638 11.4778

1.3 1.2246 4.5173 7.7569 0.5370 1.9120 3.443 2.7642 -2.1218 11.5358

1.2 1.1408 4.9650 7.3092 0.5380 2.0870 3.268 3.5735 -2.3425 11.7565

1.1 1.0906 5.4125 6.8617 0.4970 2.2525 3.1025 2.6963 -2.7397 12.1537

1 0.9998 5.8966 6.3776 0.4495 2.4305 2.9245 3.1313 -2.7547 12.1687

.9 0.8470 6.3712 5.9030 0.4920 2.6550 2.7 3.3500 -3.5360 12.95

.8 0.6953 6.8889 5.3853 0.4200 2.8550 2.5 5.4050 -4.4260 13.84

.7 0.6599 7.4088 4.8654 0.3505 3.0685 2.2865 15.2617 -5.3243 14.7383

.6 0.4746 7.9596 4.3146 0.3400 3.3550 2 9.1820 -11.4040 20.818

.5 0.4315 8.5713 3.7029 0.3125 3.5600 1.795 0 -20.5860 > 30

.4 0.3165 9.1868 3.0874 0.2740 3.8890 1.466 0 -20.5860 > 30

2 2 3.7697 0.2959 9.3114 1.3368 0.3051 3.8165 -1.0610 -3.6630 15.791

1.9 2 4.0878 0.6711 8.9362 1.4775 0.4591 3.6625 -3.5845 -6.1565 18.2845

1.8 2 3.2999 0.7089 8.8984 1.3285 0.5316 3.59 -4.1178 -6.8478 18.9758

1.7 3.1314 1.5214 8.0859 1.3250 0.6766 3.445 -4.8455 -7.1675 19.2955

1.6 2.9557 1.8668 7.7405 1.2940 0.8306 3.291 -4.3000 -6.3720 18.5

1.5 2.7710 2.3119 7.2954 1.1373 0.9009 3.2207 -5.0002 -7.1860 19.314

1.4 2.6346 2.6274 6.9799 1.0605 1.0356 3.086 -3.6315 -6.0720 18.2

1.3 2.3627 2.9885 6.6188 1.0460 1.1876 2.934 -4.1380 -6.3100 18.438

1.2 2.2259 3.3832 6.2241 1.0160 1.3316 2.79 -5.6200 -8.8220 20.95

1.1 2.0747 3.7297 5.8776 0.9575 1.4796 2.642 -7.6500 -9.5558 22.5

1 1.8764 4.1063 5.5010 0.8755 1.6231 2.4985 -6.3838 -10.3720 21.6838

.9 1.7267 4.5840 5.0233 0.8580 1.7876 2.334 -4.0670 -8.2390 20.367


A. 15

.8 1.4732 4.9999 4.6074 0.7860 1.9876 2.134 -2.5675 -9.6845 21.8125

.7 1.3760 5.4580 4.1493 0.6705 2.1551 1.9665 -0.9000 -16.9720 29.1

.6 1.0899 5.9080 3.6993 0.5820 2.3636 1.758 0 -17.8720 >30

.5 0.9386 6.4115 3.1958 0.6000 2.6141 1.5075 0 -17.8720 >30

.4 0.7417 6.9451 2.6622 0.4680 2.8496 1.272 0 -17.8720 >30

2 1.5 4.7537 .0555 8.3274 1.7799 .4116 3.3734 -15.27 0 >30

Table A.17 (RCF+ cos) Results and compared with the results of each of (RCF) and (cos companding)

y I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

1 2 4 3.0167 1.9493 12.5439 1.3219 0.8896 5.5281 2.7245 -0.3076 9.0755

.9 2.7783 2.8709 11.6223 1.2531 1.2548 5.1629 2.9981 -0.4590 9.2269

.8 2.4748 3.8585 10.6347 1.1279 1.6806 4.7371 2.8724 -0.9397 9.7076

.7 2.0799 4.9039 9.5893 0.9978 2.1340 4.2837 3.2051 -1.4110 10.1789

.6 1.7684 5.9558 8.5374 0.7710 2.5857 3.832 3.3736 -2.3970 11.1649

.5 1.4604 7.1846 7.3086 0.6732 3.1359 3.2818 3.2650 -3.8271 12.595

.4 1.1521 8.4391 6.0541 0.5450 3.6837 2.734 3.3207 -5.4714 14.2393

.3 0.8658 9.7831 4.7101 0.4597 4.2774 2.1403 4.4287 -8.1034 16.8713

.2 11.2113 3.2819 4.9241 1.4936 > -21.2321 29.6

.1 12.7916 1.7016 5.6361 .7816 > -21.2321 30

1 3 4.7448 1.4584 10.8158 2.0417 0.5467 4.8083 2.0610 -0.3250 9.739

.9 4.3655 2.2391 10.0351 1.9687 0.9077 4.4473 2.3846 -0.4264 9.8404

.8 3.9220 3.0867 9.1875 1.7506 1.2406 4.1144 2.0952 -1.0708 10.4848

.7 3.2140 3.8190 8.4552 1.5066 1.5801 3.7749 2.4776 -1.4924 10.9064

.6 2.7977 4.7661 7.5081 1.2527 2.0047 3.3503 2.4185 -2.7060 12.12

.5 2.2782 5.7834 6.4908 1.0661 2.4661 2.8889 2.5895 -3.8565 13.2705

.4 1.8853 6.9533 5.3209 0.8860 2.9620 2.393 2.2024 -5.9436 15.3576

.3 1.4064 8.1047 4.1695 0.7168 3.4718 1.8832 4.1522 -7.7338 17.1478


A. 16

.2 9.3722 2.9020 4.0344 1.3206 -17.6437 27.0577

.1 10.7498 1.5244 4.6579 .6971 > -20.5860 29.6

1 2 7.1336 1.1803 8.4270 3.0466 0.3182 3.8034 -0.7843 -0.4563 12.5843

.9 6.4694 1.6761 7.9312 2.8082 0.5138 3.6078 -0.6924 -0.7894 12.9174

.8 5.7883 2.2861 7.3212 2.5530 0.8096 3.312 -0.8804 -1.3324 13.4604

.7 5.0089 2.9470 6.6603 2.2845 1.1246 2.997 -1.0341 -2.2901 14.4181

.6 4.4710 3.7725 5.8348 1.9571 1.4757 2.6459 -0.3705 -2.7810 14.909

.5 3.6352 4.4735 5.1338 1.6068 1.7734 2.3482 -3.1728 -6.9048 19.0328

.4 2.9108 5.3119 4.2954 1.3159 2.1585 1.9631 -0.7907 -4.6413 16.7693

.3 2.2328 6.2642 3.3431 1.0410 2.5626 1.559 -8.3000 -17.4720 29.6

.2 7.3113

2.2960 3.0777 1.0439 > -17.8720 29.6

.1 8.3921 1.2152 3.5707 .5509 > -17.8720 29.6

1 1.5 8.2456 1.0679 7.3150 3.4937 0.2694 3.3563 >-18.2000 0 29.6

Table A.18 (RCF+NERF) Results and compared with the results of each of (RCF) and (NERF companding)

NERF I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

2 4 0.9191 5.2328 9.2604 0.4355 2.4177 4 3.2135 -1.6966 10.4645

3 1.6395 3.7342 8.5400 0.8685 1.7880 3.567 2.7780 -1.4860 10.9

2 2.7940 2.2218 7.3855 1.5235 1.2096 2.912 -2.3220 -3.8720 16

1.5 3.0791 1.2825 7.1004 1.8955 1.0857 2.54 -16.3220 -0.4000 > 30

Table A.19 (RCF+tanhR) Results and compared with the results of each of (RCF) and (tanhR companding)

k y I CR X X1 PAPR Y Y1 CCDF Z Z1 SNR


A. 17

OF

PAPR

(BER=

)

5 1 2 4 2.8874 0.6677 13.8255 1.1576 0.1677 6.25 3.1158 - 0.2457 9.0136

5 .8 4 2.4646 3.1106 11.3826 0.8777 1.2373 5.1804 3.3330 -0.5703 9.3382

5 .5 4 1.6787 7.2676 7.2256 0.5861 3.1177 3.3 3.5127 -3.3071 12.075

5 .2 4 0.8605 11.8091 2.6841 0.2635 5.1537 1.264 4.1231 -11.8321 20.6

10 1 -3.0836 2.4784 12.0148 -1.3540 0.8547 5.563 5.6486 -0.2321 9

10 .8 -1.7229 4.6223 9.8709 -0.7381 1.9217 4.496 5.2944 -0.8698 9.6377

10 .5 0.1334 8.2269 6.2663 -0.1059 3.4837 2.934 4.9151 -3.9459 12.7138

10 .2 0.6159 12.0589 2.4343 0.1921 5.2647 1.153 5.0159 -12.2321 21

15 1 -4.7170 4.5448 9.9484 -1.9818 1.8372 4.5805 19.8030 -1.0291 9.797

15 .8 -3.0465 6.2038 8.2894 -1.3095 2.5817 3.836 12.2656 -1.4321 10.2

15 .5 -0.5541 8.9972 5.4960 -0.3264 3.8592 2.5585 8.1255 -4.5813 13.3492

15 .2 0.4601 12.2096 2.2836 0.1342 5.3492 1.0685 4.8634 -12.9263 21.6942

20 1 -4.5883 6.5268 7.9664 -2.0457 2.5492 3.8685 19.0682 -1.7639 10.5318

20 .8 -3.3107 7.5875 6.9057 -1.4458 3.1307 3.287 18.3213 -2.5108 11.2787

20 .5 -1.1082 9.6089 4.8843 -0.5220 4.1062 2.3115 15.0955 -5.7366 14.5045

20 .2 0.4112 12.3334 2.1598 0.1085 5.3947 1.023 7.1422 -13.0291 21.797

30 1 9.4116 5.0816 3.8792 2.5385 -4.7878 13.5557

30 .8 9.6678 4.8254 4.0477 2.37 -4.8484 13.6163

30 .5 10.5681 3.9251 4.5127 1.905 -6.9271 15.695

30 .2 12.5057 1.9875 5.4807 .937 -14.4491 23.217

40 1 11.1190 3.3742 4.5622 1.8555 -19.5481 28.316

40 .8 11.0445 3.4487 4.6592 1.7585 -10.3821 19.15

40 .5 11.2677 3.2255 4.8527 1.565 -9.1149 17.8828

40 .2 12.6433 1.8499 5.5377 .88 -14.5471 23.315

5 1 2 3 4.9487 0.5100 11.7642 2.0796 0.0270 5.328 2.5768 -0.1386 9.5526

5 .8 3 4.0556 2.4826 9.7916 1.6441 0.9410 4.414 2.8712 -0.3860 9.8

5 .5 3 2.6254 5.9953 6.2789 0.9621 2.4310 2.924 2.6994 -3.4743 12.8883


A. 18

5 .2 3 1.1979 9.9275 2.3467 0.4465 4.2740 1.081 3.0276 -12.2815 21.6955

10 1 -1.6137 1.7293 10.5449 -0.6095 0.5365 4.8185 4.6886 -0.5460 9.96

10 .8 -0.5458 3.5804 8.6938 -0.1761 1.4210 3.934 3.8892 -1.6289 11.0429

10 .5 0.8648 6.7393 5.5349 0.2921 2.8190 2.536 3.9912 -4.2237 13.6377

10 .2 0.9020 10.1260 2.1482 0.3341 4.3440 1.011 3.4909 -13.1110 22.525

15 1 -3.7961 3.2467 9.0275 -1.5233 1.2330 4.122 19.2188 -0.9672 10.3812

15 .8 2 3 -2.2420 4.7893 7.4849 -0.9965 1.8320 3.523 11.2571 -1.7945 11.2085

15 .5 2 3 0.0189 7.3512 4.9230 -0.1039 3.0190 2.336 7.4021 -4.6586 14.0726

15 .2 0.7330 10.2635 2.0107 0.2677 4.4200 .935 4.5003 -12.6433 22.0573

20 1 -4.1116 4.7845 7.4897 -1.6947 1.8375 3.5175 18.1168 -2.0692 11.4832

20 .8 -2.8070 5.8722 6.4020 -1.1964 2.3174 3.0376 17.4851 -2.7009 12.1149

20 .5 -0.6404 7.8577 4.4165 -0.2755 3.2900 2.065 14.6743 -5.5117 14.9257

20 .2 0.6658 10.3690 1.9052 0.2270 4.4505 .9045 6.0819 -13.4433 22.8573

30 1 7.2887 4.9855 2.8900 2.465 -5.1985 14.6125

30 .8 7.6359 4.6383 3.0680 2.287 -5.6554 15.0694

30 .5 8.6651 3.6091 3.6140 1.741 -7.1240 16.538

30 .2 10.5174 1.7568 4.5395 .8155 -14.2860 23.7

40 1 8.8946 3.3796 3.5970 1.758 -20.5860 >30

40 .8 8.8910 3.3832 3.6190 1.736 -13.3568 22.7708

40 .5 9.2710 3.0032 3.9270 1.428 -10.2676 19.6816

40 .2 10.6357 1.6385 4.5790 .776 -15.1671 24.5811

50 1 9.8817 2.3925 4.0150 1.34 >-20.5860 >30

50 .8 9.7344 2.5398 4.0445 1.3105 >-20.5860 >30

50 .5 9.7324 2.5418 4.0990 1.256 -15.2120 24.626

50 .2 9.8883 2.3859 4.0850 1.27 >-20.5860 >30

5 1 2 7.3271 0.2215 9.3858 3.2596 -0.0264 4.148 -0.4980 -0.4994 12.6274

5 .8 2 5.9194 1.6795 7.9278 3.0001 1.0636 3.058 -0.1100 -0.6532 12.7812

5 .5 2 3.7924 4.4954 5.1119 1.6301 1.8656 2.256 -0.8703 -4.3300 16.458

5 .2 1.6223 7.6850 1.9223 0.6385 3.2326 .889 -0.9024 -13.4975 25.6255


A. 19

10 1 0.1714 0.8475 8.7598 0.3585 0.2711 3.8505 2.0108 -0.5098 12.6378

10 .8 0.8666 2.3259 7.2814 0.4979 0.8616 3.26 1.2845 -1.5196 13.6476

10 .5 1.7957 5.0033 4.6040 0.7281 2.0216 2.1 -0.7796 -6.2805 18.4085

10 .2 1.2846 7.8417 1.7656 0.5421 3.3186 .803 -2.4641 -16.3520 28.48

15 1 -2.6181 1.7578 7.8495 -0.9813 0.5416 3.58 15.0296 -2.4424 14.5704

15 .8 -1.1756 3.1888 6.4185 -0.3955 1.1996 2.922 7.5001 -2.8375 14.9655

15 .5 0.7929 5.4583 4.1490 0.3171 2.2066 1.915 1.1217 -8.2250 20.353

15 .2 1.0662 7.9298 1.6775 0.4467 3.3656 0.756 -2.1924 -16.6220 28.75

20 1 -3.4074 2.8218 6.7855 -1.2977 1.0011 3.1205 13.0242 -4.4478 16.5758

20 .8 -2.0768 3.9355 5.6718 -0.7468 1.5336 2.588 12.6110 -4.8610 16.989

20 .5 0.0043 5.8355 3.7718 -0.0085 2.3236 1.798 6.3155 -11.1565 23.2845

20 .2 0.9859 8.0222 1.5851 0.3955 3.3856 .736 -0.6608 -17.4720 29.6

30 1 4.6925 4.9148 1.7481 2.3735 -17.8720 >30

30 .8 5.2686 4.3387 1.9186 2.203 -14.4193 26.5473

30 .5 6.4575 3.1498 2.6191 1.5025 -17.3187 29.4467

30 .2 8.1430 1.4643 3.4246 .697 >-17.8720 >30

40 1 6.1219 3.4854 2.2986 1.823 >-17.8720 >>30

40 .8 6.2434 3.3639 2.4566 1.665 >-17.8720 >>30

40 .5 6.8934 2.7139 2.8286 1.293 >-17.8720 >30

40 .2 8.2333 1.3740 3.4806 .641 >-17.8720 >30

5 1 1.5 8.0365 0.0685 8.3144 3.6052 0.0729 3.5528 >-18.0900 0 >30

Table A.20 (RCF+logR) Results and compared with the results of each of (RCF) and (logR companding)

K y I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

5 1 2 4 1.2906 13.2026 0.5137 5.904 -0.2189 8.9868

5 .8 4 3.5653 10.9279 1.4507 4.967 -0.5014 9.2693

5 .5 4 7.4063 7.0869 3.1367 3.281 -3.8893 12.6572


A. 20

5 .2 4 11.5744 2.9188 5.0617 1.356 -11.8750 20.6429

10 1 4 4.4581 2.2093 12.2839 1.7353 0.8385 5.5792 3.3237 -0.4084 9.1763

10 .8 4 4.5315 4.2908 10.2024 1.6805 1.7207 4.697 3.5893 -0.8428 9.6107

10 .5 4 3.0208 7.7716 6.7216 1.1055 3.2832 3.1345 2.9930 -3.9891 12.757

10 .2 4 1.2544 11.6543 2.8389 0.4720 5.0897 1.328 3.2073 -11.6748 20.4427

20 1 4 3.3770 3.4531 11.0401 1.1940 1.3207 5.097 3.9373 -0.9108 9.6787

20 .8 4 3.5281 5.1822 9.3110 1.2650 2.1527 4.265 3.8130 -1.2571 10.025

20 .5 4 2.5891 8.1681 6.3251 1.0200 3.5097 2.908 3.5467 -4.1186 12.8865

20 .2 4 1.4327 11.7296 2.7636 0.5980 5.1237 1.294 3.3966 -11.7355 20.5034

30 1 4 3.4707 4.2536 10.2396 1.0538 1.7115 4.7062 4.9448 -1.0256 9.7935

30 .8 4 2.7496 5.7288 8.7644 0.9815 2.3792 4.0385 4.3135 -1.6186 10.3865

30 .5 4 2.3515 8.4107 6.0825 0.7820 3.5997 2.818 3.6750 -4.6399 13.4078

30 .2 4 1.0726 11.7740 2.7192 0.4210 5.1527 1.265 2.9800 -12.0321 20.8

40 1 4 2.8036 4.8378 9.6554 1.0170 2.0177 4.4 5.4633 -1.4553 10.2232

40 .8 4 2.4673 6.1338 8.3594 0.8890 2.6067 3.811 5.4110 -1.8211 10.589

40 .5 4 2.4487 8.5884 5.9048 0.8983 3.6960 2.7217 4.1120 -4.7321 13.5

40 .2 4 1.1876 11.8120 2.6812 0.4235 5.1627 1.255 3.2819 -11.8370 20.6049

50 1 4 2.4698 5.3095 9.1837 0.8367 2.1744 4.2433 6.1574 -1.6547 10.4226

50 .8 4 2.2419 6.4658 8.0274 0.7788 2.7045 3.7132 5.0437 -2.4484 11.2163

50 .5 4 2.0257 8.7381 5.7551 0.7408 3.7465 2.6712 3.7357 -4.9204 13.6883

50 .2 4 1.0953 11.8360 2.6572 0.4230 5.1727 1.245 3.5624 -12.0697 20.8376

70 1 4 1.8443 5.9819 8.5113 0.7005 2.5262 3.8915 9.0967 -2.1354 10.9033

70 .8 4 2.2212 6.9428 7.5504 0.8415 2.9592 3.4585 6.4961 -2.4610 11.2289

70 .5 4 2.1335 8.9517 5.5415 0.8038 3.8555 2.5622 4.5018 -4.8583 13.6262

70 .2 4 1.3557 11.8736 2.6196 0.4858 5.1985 1.2192 3.0288 -12.0833 20.8512

90 1 4 1.6931 6.4872 8.0060 0.7215 2.7192 3.6985 18.2150 -3.0171 11.785

90 .8 4 1.8910 7.3116 7.1816 0.6245 3.0922 3.3255 7.8815 -3.3506 12.1185

90 .5 4 1.7246 9.1252 5.3680 0.6245 3.9182 2.4995 5.0814 -5.1321 13.9

90 .2 4 1.1122 11.9078 2.5854 0.3890 5.2067 1.211 3.7194 -11.8727 20.6406


A. 21

5 1 3 1.0477 11.2265 0.2240 5.131 -0.1171 9.5311

5 .8 3 2.9539 9.3203 1.0718 4.2832 -1.1021 10.5161

5 .5 3 6.1589 6.1153 2.5305 2.8245 -3.5519 12.9659

5 .2 3 9.7371 2.5371 4.1768 1.1782 -11.7935 21.2075

10 1 3 6.2371 1.7693 10.5049 2.6175 0.6580 4.697 2.7294 -0.3566 9.7706

10 .8 3 5.9788 3.5191 8.7551 2.4579 1.4354 3.9196 3.0403 -0.7457 10.1597

10 .5 3 3.9290 6.4608 5.8134 1.5920 2.7070 2.648 2.0783 -4.2577 13.6717

10 .2 3 1.6290 9.8099 2.4643 0.6580 4.2130 1.142 2.4248 -11.8112 21.2252

20 1 3 4.8603 2.7174 9.5568 1.9060 0.9700 4.385 3.1218 -1.0802 10.4942

20 .8 3 4.7576 4.1927 8.0815 1.8177 1.6427 3.7123 2.9968 -1.4272 10.8412

20 .5 3 3.4143 6.7743 5.4999 1.3930 2.8200 2.535 2.6157 -4.4035 13.8175

20 .2 3 1.7983 9.8762 2.3980 0.7803 4.2433 1.1117 2.6000 -11.8860 21.3

30 1 3 4.8063 3.3702 8.9040 1.7293 1.3243 4.0307 4.1696 -1.1547 10.5687

30 .8 3 3.9069 4.6671 7.6071 1.4890 1.8240 3.531 3.2438 -2.0422 11.4562

30 .5 3 3.1544 6.9946 5.2796 1.1335 2.8885 2.4665 2.9597 -4.7091 14.1231

30 .2 3 1.4254 9.9078 2.3664 0.5853 4.2543 1.1007 2.4858 -11.8802 21.2942

40 1 3 4.0178 3.8330 8.4412 1.5754 1.5134 3.8416 4.3566 -1.9159 11.3299

40 .8 3 3.5545 5.0020 7.2722 1.4000 2.0550 3.3 4.4867 -2.0993 11.5133

40 .5 3 3.2209 7.1416 5.1326 1.2790 3.0140 2.341 3.6694 -4.5286 13.9426

40 .2 3 1.5353 9.9407 2.3335 0.5945 4.2710 1.084 2.7652 -11.7076 21.1216

50 1 3 3.6384 4.2591 8.0151 1.4450 1.7200 3.635 4.5271 -2.6389 12.0529

50 .8 3 3.2426 5.2475 7.0267 1.2997 2.1627 3.1923 3.6600 -3.1860 12.6

50 .5 3 2.7517 7.2451 5.0291 1.1300 3.0730 2.282 3.0435 -4.9665 14.3805

50 .2 3 1.4373 9.9590 2.3152 -0.1060 3.5810 1.774 2.9000 -12.0860 21.5

70 1 3 2.9375 4.8561 7.4181 1.1777 1.9407 3.4143 7.5555 -3.0305 12.4445

70 .8 3 3.2059 5.7085 6.5657 1.2800 2.3350 3.02 5.2071 -3.1039 12.5179

70 .5 3 2.8683 7.4675 4.8067 1.1529 3.1419 2.2131 3.7626 -4.9514 14.3654

70 .2 3 1.6943 9.9932 2.2810 0.6430 4.2930 1.062 2.2988 -12.1672 21.5812

90 1 3 2.6972 5.2723 7.0019 1.1900 2.1250 3.23 16.9738 -3.6122 13.0262


A. 22

90 .8 3 2.8076 6.0092 6.2650 1.0630 2.4680 2.887 6.3905 -4.1955 13.6095

90 .5 3 2.4246 7.6062 4.6680 0.9726 3.2036 2.1514 4.4034 -5.1640 14.578

90 .2 3 1.4463 10.0229 2.2513 0.5540 4.3090 1.046 3.1088 -11.8372 21.2512

5 1 2 0.6047 9.0026 0.2136 3.908 -0.6420 12.77

5 .8 2 2.1294 7.4779 0.7711 3.3505 -1.7059 13.8339

5 .5 2 4.6828 4.9245 1.9060 2.2156 -5.2076 17.3356

5 .2 2 7.5585 2.0488 3.2011 0.9205 -13.3424 25.4704

10 1 2 8.2868 1.1521 8.4552 3.4773 0.2844 3.8372 -0.1639 -0.5359 12.6639

10 .8 2 7.6483 2.5217 7.0856 3.2775 1.0216 3.1 -0.9584 -2.0304 14.1584

10 .5 2 5.0356 4.9005 4.7068 2.1470 2.0286 2.093 -2.4240 -6.0460 18.174

10 .2 2 2.1069 7.6209 1.9864 0.9035 3.2251 .8965 -2.1900 -13.7120 25.84

20 1 2 6.6065 1.7967 7.8106 2.9073 0.7379 3.3837 -0.8286 -2.3166 14.4446

20 .8 2 6.3082 3.0764 6.5309 2.6340 1.2256 2.896 -2.3474 -4.0574 16.1854

20 .5 2 4.4669 5.1600 4.4473 1.9210 2.1146 2.007 -1.8076 -6.1128 18.2408

20 .2 2 2.2520 7.6630 1.9443 1.0097 3.2393 .8823 -2.2310 -14.0030 26.131

30 1 2 6.3973 2.2943 7.3130 2.5700 0.9316 3.19 0.5122 -2.0981 14.2261

30 .8 2 5.3452 3.4385 6.1688 2.2390 1.3406 2.781 -0.7515 -3.3235 15.4515

30 .5 2 4.2164 5.3897 4.2176 1.6813 2.2029 1.9187 -1.0756 -6.0304 18.1584

30 .2 2 1.9076 7.7231 1.8842 0.8307 3.2663 .8553 -1.9290 -13.5810 25.709

40 1 2 5.6362 2.7845 6.8228 2.3297 1.0343 3.0873 -1.4135 -4.9720 17.1

40 .8 2 4.8415 3.6221 5.9852 2.0360 1.4576 2.664 -1.9750 -5.8470 17.975

40 .5 2 4.2284 5.4822 4.1251 1.7700 2.2716 1.85 -1.3729 -6.8569 18.9849

40 .2 2 1.9958 7.7343 1.8730 0.8282 3.2713 .8503 -1.6906 -13.4494 25.5774

50 1 2 5.0052 2.9590 6.6483 2.1132 1.1548 2.9668 -1.6049 -6.0569 18.1849

50 .8 2 4.5085 3.8465 5.7608 1.9000 1.5296 2.592 -0.9329 -5.0649 17.1929

50 .5 2 3.7058 5.5323 4.0750 1.5884 2.2980 1.8236 -2.2943 -7.5903 19.7183

50 .2 2 1.8782 7.7330 1.8743 0.8302 3.2838 .8378 -2.1000 -14.3720 26.5

70 1 2 4.2424 3.4941 6.1132 1.8590 1.3886 2.733 -0.4088 -8.2808 20.4088

70 .8 2 4.3524 4.1881 5.4192 1.8465 1.6681 2.4535 -1.8350 -7.4320 19.56


A. 23

70 .5 2 3.7415 5.6738 3.9335 1.5870 2.3426 1.779 -2.1860 -8.1860 20.314

70 .2 2 2.1221 7.7541 1.8532 0.8657 3.2823 .8393 -3.5928 -15.3448 27.4728

90 1 2 3.8737 3.7819 5.8254 1.7740 1.4756 2.646 0.8250 -17.0470 29.175

90 .8 2 3.9332 4.4679 5.1394 1.6365 1.8081 2.3135 -4.0838 -11.9558 24.0838

90 .5 2 3.2689 5.7836 3.8237 1.3934 2.3910 1.7306 -2.2159 -9.0693 21.1973

90 .2 2 1.8688 7.7785 1.8288 0.7643 3.2859 .8357 -3.1778 -15.4098 27.5378

5 1 1.5 11.4446 0.4088 7.9741 5.1875 0.2732 3.3525 >-17.9300 0 >30

A.6 Hybrid RFC with companding Results:

X = PAPR (Companding ) – PAPR (Companding +RFC)

Y =CCDF of PAPR (Companding) - CCDF of PAPR (Companding + RFC)

Z= SNR (BER= ) (Companding) – SNR (BER= ) (Companding +RFC)

X1 == PAPR (RFC) – PAPR (Companding + RFC)

Y1 =CCDF of PAPR (RFC) - CCDF of PAPR (Companding + RFC)

Z1= SNR (BER= ) (RFC) –SNR (BER= ) (Companding + RFC)

Table A.21 (RFC+A) Results and compared with the results of each of (RFC) and (A companding)

A I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

5 4 4 10.6770 5.9547 8.2291 2.9485 2.3735 3.6915 5.5216 -2.4071 8.0784

10 4 4 8.1290 7.6211 6.5627 1.8080 3.1430 2.922 6.2537 -4.1950 9.8663

20 4 4 8.8708 8.7281 5.4557 1.7175 3.6565 2.4085 5.8070 -6.0977 11.769

30 4 4 9.4736 9.2173 4.9665 1.5030 3.8830 2.182 5.2467 -7.0820 12.7533

40 4 4 7.1833 9.5126 4.6712 1.2752 4.0202 2.0448 5.4416 -8.0871 13.7584

50 4 4 8.1376 9.7186 4.4652 1.3045 4.1095 1.9555 5.6391 -8.4466 14.1179

60 4 4 8.2792 9.8724 4.3114 1.2560 4.2210 1.844 5.5909 -8.8538 14.5251

70 4 4 7.9605 9.9940 4.1898 1.0095 4.2375 1.8275 6.3199 -9.1388 14.8101


A. 24

80 4 4 7.7809 10.0909 4.0929 0.9546 4.2654 1.7996 6.2943 -9.2936 14.9649

87.6 4 4 7.7519 10.1538 4.0300 0.9755 4.3175 1.7475 6.2175 -9.4832 15.1545

90 4 4 7.7326 10.1647 4.0191 1.0037 4.3297 1.7353 5.8577 -9.6257 15.297

100 4 4 7.4070 10.2365 3.9473 0.8784 4.3634 1.7016 6.1995 -9.7492 15.4205

120 4 4 7.5977 10.3714 3.8124 0.8928 4.4228 1.6422 6.0463 -10.1454 15.8167

140 4 4 10.4564 3.7274 4.4580 1.607 -10.5325 16.2038

160 4 4 10.5338 3.6500 4.4941 1.5709 -10.8208 16.4921

180 4 4 9.1488 5.0350 4.5362 1.5288 -10.8888 16.5601

200 4 4 10.6436 3.5402 4.5529 1.5121 -11.0074 16.6787

5 4 3 11.8201 4.7330 7.0860 3.6748 1.8988 2.9652 4.7945 -2.7670 8.8055

10 4 3 8.8945 6.0218 5.7972 2.3533 2.4873 2.3767 5.2186 -4.8629 10.9014

20 4 3 9.4051 6.8976 4.9214 2.1496 2.8876 1.9764 4.5924 -6.9451 12.9836

30 4 3 9.9401 7.3190 4.5000 1.9007 3.0797 1.7843 4.4688 -7.4927 13.5312

40 4 3 7.5889 7.5534 4.2656 1.6440 3.1880 1.676 5.0920 -8.0695 14.108

50 4 3 8.5004 7.7166 4.1024 1.6596 3.2636 1.6004 4.5339 -9.1846 15.2231

60 4 3 8.6499 7.8783 3.9407 1.5366 3.3006 1.5634 4.8399 -9.2376 15.2761

70 4 3 8.2973 7.9660 3.8530 1.3512 3.3782 1.4858 5.5438 -9.5477 15.5862

80 4 3 8.1005 8.0457 3.7733 1.3053 3.4151 1.4489 5.2229 -9.9978 16. 0363

87.6 4 3 8.0606 8.0977 3.7213 1.2983 3.4393 1.4247 5.2748 -10.0587 16.0972

90 4 3 8.0332 8.1005 3.7185 1.3020 3.4270 1.437 5.1440 -9.9722 16.0107

100 4 3 7.5943 8.0590 3.7600 1.1672 3.4512 1.4128 5.1326 -10.4489 16.4874

120 4 3 7.7448 8.1537 3.6653 1.1669 3.4959 1.3681 5.1104 -10.7141 16.7526

5 4 2 13.2210 2.7949 5.6851 4.5796 1.2446 2.0604 0.5206 -4.6524 13.0794

10 4 2 10.0137 3.8020 4.6780 3.1030 1.6780 1.627 1.2700 -6.4230 14.85

20 4 2 9.3891 3.5426 4.9374 2.1530 1.3320 1.973 4.7008 -4.4482 12.8752

30 4 2 9.8898 3.9297 4.5503 1.8913 1.5113 1.7937 3.9879 -5.5851 14.0121

40 4 2 8.2320 4.8575 3.6225 2.1484 2.1334 1.1716 0.8444 -9.9286 18.3556


A. 25

50 4 2 9.0584 4.9356 3.5444 2.1234 2.1684 1.1366 1.3836 -9.9464 18.3734

60 4 2 8.5255 4.4149 4.0651 1.5636 1.7686 1.5364 4.9809 -6.7081 15.1351

70 4 2 8.1824 4.5121 3.9679 1.3459 1.8139 1.4911 5.2696 -7.4334 15.8604

80 4 2 8.4834 5.0896 3.3904 1.7303 2.2811 1.0239 1.5838 -11.2484 19.6754

87.6 4 2 8.1676 4.8657 3.6143 1.7146 2.2966 1.0084 2.0854 -10.8596 19.2866

90 4 2 8.0728 4.8011 3.6789 1.2991 1.8651 1.4399 5.3091 -7.4186 15.8456

100 4 2 7.7330 4.8587 3.6213 1.1672 1.8922 1.4128 5.7459 -7.4471 15.8741

120 4 2 8.1811 5.2510 3.2290 1.5670 2.3370 .968 2.3035 -11.1325 19.5595

5 4 1.5 14.3373 1.8221 4.5688 5.2036 0.9794 1.4364 -16 -1.3800 29.6

Table A.22 (RFC+ ) Results and compared with the results of each of (RFC) and ( companding)

MU I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

5 4 4 8.5326 5.2832 8.9006 2.4646 2.1136 3.9514 5.5832 -2.0818 7.7531

10 4 4 8.9351 6.5719 7.6119 2.3622 2.6572 3.4078 6.1305 -2.8382 8.5095

20 4 4 10.1373 7.0638 7.1200 2.1758 3.1628 2.9022 6.2688 -4.3099 9.9812

30 4 4 8.6956 8.2438 5.9400 1.7481 3.4131 2.6519 6.0706 -5.4231 11.0944

40 4 4 8.2595 8.5980 5.5858 1.5164 3.5814 2.4836 6.3302 -5.7485 11.4198

50 4 4 8.5378 8.8594 5.3244 1.4583 3.7233 2.3417 6.4429 -6.1558 11.8271

60 4 4 8.5780 9.0291 5.1547 1.3896 3.7946 2.2704 6.2734 -6.8323 12.5036

70 4 4 9.2492 9.2093 4.9745 1.4627 3.8817 2.1833 6.1287 -7.2000 12.8713

80 4 4 11.6899 9.3236 4.8602 1.2103 3.9453 2.1197 6.2719 -7.5308 13.2021

90 4 4 6.8796 8.2050 5.9788 1.2110 3.9760 2.089 6.2989 -7.6298 13.3011

100 4 4 7.7661 9.5357 4.6481 1.0783 4.0233 2.0417 6.2706 -8.0581 13.7294

120 4 4 6.3834 8.4530 5.7308 1.0264 4.0914 1.9736 6.2921 -8.3166 13.9879

140 4 4 8.3693 9.8152 4.3686 1.1766 4.1816 1.8834 6.1176 -8.7111 14.3824

160 4 4 7.9319 9.9074 4.2764 0.9970 4.1960 1.869 5.8882 -8.9155 14.5868

180 4 4 8.3988 9.9968 4.1870 1.0647 4.2497 1.8153 6.2099 -9.1688 14.8401


A. 26

200 4 4 8.1269 10.0684 4.1154 0.9662 4.2712 1.7938 5.9880 -9.3407 15.012

220 4 4 7.2315 10.1431 4.0407 0.9042 4.3242 1.7408 6.0007 -9.5665 15.2378

240 4 4 11.9179 10.1873 3.9965 1.1064 4.3214 1.7436 6.2370 -9.6917 15.363

255 4 4 7.2789 9.7193 4.4645 0.9756 4.3606 1.7044 6.1604 -9.6363 15.3076

5 4 3 9.7665 4.1523 7.6667 3.2220 1.6700 3.194 4.8568 -2.4410 8.4795

10 4 3 9.9560 5.2280 6.5910 3.0278 2.1218 2.7422 4.9688 -3.6327 9.6712

20 4 3 11.5810 6.1427 5.6763 2.7424 2.5284 2.3356 5.0238 -5.1877 11.2262

30 4 3 9.3580 6.5414 5.2776 2.2537 2.7177 2.1463 5.0583 -6.0682 12.1067

40 4 3 8.8961 6.8698 4.9492 1.9936 2.8576 2.0064 5.1076 -6.6039 12.6424

50 4 3 9.0117 6.9685 4.8505 1.8652 2.9292 1.9348 5.0674 -7.1641 13.2026

60 4 3 9.0525 7.1388 4.6802 1.8009 3.0049 1.8591 5.4896 -7.2489 13.2874

70 4 3 9.7253 7.3206 4.4984 1.8636 3.0816 1.7824 5.2608 -7.7007 13.7392

80 4 3 12.0968 7.3657 4.4533 1.5756 3.1096 1.7544 5.2740 -8.1615 14.2

90 4 3 8.5998 7.5604 4.2586 1.5980 3.1620 1.702 5.3629 -8.1986 14.2371

100 4 3 8.2039 7.6087 4.2103 1.4553 3.1993 1.6647 5.8642 -8.0973 14.1358

120 4 3 7.9277 7.6325 4.1865 1.3889 3.2529 1.6111 5.4133 -8.8282 14.8667

140 4 3 8.7502 7.8313 3.9877 1.4974 3.3014 1.5626 5.2935 -9.1680 15.2065

160 4 3 8.2815 7.8922 3.9268 1.3435 3.3415 1.5225 5.3149 -9.1216 15.1601

180 4 3 8.7284 7.9616 3.8574 1.3896 3.3736 1.4904 5.2071 -9.8044 15.8429

200 4 3 8.4567 8.0334 3.7856 1.2902 3.3942 1.4698 5.2732 -9.6883 15.7268

220 4 3 7.5390 8.0858 3.7332 1.1998 3.4188 1.4452 5.2154 -9.9846 16.0231

240 4 3 12.2278 8.1324 3.6866 1.4267 3.4407 1.4233 5.5058 -10.0557 16.0942

255 4 3 8.0884 8.1640 3.6550 1.2715 3.4555 1.4085 5.3381 -10.0914 16.1299

5 4 2 11.4573 2.5041 5.9759 4.2422 1.1312 2.1738 1.6003 -3.3090 11.736

10 4 2 11.2272 3.1602 5.3198 3.8944 1.4294 1.8756 1.4241 -4.7889 13.2159

20 4 2 12.5123 3.7350 4.7450 3.4619 1.6889 1.6161 1.3077 -6.5153 14.9423

30 4 2 10.1818 4.0262 4.4538 2.9164 1.8214 1.4836 1.2065 -7.5315 15.9585

40 4 2 9.4679 4.1026 4.3774 2.5916 1.8966 1.4084 1.9988 -7.3242 15.7512

50 4 2 9.6186 4.2364 4.2436 2.4543 1.9593 1.3457 2.0125 -7.8305 16.2575


A. 27

60 4 2 9.5917 4.3390 4.1410 2.3625 2.0075 1.2975 2.1959 -8.1541 16.5811

70 4 2 10.1649 4.4212 4.0588 2.3873 2.0463 1.2587 1.3881 -9.1849 17.6119

80 4 2 12.3265 4.2564 4.2236 2.1150 2.0900 1.215 1.8028 -9.2442 17.6712

90 4 2 8.6925 4.3141 4.1659 2.1121 2.1171 1.1879 1.7990 -9.3740 17.801

100 4 2 8.2980 4.3638 4.1162 1.9557 2.1407 1.1643 2.1021 -9.4709 17.8979

120 4 2 8.0803 4.4461 4.0339 1.8746 2.1796 1.1254 2.2005 -9.6525 18.0795

140 4 2 9.2679 5.0100 3.4700 1.9463 2.1913 1.1137 2.0155 -10.0575 18.4845

160 4 2 8.7930 5.0647 3.4153 1.7787 2.2177 1.0873 1.7904 -10.2576 18.6846

180 4 2 9.2168 5.1110 3.3690 1.8151 2.2401 1.0649 2.1229 -10.5001 18.9271

200 4 2 8.9133 5.1510 3.3290 1.7145 2.2595 1.0455 1.8744 -10.6986 19.1256

220 4 2 7.9705 5.1783 3.3017 1.6223 2.2823 1.0227 2.4273 -10.7564 18.8112

240 4 2 12.5551 5.1207 3.3593 1.8317 2.2867 1.0183 2.4166 -10.3842 19.1834

255 4 2 8.3361 5.0727 3.4073 1.6868 2.3118 .9932 2.1590 -10.8820 19.309

5 4 1.5 12.4989 1.4566 4.9343 4.8033 0.8031 1.6127 -16.6637 -1.7800 >30

Table A.23 (RFC+ RCT) Results and compared with the results of each of (RFC) and (RCT).

R I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

.9 4 4 8.7058 1.0265 13.1573 3.7793 0.2943 5.7707 5.7109 -0.2943 5.9656

.8 4 4 8.2277 1.2804 12.9034 3.3454 0.7289 5.3361 5.8086 -0.5071 6.1784

.7 4 4 7.2828 3.3375 10.8463 3.2633 1.2703 4.7947 5.9381 -0.8043 6.4756

.6 4 4 5.7647 4.5938 9.5900 2.4276 1.8101 4.2549 6.2980 -1.4307 7.102

.5 4 4 5.6633 5.9207 8.2631 2.1581 2.3881 3.6769 6.0880 -2.6907 8.362

.4 4 4 4.7000 7.3546 6.8292 1.7575 3.0010 3.064 6.0777 -4.3960 10.0673

.3 4 4 3.2463 8.8772 5.3066 1.3323 3.6873 2.3777 5.6848 -7.1689 12.8402

.2 4 4 2.3157 10.5107 3.6731 0.9217 4.4122 1.6528 5.9429 -10.6358 16.3071

.1 4 4 0.9577 12.2689 1.9149 0.3934 5.1904 .8746 5.5206 -17.1081 22.7794

.9 3 10.8872 0.8431 10.9759 4.9102 0.2242 4.6398 5.2180 -0.4200 6.4585


A. 28

.8 4 3 11.0343 1.7222 10.0968 4.3707 0.5532 4.3108 5.4803 -0.4682 6.5067

.7 3 9.0586 2.7485 9.0705 4.1650 0.9710 3.893 5.2811 -1.0941 7.1326

.6 3 7.3092 3.7735 8.0455 3.2110 1.3925 3.4715 5.4153 -1.9462 7.9847

.5 3 6.9813 4.8739 6.9451 2.8005 1.8295 3.0345 5.1927 -3.2188 9.2573

.4 3 5.7331 6.0229 5.7961 2.2985 2.3410 2.523 5.1535 -4.9530 10.9915

.3 3 4.0685 7.3346 4.4844 1.7236 2.8776 1.9864 5.5226 -6.9639 13.0024

.2 3 2.8625 8.6927 3.1263 1.1770 3.4665 1.3975 5.3988 -10.8127 16.8512

.1 3 1.2197 10.1661 1.6529 0.5409 4.1369 .7271 4.8168 -17.4447 23.4832

.9 2 13.9172 0.5341 7.9459 6.4071 0.1621 3.1429 2.6339 -0.6156 9.0426

.8 2 13.9228 1.2717 7.2083 5.7633 0.3868 2.9182 2.8523 -0.7077 9.1347

.7 2 11.5339 1.8848 6.5952 5.4034 0.6504 2.6546 2.9250 -1.0617 9.4887

.6 2 9.4644 2.5897 5.8903 4.2677 0.8902 2.4148 3.1766 -1.7964 10.2234

.5 2 8.8441 3.3977 5.0823 3.7262 1.1962 2.1088 3.0342 -2.9888 11.4158

.4 2 7.2554 4.2062 4.2738 3.0466 1.5301 1.7749 2.3582 -5.3598 13.7868

.3 2 5.2124 5.1395 3.3405 2.2784 1.8734 1.4316 2.8951 -7.2029 15.6299

.2 2 3.6394 6.1306 2.3494 1.5677 2.2982 1.0068 3.4017 -10.4213 18.8483

.1 2 1.6257 7.2331 1.2469 0.7444 2.7814 .5236 2.6712 -17.2018 25.6288

.9 1.5 15.9263 0.4541 5.9368 7.3321 0.1979 2.2179 -17.0065 -0.4630 28.683

Table A.24 (RFC+AEXP) Results and compared with the results of each of (RFC) and (AEXP companding)

AEXP I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

2 4 4 1.9702 3.0729 11.1109 0.3967 1.3084 4.7566 7.5446 -1.5141 7.1854

1.9 4 4 2.2773 3.4371 10.7467 0.5200 1.4450 4.62 6.7516 -2.2771 7.9484

1.8 4 4 1.8198 3.8053 10.3785 0.4753 1.6218 4.4432 7.3982 -1.7885 7.4598

1.7 4 4 1.2434 4.2099 9.9739 0.4379 1.7329 4.3321 6.4435 -2.3352 8.0065

1.6 4 4 1.1310 4.6186 9.5652 0.4047 1.8847 4.1803 6.2825 -2.2462 7.9175

1.5 4 4 0.9692 5.0866 9.0972 0.3714 2.0784 3.9866 6.8105 -1.8320 7.5033


A. 29

1.4 4 4 0.9623 5.5316 8.6522 0.3238 2.2423 3.8227 7.1086 -1.7886 7.4599

1.3 4 4 0.7656 5.9679 8.2159 0.3317 2.4167 3.6483 7.0868 -1.5419 7.2132

1.2 4 4 0.7023 6.4361 7.7477 0.0659 2.3249 3.7401 7.6358 -2.0229 7.6942

1.1 4 4 0.6813 6.9128 7.2710 0.3208 2.7863 3.2787 6.8206 -2.3581 8.0294

1 4 0.6085 7.4149 6.7689 0.2994 2.9904 3.0746 7.4259 -2.2028 7.8741

.9 4 0.4981 7.9319 6.2519 0.3327 3.2057 2.8593 7.5934 -3.0353 8.7066

.8 4 0.3743 8.4775 5.7063 0.2977 3.4427 2.6223 10.0902 -3.4835 9.1548

.7 4 0.3806 9.0391 5.1447 0.2528 3.6808 2.3842 19.5598 -4.7689 10.4402

.6 4 0.2386 9.6332 4.5506 0.2178 3.9428 2.1222 18.5802 -5.7485 11.4198

.5 4 0.2162 10.2656 3.9182 0.2695 4.2270 1.838 15.5774 -8.7513 14.4226

.4 4 0.1528 10.9327 3.2511 0.2031 4.5281 1.5369 0 -24.3287 >30

2 3 2.8409 1.5788 10.2402 0.8554 0.5661 4.2979 6.6001 -2.0914 8.1299

1.9 3 3.1584 1.9534 9.8656 0.9622 0.6862 4.1778 6.4858 -2.1757 8.2142

1.8 3 2.6683 2.2890 9.5300 0.8424 0.7879 4.0761 6.5050 -2.3145 8.353

1.7 3 2.0759 2.6776 9.1414 0.8486 0.9426 3.9214 6.2358 -2.1757 8.2142

1.6 3 1.9290 3.0518 8.7672 0.8007 1.0797 3.7843 5.9424 -2.2191 8.2576

1.5 3 1.6787 3.4313 8.3877 0.6796 1.1856 3.6784 5.9209 -2.3544 8.3929

1.4 3 1.6218 3.8263 7.9927 0.6111 1.3286 3.5354 6.0119 -2.5181 8.5566

1.3 3 1.4161 4.2536 7.5654 0.6129 1.4969 3.3671 5.7748 -2.4867 8.5252

1.2 3 1.2859 4.6549 7.1641 0.5822 1.6402 3.2238 6.7358 -2.5557 8.5942

1.1 3 1.2248 5.0915 6.7275 0.5470 1.8115 3.0525 6.2308 -2.5807 8.6192

1 3 1.1195 5.5611 6.2579 0.5258 2.0158 2.8482 6.3021 -2.9594 8.9979

.9 3 0.9545 6.0235 5.7955 0.5417 2.2137 2.6503 7.0545 -3.2070 9.2455

.8 3 0.7515 6.4899 5.3291 0.5069 2.4509 2.4131 9.2588 -3.9477 9.9862

.7 3 0.7338 7.0275 4.7915 0.4166 2.6436 2.2204 19.1217 -4.8398 10.8783

.6 3 0.5422 7.5720 4.2470 0.3589 2.8829 1.9811 17.5183 -6.4432 12.4817

.5 3 0.4644 8.1490 3.6700 0.4074 3.1639 1.7001 1.3672 -22.5943 28.6328

.4 3 0.3578 8.7729 3.0461 0.3188 3.4428 1.4212 0 -23.9615 >30

2 2 4.4446 -0.1565 8.6365 1.8041 -0.0442 3.3492 4.4356 -1.8674 10.2944


A. 30

1.9 2 4.7584 0.2144 8.2656 1.9134 0.0784 3.2266 2.6868 -3.5862 12.0132

1.8 2 4.2805 0.5622 7.9178 1.7817 0.1682 3.1368 2.7087 -3.7223 12.1493

1.7 2 3.5883 0.8510 7.6290 1.6806 0.2156 3.0894 1.2358 -4.7872 13.2142

1.6 2 3.4069 1.1907 7.2893 1.5898 0.3098 2.9952 0.5798 -5.1932 13.6202

1.5 2 3.2267 1.6403 6.8397 1.5115 0.4585 2.8465 0.8877 -4.9991 13.4261

1.4 2 3.0173 1.8828 6.5972 1.3648 0.5233 2.7817 0.9430 -5.1985 13.6255

1.3 2 2.7391 2.2376 6.2424 1.3171 0.6421 2.6629 0.6155 -5.2575 13.6845

1.2 2 2.6453 2.6753 5.8047 1.3019 0.8009 2.5041 1.5764 -5.3266 13.7536

1.1 2 2.5080 3.0357 5.4443 1.2229 0.9284 2.3766 1.1190 -5.3040 13.731

1 2 2.2867 3.3893 5.0907 1.1397 1.0707 2.2343 0.2170 -6.6560 15.083

.9 2 2.0616 3.7916 4.6884 1.1004 1.2134 2.0916 2.2184 -5.6546 14.0816

.8 2 1.7864 4.1858 4.2942 0.9863 1.3713 1.9337 3.6618 -7.1562 15.5832

.7 2 1.6340 4.5887 3.8913 0.8742 1.5422 1.7628 12.9485 -8.6245 17.0515

.6 2 1.3458 5.0366 3.4434 0.7287 1.6937 1.6113 1.6400 -19.9330 28.36

.5 2 1.1395 5.4851 2.9949 0.7272 1.9247 1.3803 0 -21.5730 >30

.4 2 0.9063 5.9824 2.4976 0.5759 2.1409 1.1641 0 -21.5730 >30

2 1.5 5.8486 -0.8416 7.2325 2.6752 -0.0623 2.4781 -15.2700 -1.7800 >30

Table A.25 (RFC+ cos) Results and compared with the results of each of (RFC) and (cos companding)

Cos y I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

1 4 4 3.3647 1.9879 12.1959 1.4804 0.6954 5.3696 5.9268 -0.2019 5.8732

.9 4 2.3785 2.1617 12.0221 1.4216 1.0706 4.9944 6.1069 -0.4468 6.1181

.8 4 2.4468 3.5211 10.6627 1.2605 1.4605 4.6045 5.9421 -0.9666 6.6379

.7 4 2.3786 4.8932 9.2906 1.1151 1.8986 4.1664 6.1775 -1.5352 7.2065

.6 4 2.0945 5.9725 8.2113 0.9266 2.3886 3.6764 6.4601 -2.4071 8.0784

.5 4 1.6828 7.0976 7.0862 0.7674 2.8774 3.1876 6.2525 -3.9362 9.6075

.4 4 1.2488 8.2264 5.9574 0.6048 3.3908 2.6742 6.2306 -5.6581 11.3294


A. 31

.3 4 1.0157 9.6236 4.5602 0.5335 3.9985 2.0665 7.6706 -7.9581 13.6294

.2 4 11.0172 3.1666 4.6068 1.4582 >-23.9287 >30

.1 4 12.5275 1.6563 5.3006 .7644 >>-

23.9287

>>30

1 3 5.3429 1.6013 10.2177 2.3797 0.3937 4.4703 5.5047 -0.2568 6.2953

.9 3 4.9243 2.3427 9.4763 2.2367 0.6847 4.1793 5.4913 -0.6952 6.7337

.8 3 4.4042 3.1137 8.7053 2.0116 1.0106 3.8534 5.5914 -0.9501 6.9886

.7 3 3.7962 3.9460 7.8730 1.8062 1.3887 3.4753 5.4577 -1.8878 7.9263

.6 3 3.3283 4.8415 6.9775 1.4903 1.7513 3.1127 5.9005 -2.5995 8.638

.5 3 2.7294 5.7794 6.0396 1.2001 2.1091 2.7549 5.6383 -4.1832 10.2217

.4 3 2.1840 6.7968 5.0222 1.0164 2.6014 2.2626 5.8226 -5.6989 11.7374

.3 3 1.6589 7.9020 3.9170 0.8270 3.0910 1.773 7.1450 -8.1165 14.155

.2 3 9.0885 2.7305 3.6135 1.2505 -19.2430 25.2815

.1 3 10.3771 1.4419 4.2062 .6578 -23.5615 29.6

1 2 8.4031 1.3225 7.1575 3.7063 0.1613 3.1437 3.3286 -0.0444 8.4714

.9 2 7.7215 1.8009 6.6791 3.4932 0.3822 2.9228 3.1239 -0.6741 9.1011

.8 2 6.8456 2.2161 6.2639 3.1303 0.5703 2.7347 2.5218 -1.6312 10.0582

.7 2 6.0573 2.8681 5.6119 2.7832 0.8067 2.4983 3.0995 -1.8575 10.2845

.6 2 5.2889 3.4631 5.0169 2.3470 1.0490 2.256 2.6977 -3.4138 11.8408

.5 2 4.4202 4.1312 4.3488 2.0022 1.3522 1.9528 3.0493 -4.3837 12.8107

.4 2 3.5833 4.8571 3.6229 1.6358 1.6618 1.6432 2.9223 -6.2107 14.6377

.3 2 2.6995 5.6036 2.8764 1.3013 2.0063 1.2987 4.0627 -8.8103 17.2373

.2 2 6.4762 2.0038 2.3977 .9073 -21.1730 29.6

.1 2 7.4125 1.0675 2.8196 .4854 -21.1730 29.6

1 1.5 10.1993 1.0296 5.3613 4.5714 0.1372 2.2786 -17.8000 -1.3800 29.6


A. 32

Table A.26 (RFC+NERF) Results and compared with the results of each of (RFC) and (NERF companding)

NERF I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

4 4 0.9191 5.2328 9.0903 0.4355 2.4177 3.8875 3.2135 -1.6966 7.2096

4 3 1.6395 3.7342 8.2522 0.8685 1.7880 3.4578 2.7780 -1.4860 7.7814

4 2 2.7940 2.2218 6.9153 1.5235 1.2096 2.5844 -2.3220 -3.8720 11.883

4 1.75 3.7575 6.4220 2.1590 2.2765 -5.7976 19.4756

4 1.5 3.0791 1.2825 5.7098 1.8955 1.0857 1.9541 -16.3220 -0.4000 >30

Table A.27 (RFC+tanhR) Results and compared with the results of each of (RFC) and (tanhR companding)

k y I CR X X1 PAPR Y Y1 CCDF

OF

PAPR

Z Z1 SNR

(BER=

)

5 1 4 4 2.6944 0.1653 14.0185 1.2590 -0.0836 6.1486 6.3440 -0.1141 5.7854

5 .8 4 4 2.1182 2.4548 11.7290 0.8791 0.8860 5.179 6.2949 -0.7050 6.3763

5 .5 4 4 1.2755 6.5550 7.6288 0.4358 2.6147 3.4503 6.8333 -3.0831 8.7544

5 .2 4 4 0.6910 11.3302 2.8536 0.2234 4.7609 1.3041 7.2231 -11.8287 17.5

10 1 4 -4.5995 0.6531 13.5307 -1.7440 0.1120 5.953 8.8688 -0.1085 5.7798

10 .8 4 4 -2.9960 3.0398 11.1440 -1.2074 1.0997 4.9653 8.7788 -0.4820 6.1533

10 .5 4 4 -0.6902 7.0939 7.0899 -0.3799 2.8570 3.208 8.5310 -3.4266 9.0979

10 .2 4 0.4108 11.5444 2.6394 0.1330 4.8529 1.2121 8.5822 -11.7624 17.4337

15 1 4 4 -7.5448 1.4076 12.7762 -3.0363 0.4300 5.635 23.6963 -0.2324 5.9037

15 .8 4 4 -5.2051 3.7358 10.4480 -2.1608 1.3777 4.6873 15.9471 -0.8472 6.5185

15 .5 4 4 -1.6597 7.5822 6.6016 -0.7619 3.0710 2.994 11.7960 -4.0074 9.6787

15 .2 4 4 0.2390 11.6791 2.5047 0.0530 4.9153 1.1497 8.9374 -11.9489 17.6202

20 1 4 4 -8.4594 2.3463 11.8375 -3.4260 0.8162 5.2488 23.5747 -0.3540 6.0253

20 .8 4 4 -6.0915 4.4973 9.6865 -2.5352 1.6886 4.3764 22.8019 -1.1268 6.7981

20 .5 4 4 -2.7713 7.6364 6.5474 -1.0317 3.2438 2.8212 19.6576 -4.2711 9.9424


A. 33

20 .2 4 4 0.1722 11.7850 2.3988 0.0269 4.9604 1.1046 10.7485 -12.5194 18.1907

30 1 4 4 4.3979 9.7859 1.5013 4.5637 -1.1109 6.7822

30 .8 4 5.9931 8.1907 2.2362 3.8288 -1.9585 7.6298

30 .5 4 8.7585 5.4253 3.5268 2.5382 -4.5051 10.1764

30 .2 4 11.9430 2.2408 5.0263 1.0387 -12.8835 18.5548

40 1 4 6.3037 7.8801 2.2698 3.7952 -1.9663 7.6376

40 .8 4 7.3308 6.8530 2.7876 3.2774 -2.6475 8.3188

40 .5 4 9.3689 4.8149 3.7886 2.2764 -5.6472 11.3185

40 .2 4 12.0608 2.1230 5.0792 .9858 -13.2070 18.8783

5 1 3 5.0723 0.1784 11.6406 2.4801 -0.0635 4.9275 5.9113 -0.1796 6.2181

5 .8 3 4.0340 2.0058 9.8132 1.8728 0.6787 4.1853 5.8850 -0.7477 6.7862

5 .5 3 2.4146 5.3293 6.4897 1.0524 2.0303 2.8337 6.4305 -3.1187 9.1572

5 .2 3 1.0832 9.3576 2.4614 0.4255 3.7620 1.102 6.8634 -11.8212 17.8597

10 1 3 -2.4226 0.4652 11.3538 -0.6289 0.0261 4.8379 8.6281 0.0180 6.0205

10 .8 3 -1.2989 2.3721 9.4469 -0.3228 0.7833 4.0807 8.4958 -0.3978 6.4363

10 .5 3 0.3078 5.7271 6.0919 0.1424 2.1783 2.6857 7.5664 -4.0240 10.0625

10 .2 3 0.7564 9.5252 2.2938 0.3116 3.8305 1.0335 6.5865 -13.3909 19.4294

15 1 3 -5.6505 0.9371 10.8819 -2.0568 0.2085 4.6555 23.2474 -0.3141 6.3526

15 .8 3 -3.7325 2.8436 8.9754 -0.3544 1.9831 2.8809 15.3959 -1.0312 7.0697

15 .5 3 -0.7879 6.0892 5.7298 -0.2991 2.3328 2.5312 11.6768 -3.7594 9.7979

15 .2 3 0.5623 9.6376 2.1814 0.2234 3.8847 .9793 8.0920 -12.4271 18.4656

20 1 3 -6.7832 1.6577 10.1613 -2.6056 0.4356 4.4284 22.9245 -0.6370 6.6755

20 .8 3 -4.7701 3.4539 8.3651 -1.8404 1.1824 3.6816 22.1648 -1.3967 7.4352

20 .5 3 -1.5784 6.4645 5.3545 -0.6063 2.4682 2.3958 19.1135 -4.4480 10.4865

20 .2 3 0.4900 9.7380 2.0810 0.1826 3.9151 .9489 9.8503 -13.0504 19.0889

30 1 3 3.0272 8.7918 0.9312 3.9328 -0.8914 6.9299

30 .8 3 4.4995 7.3195 1.5702 3.2938 -2.1879 8.2264

30 .5 3 7.0108 4.8082 2.6746 2.1894 -5.1420 11.1805

30 .2 3 9.8546 1.9644 3.9733 .8907 -12.8308 18.8693


A. 34

40 1 3 4.5177 7.3013 1.5349 3.3291 -2.1879 8.2264

40 .8 3 5.5636 6.2554 2.0098 2.8542 -2.9035 8.942

40 .5 3 7.5154 4.3036 2.9066 1.9574 -6.1790 12.2175

40 .2 3 9.9583 1.8607 4.0222 .8418 -13.4003 19.4388

50 1 3 5.8206 5.9984 2.0640 2.8 -3.7195 9.758

50 .8 3 6.4840 5.3350 2.3953 2.4687 -4.0708 10.1093

50 .5 3 7.9373 3.8817 3.0958 1.7682 -7.0648 13.1033

50 .2 3 10.0415 1.7775 4.0640 .8 -14.0866 20.1251

5 1 2 8.0411 -0.1918 8.6718 4.0083 -0.0943 3.3993 3.4383 -0.2641 8.6911

5 .8 2 6.4911 1.1239 7.3561 3.1445 0.3914 2.9136 3.3481 -0.8961 9.3231

5 .5 2 3.9618 3.5375 4.9425 1.8462 1.2651 2.0399 3.0606 -4.1001 12.5271

5 .2 2 1.6309 6.5663 1.9137 0.7088 2.4863 .8187 3.0185 -13.2776 21.7046

10 1 2 0.6809 0.2297 8.2503 0.9295 0.0255 3.2795 6.0106 -0.2110 8.638

10 .8 2 1.2029 1.5349 6.9451 0.9545 0.5016 2.8034 5.1929 -1.3122 9.7392

10 .5 2 1.8224 3.9027 4.5773 0.9068 1.3837 1.9213 4.9274 -4.2745 12.7015

10 .2 2 1.2947 6.7245 1.7555 0.5888 2.5487 .7563 4.2034 -13.3855 21.8125

15 1 2 -2.8631 0.3855 8.0945 -0.6314 0.0749 3.2301 20.9592 -0.2138 8.6408

15 .8 2 -1.5051 1.7320 6.7480 -0.2065 0.5720 2.733 12.6710 -1.3676 9.7946

15 .5 2 0.5603 4.0984 4.3816 0.4052 1.4781 1.8269 8.2439 -4.8038 13.2308

15 .2 2 1.0594 6.7957 1.6843 0.4813 2.5836 .7214 3.5147 -14.6159 23.0429

20 1 2 -4.3947 0.7072 7.7728 -1.3229 0.1593 3.1457 20.4398 -0.7332 9.1602

20 .8 2 -2.8497 2.0353 6.4447 -0.8356 0.6282 2.6768 19.6499 -1.5231 9.9501

20 .5 2 -0.3872 4.3167 4.1633 0.0055 1.5210 1.784 16.6531 -4.5199 12.9469

20 .2 2 0.9528 6.8618 1.6182 0.4239 2.5974 .7076 5.9826 -14.5296 22.9566

30 1 2 1.4478 7.0322 0.3891 2.9159 -1.0211 9.4481

30 .8 2 2.6476 5.8324 0.8513 2.4537 -2.8904 11.3174

30 .5 2 4.6783 3.8017 1.6656 1.6394 -6.2576 14.6846


A. 35

30 .2 2 6.9505 1.5295 2.6311 .6739 -16.7147 25.1417

40 1 2 2.2898 6.1902 0.6198 2.6852 -2.3539 10.7809

40 .8 2 3.2772 5.2028 1.0325 2.2725 -3.5287 11.9557

40 .5 2 4.9997 3.4803 1.7771 1.5279 -7.5339 15.9609

40 .2 2 7.0196 1.4604 2.6697 .6353 -16.0484 24.4754

5 1 1.5 10.4523 0.1303 6.2606 4.9898 -0.0020 2.4178 -16.3460 -0.2554 28.4754

Table A.28 (RFC+logR) Results and compared with the results of each of (RFC) and (logR companding)

k y I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

5 1 4 4 0.7014 13.4824 0.1402 5.9248 -0.0429 5.7142

5 .8 4 3.0106 11.1732 1.1054 4.9596 -0.7108 6.3821

5 .5 4 6.9250 7.2588 2.7955 3.2695 -3.5314 9.2027

5 .2 4 11.2475 2.9363 4.7255 1.3395 -11.5564 17.2277

10 1 4 2.7526 0.1944 13.9894 1.5031 0.2536 5.8114 6.6311 -0.1976 5.8689

10 .8 4 4.0614 3.5113 10.6725 1.6195 1.3070 4.758 6.5858 -0.9429 6.6142

10 .5 4 2.7840 7.2254 6.9584 1.1077 2.9327 3.1323 6.4264 -3.6523 9.3236

10 .2 4 1.2286 11.3191 2.8647 0.4931 4.7581 1.3069 6.4737 -11.5050 17.1763

20 1 4 2.3809 2.1476 12.0362 0.9625 0.7365 5.3285 7.5177 -0.4270 6.0983

20 .8 4 2.0822 3.4269 10.7569 1.0705 1.6055 4.4595 7.1717 -0.9950 6.6663

20 .5 4 1.8293 7.0989 7.0849 0.9371 3.0741 2.9909 6.6767 -4.0852 9.7565

20 .2 4 1.2155 11.2030 2.9808 0.6147 4.7877 1.2773 6.5743 -11.6544 17.3257

30 1 4 2.3488 2.8223 11.3615 0.7103 1.0153 5.0497 8.5090 -0.5580 6.2293

30 .8 4 1.9860 4.6558 9.5280 0.7510 1.7960 4.269 7.5945 -1.4342 7.1055

30 .5 4 2.0414 7.7912 6.3926 0.7017 3.1667 2.8983 7.3627 -4.0488 9.7201

30 .2 4 1.0410 11.4330 2.7508 0.4266 4.8056 1.2594 6.3249 -11.7838 17.4551

40 1 4 1.6240 3.3488 10.8350 0.5382 1.1862 4.8788 8.9930 -1.0222 6.6935


A. 36

40 .8 4 1.6777 5.0348 9.1490 0.5574 1.9224 4.1426 8.7135 -1.6152 7.2865

40 .5 4 2.1372 7.9675 6.2163 0.7850 3.2300 2.835 7.8613 -4.0794 9.7507

40 .2 4 1.1544 11.4694 2.7144 0.4373 4.8238 1.2412 6.3458 -11.8697 17.541

50 1 4 1.2444 3.7747 10.4091 0.4157 1.4007 4.6643 10.3272 -0.5815 6.2528

50 .8 4 1.4152 5.3297 8.8541 0.5253 2.0983 3.9667 9.0464 -1.5423 7.2136

50 .5 4 1.6948 8.0978 6.0860 0.6536 3.3066 2.7584 6.9655 -4.7872 10.4585

50 .2 4 1.0611 11.4924 2.6914 0.4371 4.8341 1.2309 6.8945 -11.8342 17.5055

70 1 4 0.6110 4.4392 9.7446 0.2321 1.7051 4.3599 12.8556 -1.4731 7.1444

70 .8 4 1.3851 5.7973 8.3865 0.5107 2.2757 3.7893 10.3224 -1.7313 7.4026

70 .5 4 1.7977 8.3065 5.8773 0.6907 3.3897 2.6753 7.7414 -4.7153 10.3866

70 .2 4 1.3230 11.5315 2.6523 0.4884 4.8484 1.2166 6.1660 -12.0427 17.7149

90 1 4 0.4691 4.9538 9.2300 0.2755 1.9205 4.1445 22.5168 -1.8119 7.4832

90 .8 4 1.0355 6.1467 8.0371 0.3334 2.4484 3.6166 12.1155 -2.2132 7.8845

90 .5 4 1.3705 8.4617 5.7221 0.5281 3.4691 2.5959 8.8508 -4.4593 10.1306

90 .2 4 1.0737 11.5599 2.6239 0.4024 4.8674 1.1976 6.5459 -12.1428 17.8141

5 1 3 0.5230 11.2960 0.0727 4.7913 -0.2954 6.3339

5 .8 3 2.4889 9.3301 0.8322 4.0318 -1.0780 7.1165

5 .5 3 5.6872 6.1318 2.1320 2.732 -3.8176 9.8561

5 .2 3 9.3116 2.5074 3.7351 1.1289 -11.9709 18.0094

10 1 3 5.9143 0.9913 10.8277 2.6999 0.2494 4.6146 6.2141 -0.2474 6.2859

10 .8 3 5.6943 2.7794 9.0396 2.4796 0.9661 3.8979 6.3045 -0.8570 6.8955

10 .5 3 3.8312 5.9078 5.9112 1.6309 2.2549 2.6091 5.9720 -3.7395 9.778

10 .2 3 1.6255 9.3512 2.4678 0.7015 3.7655 1.0985 5.5242 -12.0873 18.1258

20 1 3 4.2590 1.6609 10.1581 1.9290 0.5020 4.362 7.1443 -0.4332 6.4717

20 .8 3 4.4077 3.3876 8.4314 1.8493 1.1833 3.6807 6.7307 -1.0688 7.1073

20 .5 3 3.2711 6.1759 5.6431 1.4476 2.3836 2.4804 6.3066 -4.0881 10.1266

20 .2 3 1.8024 9.4251 2.3939 0.8319 3.8039 1.0601 6.2466 -11.6149 17.6534

30 1 3 4.0813 2.1900 9.6290 1.5873 0.6913 4.1727 7.9106 -0.7892 6.8277

30 .8 3 3.4342 3.7392 8.0798 1.4834 1.3274 3.5366 6.9501 -1.7114 7.7499


A. 37

30 .5 3 2.9839 6.3689 5.4501 1.2057 2.4697 2.3943 6.8281 -4.2162 10.2547

30 .2 3 1.4195 9.4467 2.3723 0.6260 3.8040 1.06 5.7628 -11.9787 18.0172

40 1 3 3.1671 2.5271 9.2919 1.3860 0.8330 4.031 8.5830 -1.0650 7.1035

40 .8 3 3.0396 4.0319 7.7871 1.2730 1.4370 3.427 8.2716 -1.6899 7.7284

40 .5 3 3.0029 6.4684 5.3506 1.2607 2.5047 2.3593 6.8099 -4.7636 10.8021

40 .2 3 1.5260 9.4762 2.3428 0.6341 3.8196 1.0444 5.7427 -12.1056 18.1441

50 1 3 2.7298 2.8953 8.9237 1.2264 1.0104 3.8536 9.2743 -1.2672 7.3057

50 .8 3 2.6866 4.2363 7.5827 1.1700 1.5420 3.322 8.1233 -2.0982 8.1367

50 .5 3 2.5838 6.6220 5.1970 1.1024 2.5544 2.3096 6.8099 -4.5756 10.6141

50 .2 3 1.4481 9.5146 2.3044 0.6294 3.8254 1.0386 6.4055 -11.9560 17.9945

70 1 3 1.9549 3.4183 8.4007 0.9412 1.2132 3.6508 12.6132 -1.3483 7.3868

70 .8 3 2.6206 4.6680 7.1510 1.1170 1.6810 3.183 9.5620 -2.1245 8.163

70 .5 3 2.6296 6.7736 5.0454 1.1271 2.6251 2.2389 6.9099 -5.1796 11.2181

70 .2 3 1.7008 9.5445 2.2745 0.6822 3.8412 1.0228 5.7203 -12.1212 18.1597

90 1 3 1.7931 3.9130 7.9060 0.9073 1.3513 3.5127 22.0765 -1.8850 7.9235

90 .8 3 2.1595 4.9059 6.9131 0.9197 1.8337 3.0303 11.0954 -2.8661 8.9046

90 .5 3 2.1755 6.9019 4.9171 0.9443 2.6843 2.1797 7.7612 -5.1817 11.2202

90 .2 3 1.4527 9.5741 2.2449 0.5923 3.8563 1.0077 6.0773 -12.2442 18.2827

5 1 2 0.2534 8.2266 -0.0018 3.3068 0.1415 8.2855

5 .8 2 1.5881 6.8919 0.5126 2.7924 -0.8028 9.2298

5 .5 2 3.8544 4.6256 1.3946 1.9104 -4.3434 12.7704

5 .2 2 6.5880 1.8920 2.4961 .8089 -13.2386 21.6656

10 1 2 8.8239 0.5619 7.9181 4.1088 0.0993 3.2057 3.8634 -0.2096 8.6366

10 .8 2 8.1036 1.8497 6.6303 3.6671 0.5946 2.7104 3.3864 -1.3866 9.8136

10 .5 2 5.3311 4.0687 4.4113 2.3911 1.4561 1.8489 2.5828 -4.7402 13.1672

10 .2 2 2.2409 6.6276 1.8524 1.0076 2.5126 .7924 2.6103 -12.6127 21.0397

20 1 2 6.7883 0.8512 7.6288 3.2716 0.2856 3.0194 4.5313 -0.6577 9.0847

20 .8 2 6.4508 2.0917 6.3883 2.9631 0.7381 2.5669 3.2202 -2.1908 10.6178


A. 38

20 .5 2 4.6377 4.2035 4.2765 2.1625 1.5395 1.7655 2.9487 -5.0575 13.4845

20 .2 2 2.3661 6.6498 1.8302 1.1165 2.5295 .7755 2.0485 -13.4245 21.8515

30 1 2 6.3349 1.1046 7.3754 2.8383 0.3833 2.9217 4.4838 -1.8275 10.2545

30 .8 2 5.3149 2.2809 6.1991 2.5214 0.8064 2.4986 3.8255 -2.4475 10.8745

30 .5 2 4.2482 4.2942 4.1858 1.8696 1.5746 1.7304 3.0074 -5.6484 14.0754

30 .2 2 1.9742 6.6624 1.8176 0.9202 2.5392 .7658 1.9224 -13.4306 21.8576

40 1 2 5.2026 1.2236 7.2564 2.5925 0.4805 2.8245 5.9716 -1.2879 9.7149

40 .8 2 4.7025 2.3558 6.1242 2.2848 0.8898 2.4152 4.2817 -3.2913 11.7183

40 .5 2 4.3637 4.4902 3.9898 1.9292 1.6142 1.6908 3.3653 -5.8197 14.2467

40 .2 2 2.0398 6.6510 1.8290 0.9285 2.5550 .75 1.5957 -13.8641 22.2911

50 1 2 4.9773 1.8038 6.6762 2.3391 0.5641 2.7409 6.0264 -2.1266 10.5536

50 .8 2 4.5914 2.8021 5.6779 2.1381 0.9511 2.3539 4.6848 -3.1482 11.5752

50 .5 2 3.8064 4.5056 3.9744 1.7476 1.6406 1.6644 3.3485 -5.6485 14.0755

50 .2 2 1.9923 6.7198 1.7602 0.9215 2.5585 .7465 2.5252 -13.4478 21.8748

70 1 2 3.9401 2.0645 6.4155 1.9476 0.6606 2.6444 7.8528 -3.7202 12.1472

70 .8 2 4.3097 3.0181 5.4619 2.0283 1.0333 2.2717 5.0058 -4.2922 12.7192

70 .5 2 3.8191 4.6241 3.8559 1.7605 1.6995 1.6055 3.1049 -6.5961 15.0231

70 .2 2 2.1731 6.6778 1.8022 0.9599 2.5599 .7451 2.0422 -13.4108 21.8378

90 1 2 3.5714 2.3523 6.1277 1.9041 0.7891 2.5159 16.2341 -5.3389 13.7659

90 .8 2 3.7503 3.1577 5.3223 1.7519 1.1069 2.1981 7.4641 -4.1089 12.5359

90 .5 2 3.3643 4.7517 3.7283 1.5721 1.7531 1.5519 3.3538 -7.2006 15.6276

90 .2 2 1.9762 6.7586 1.7214 0.8578 2.5628 .7422 1.4370 -14.4960 22.923

5 1 1.5 0.0422 6.3487 0.1114 2.3044 8.7803 19.4397


A. 39

A.7 Hybrids RCF with companding Results:

X = PAPR (Pre-coding ) – PAPR (Pre-coding +RCF)

Y =CCDF of PAPR (Pre-coding) - CCDF of PAPR (Pre-coding +RCF)

Z= SNR (BER= ) (Pre-coding) – SNR (BER= ) (Pre-coding +RCF)

X1 == PAPR (RCF) – PAPR (Pre-coding +RCF)

Y1 =CCDF of PAPR (RCF) - CCDF of PAPR (Pre-coding +RCF)

Z1= SNR (BER= ) (RCF) – SNR (BER= ) (Pre-coding +RCF)

Table A.29 (Pre-coding +RCF) Results and compared with the results of each of (Pre-coding) and (RCF).

Pre-

coding

I CR X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER=

)

WHT 1 4 8.8659 0.1148 13.9718 3.8833 0.0031 6.0213 -0.1058 0.0842 11.7358

WHT 1 3 11.4801 0.2126 11.3576 5.1262 0.0066 4.7784 -0.6262 0.1149 12.2562

WHT 1 2 14.8677 0.2786 7.9700 6.8155 0.0295 3.0891 -2.5798 0.1658 14.2098

WHT 1 1.5 16.6813 0.2274 6.1564 7.8561 0.8515 2.0485 -13.3450 4.5250 24.975

WHT 1 1.3 17.6205 5.2172 8.2340 1.6706 -18.3700 >30

WHT 1 1.1 18.3732 4.4645 8.6977 1.2069 -18.3700 >30

DCT 1 4 4.2276 0.2032 13.8834 1.6434 0.0038 6.0206 0.0853 0.2773 11.5427

DCT 1 3 6.9879 0.4471 11.1231 2.8918 0.0128 4.7722 0.0641 0.8072 11.5639

DCT 1 2 10.3525 0.4901 7.7585 4.6089 0.0635 3.0551 -1.8862 0.8614 13.5142

DCT 1 1.5 12.1038 0.3766 6.0072 5.6285 0.8645 2.0355 -5.2472 12.6248 16.8752

DCT 1 1.3 12.7949 5.3161 6.0657 1.5983 -10.2117 21.8397

DCT 1 1.1 13.3286 4.7824 6.4410 1.223 -18.3720 >30

DST 1 4 3.5817 0.2034 13.8832 1.5024 0.0038 6.0206 -0.1290 0.0630 11.757

DST 1 3 6.3420 0.4473 11.1229 2.7508 0.0128 4.7722 -0.3210 0.4221 11.949

DST 1 2 9.7467 0.5304 7.7182 4.4656 0.0612 3.0574 -1.4552 1.2924 13.0832


A. 40

DST 1 1.5 11.6978 0.6167 5.7671 5.4920 0.8690 2.031 -5.1509 12.7211 16.7789

DST 1 1.3 12.7190 4.7459 5.9291 1.5939 -10.4894 22.1174

DST 1 1.1 13.0431 4.4218 6.3389 1.1841 -18.3720 >30

DHT 1 4 0.0069 7.1348 6.9518 0.3397 3.0141 3.0103 0.0759 0.3139 11.5061

DHT 1 3 0.0115 4.6230 6.9472 0.3397 1.7747 3.0103 0.0272 0.8163 11.5548

DHT 1 2 0.0008 1.2907 6.9579 0.3397 0.1083 3.0103 -0.0615 2.7321 11.6435

DHT 1 1.5 1.7911 1.2162 5.1676 1.4546 1.0046 1.8954 -1.0130 16.9050 12.595

DHT 1 1.3 2.3945 4.5642 1.6229 1.7271 -1.4779 13.0599

DHT 1 1.1 0.0323 6.9264 1.4687 1.8813 -1.5015 13.0835

DCT Pilot 4 4.1673 0.2536 13.9437 1.5259 0.1099 6.1381 2.4153 0.1473 9.2127

DCT Pilot 3 6.7675 0.5734 11.3435 2.6731 0.2341 4.9909 2.0345 0.2480 9.5935

DCT Pilot 2 9.3391 0.6713 8.7719 3.9837 0.1993 3.6803 -1.2236 0.1484 12.8516

DCT Pilot 1.5 10.5003 0.4001 7.6107 4.5440 0.2368 3.12 -10.8087 7.1633 22.4367

DST Pilot 4 3.5404 0.2728 13.9245 1.3769 0.1019 6.1461 2.2559 -0.0121 9.3721

DST Pilot 3 6.1023 0.5543 11.3626 2.4963 0.1983 5.0267 1.7912 0.0047 9. 8368

DST Pilot 2 8.8583 0.8366 8.6066 3.8617 0.2183 3.6613 -1.1227 0.2493 12.7507

DST Pilot 1.5 9.9215 0.4674 7.5434 4.4768 0.3106 3.0462 -8.0414 9.9306 19.6694

DHT Pilot 4 -6.9472 0.2914 13.9059 -2.7662 0.1318 6.1162 1.9934 -0.2286 9.5886

DHT Pilot 3 -4.2539 0.7043 11.2126 -1.6302 0.2448 4.9802 1.8664 0.1259 9.7156

DHT Pilot 2 -1.3648 1.1197 8.3235 -0.2838 0.2458 3.6338 0.2611 1.6791 11.3209

DHT Pilot 1.5 -0.5168 0.5353 7.4755 0.1727 0.1795 3.1773 -2.5626 15.4554 14.1446

WHT 2 4 8.4655 0.1210 14.3722 3.5038 0.0169 6.4008 2.8937 0.0316 8.7363

WHT 2 3 10.7834 0.2199 12.0543 4.6331 0.0835 5.2715 1.9358 -0.2802 9.6942

WHT 2 2 13.4396 0.2092 9.3981 5.9192 0.1362 3.9854 -0.8971 -0.3991 12.5271

WHT 2 1.5 14.6473 0.1925 8.1904 6.3965 0.1176 3.5081 -18.3700 -0.4000 >30

DCT 2 4 3.9078 0.2900 14.2032 1.3830 0.1367 6.281 2.7806 -0.0795 8.8474

DCT 2 3 6.3048 0.4680 11.8062 2.4546 0.1456 5.2094 2.3883 0.1743 9.2397

DCT 2 2 9.1329 0.6292 8.9781 3.6787 0.1363 3.9853 -0.3089 0.1911 11.9369

DCT 2 1.5 10.1414 0.4133 7.9696 4.2608 0.2225 3.4032 -7.3951 10.5769 19.0231


A. 41

DST 2 4 3.2174 0.2457 14.2475 1.2382 0.1329 6.2848 2.8864 0.0263 8.7416

DST 2 3 5.7121 0.5214 11.7528 2.3688 0.2008 5.1542 2.6270 0.4130 9.001

DST 2 2 8.4680 0.6104 8.9969 3.6055 0.2041 3.9175 -0.3350 0.1650 11.963

DST 2 1.5 9.5290 0.4470 7.9359 4.1101 0.2128 3.4129 -7.2475 10.7245 18.8755

DHT 2 4 -7.2690 0.2655 14.2277 -2.9015 0.1662 6.2515 2.7701 -0.0440 8.8119

DHT 2 3 -4.7567 0.5588 11.7154 -1.7979 0.2071 5.1479 2.6878 0.5198 8.8942

DHT 2 2 -1.9188 0.7298 8.8775 -0.6022 0.1694 3.9522 0.7115 1.2575 10.8705

DHT 2 1.5 -0.5578 0.8664 7.5165 -0.0681 0.2076 3.4181 -0.8762 17.1418 12.4582

A.8 Hybrid Pre-coding with Companding Results:

X = PAPR (Pre-coding ) – PAPR (Pre-coding + Companding)

Y =CCDF of PAPR (Pre-coding) - CCDF of PAPR(Pre-coding + Companding)

Z= SNR (BER= ) (Pre-coding) – SNR (BER= ) (Pre-coding + Companding)

X1 == PAPR (Companding) – PAPR (Pre-coding + Companding)

Y1 =CCDF of PAPR (Companding) - CCDF of PAPR (Pre-coding + Companding)

Z1= SNR (BER= ) (Companding) – SNR (BER= ) (Pre-coding + Companding)

Table A.30 (Pre-coding +A) Results and compared with the results of each of (Pre-coding) and (A companding).

Precoding A X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

WHT 5 6.4003 2.4687 16.4374 4.2596 0.9950 5.645 -2.2393 -0.2693 13.8693

WHT 10 8.2527 0.1067 14.5850 5.5140 0.3394 4.3906 -4.7003 -0.2103 16.3303

WHT 15 9.0269 0.5157 13.8108 6.0383 0.2597 3.8663 -6.2322 -0.2862 17.8622

WHT 20 9.4809 1.0833 13.3568 6.3435 0.1239 3.5611 -6.6714 -0.3014 18.3014

WHT 30 10.0185 -0.9647 12.8192 6.7024 0.1178 3.2022 -8.0292 -0.4592 19.6592

WHT 35 10.1382 0.2105 12.6995 6.9423 0.4507 2.9623 -8.1607 -0.5907 19.7907

WHT 40 10.2848 0.0499 12.5529 7.0335 0.3889 2.8711 -8.5875 -0.4605 20.2175


A. 42

WHT 50 10.5114 0.2643 12.3263 7.1740 0.3694 2.7306 -10.0027 -1.5167 21.6327

WHT 70 10.8155 0.1281 12.0222 7.3616 0.2940 2.543 -9.5360 -0.0360 21.166

WHT 87.6 10.9967 -0.0591 11.8410 7.4729 0.2913 2.4317 -10.0027 -0.2607 21.6327

WHT 100 11.0811 -0.4023 11.7566 7.5333 0.2087 2.3713 -10.6669 -0.6769 22.2969

WHT 120 11.2105 -0.2171 11.6272 7.6154 0.2458 2.2892 -10.9249 -0.6919 22.5549

DCT 5 6.2514 7.0465 11.8596 3.4636 2.4396 4.2004 -2.4773 -0.5053 14.1053

DCT 10 7.7579 4.3386 10.3531 4.3642 1.4302 3.2998 -4.6072 -0.1152 16.2352

DCT 15 8.3981 4.6136 9.7129 4.7366 1.1986 2.9274 -6.0249 -0.0769 17.6529

DCT 20 8.7766 5.1057 9.3344 4.9509 0.9719 2.7131 -6.7160 -0.3440 18.344

DCT 30 9.5223 3.2658 8.5887 5.2077 0.8637 2.4563 -7.3912 0.1808 19.0192

DCT 35 9.6724 4.4714 8.4386 5.2937 1.0427 2.3703 -7.7982 -0.2262 19.4262

DCT 40 9.5424 4.0342 8.5686 5.4200 1.0160 2.244 -8.0486 0.0804 19.6766

DCT 50 9.9838 4.4634 8.1272 5.4722 0.9082 2.1918 -8.6099 -0.1219 20.2379

DCT 70 9.9556 3.9949 8.1554 5.5693 0.7423 2.0947 -9.0955 0.4065 20.7235

DCT 87.6 10.1086 3.7795 8.0024 5.6578 0.7168 2.0062 -9.5582 0.1858 21.1862

DCT 100 10.6923 3.9356 7.4187 5.7713 0.6873 1.8927 -9.7262 0.2658 21.3542

DCT 120 10.3032 3.6023 7.8078 5.7703 0.6413 1.8937 -10.0745 0.1605 21.7025

DST 5 6.1302 7.5714 11.3347 3.3458 2.4628 4.1772 -2.7785 -0.8065 14.4065

DST 10 7.6475 4.8743 9.8174 4.2613 1.4683 3.2617 -4.8245 -0.3325 16.4525

DST 15 8.2865 5.1481 9.1784 4.6392 1.2422 2.8838 -6.4536 -0.5056 18.0816

DST 20 8.6628 5.6380 8.8021 4.8599 1.0219 2.6631 -7.0850 -0.7130 18.713

DST 30 9.1103 3.4999 8.3546 5.1202 0.9172 2.4028 -8.3413 -0.7693 19.9693

DST 35 8.2173 3.6624 9.2476 5.1115 1.0015 2.4115 -8.2486 -0.6766 19.8766

DST 40 8.3394 3.4773 9.1255 5.1855 0.9225 2.3375 -8.4777 -0.3487 20.1057

DST 50 8.5285 3.6542 8.9364 5.2998 0.8768 2.2232 -9.0263 -0.5383 20.6543

DST 70 8.7829 3.4683 8.6820 5.4526 0.7666 2.0704 -9.6570 -0.1550 21.285

DST 87.6 8.9348 3.2518 8.5301 5.5434 0.7434 1.9796 -10.1584 -0.4144 21.7864

DST 100 9.1724 3.0618 8.2925 5.6106 0.6676 1.9124 -10.2688 -0.2768 21.8968


A. 43

DST 120 9.2816 3.2268 8.1833 5.6740 0.6860 1.849 -10.5360 -0.3010 22.164

DHT 5 2.2752 14.2226 4.6835 0.9293 4.2193 2.4207 -0.8663 1.1517 12.4483

DHT 10 2.6637 10.3967 4.2950 1.0712 2.4512 2.2788 -1.5606 2.9774 13.1426

DHT 15 2.8227 10.1905 4.1360 1.1343 1.9103 2.2157 -2.2378 3.7562 13.8198

DHT 20 2.9158 10.3972 4.0429 1.1712 1.5062 2.1788 -2.7303 3.6877 14.3123

DHT 30 3.0324 7.9282 3.9263 1.1238 1.0938 2.2262 -3.4581 4.1599 15.0401

DHT 35 3.0690 9.0203 3.8897 1.1358 1.1988 2.2142 -3.6582 3.9598 15.2402

DHT 40 3.0988 8.7429 3.8599 1.1456 1.0556 2.2044 -3.7728 4.4022 15.3548

DHT 50 3.1450 8.7769 3.8137 1.1608 0.9108 2.1892 -4.0772 4.4568 15.6592

DHT 70 3.2239 8.4155 3.7348 1.1915 0.6785 2.1585 -4.4893 5.0587 16.0713

DHT 87.6 3.2613 8.0845 3.6974 1.2050 0.5780 2.145 -4.6584 5.1316 16.2404

DHT 100 3.2819 7.6775 3.6768 1.2125 0.4425 2.1375 -4.7606 5.2774 16.3426

DHT 120 3.3088 7.7602 3.6499 1.2224 0.4074 2.1276 -4.9040 5.3770 16.486

Table A.31(Pre-coding + ) Results and compared with the results of each of (Pre-coding) and ( companding).

Precoding X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

WHT 5 6.3192 0.9147 16.5185 4.1987 0.7101 5.7059 -1.8052 -0.0989 13.4352

WHT 10 8.7224 2.4317 14.1153 5.1385 1.0039 4.7661 -3.1978 -0.1878 14.8278

WHT 20 9.9641 4.3837 12.8736 5.8731 1.0465 4.0315 -4.9404 -0.3204 16.5704

WHT 30 8.8772 0.6751 13.9605 6.1212 0.6166 3.7834 -5.8312 -0.2962 17.4612

WHT 40 9.2710 0.2786 13.5667 6.3771 0.4725 3.5275 -6.7931 -0.6731 18.4231

WHT 50 10.2718 1.2963 12.5659 6.6101 0.5055 3.2945 -7.0780 -0.4380 18.708

WHT 60 8.9088 -0.1962 13.9289 6.4984 0.2538 3.4062 -7.2855 -0.1385 18.9155


A. 44

WHT 70 9.0814 0.4674 13.7563 6.6125 0.3539 3.2921 -7.4804 -0.1104 19.1104

WHT 80 11.7882 5.5006 11.0495 7.0090 0.4344 2.8956 -8.0466 -0.2026 19.6766

WHT 90 11.9079 1.9286 10.9298 7.0824 0.4778 2.8222 -8.1737 -0.2037 19.8037

WHT 100 10.9096 0.4861 11.9281 6.9954 0.2108 2.9092 -8.4410 -0.0710 20.071

WHT 120 11.5372 0.8137 11.3005 7.1759 0.2713 2.7287 -9.1505 -0.5005 20.7805

WHT 140 11.6736 1.5738 11.1641 7.2552 0.4106 2.6494 -9.4242 -0.5542 21.0542

WHT 160 9.8688 -0.7606 12.9689 7.1305 0.0919 2.7741 -9.2081 -0.3631 20.8381

WHT 180 9.9644 -0.2875 12.8733 7.1930 0.1684 2.7116 -9.8181 -0.3981 21.4481

WHT 200 11.3965 0.8011 11.4412 7.3336 0.1890 2.571 -10.0901 -0.7201 21.7201

WHT 220 11.5960 0.0305 11.2417 7.3780 0.1184 2.5266 -10.3356 -0.7271 21.9656

WHT 240 11.6595 4.7362 11.1782 7.4171 0.3625 2.4875 -10.5607 -0.5907 22.1907

WHT 250 12.1288 1.7269 10.7089 7.5244 0.3428 2.3802 -10.3413 -0.3713 21.9713

WHT 255 12.1429 1.0486 10.6948 7.5329 0.3083 2.3717 -10.4272 -0.5892 22.0572

WHT 260 11.3810 0.5296 11.4567 7.5448 0.3062 2.3598 -10.3295 -0.2595 21.9595

WHT 280 11.8364 0.5878 11.0013 7.4979 0.1933 2.4067 -10.6842 -0.4142 22.3142

WHT 300 11.8831 0.9527 10.9546 7.5278 0.2332 2.3768 -10.8583 -0.4883 22.4883

WHT 320 11.8574 1.3900 10.9803 7.5393 0.3347 2.3653 -11.0382 -0.5432 22.6682

WHT 500 12.1312 0.4886 10.7065 7.7131 0.1085 2.1915 -11.5297 -0.2397 23.1597

WHT 700 12.4720 0.4561 10.3657 7.8745 0.1399 2.0301 -11.9464 -0.0764 23.5764

WHT 1000 13.5600 3.6176 9.2777 8.0035 0.3789 1.9011 -12.5842 -0.4502 24.2142

DCT 5 5.2168 4.5390 12.8942 3.2318 1.9838 4.4322 -1.8527 -0.1444 13.4807

DCT 10 6.4678 4.9038 11.6432 3.8305 1.9365 3.8335 -3.3271 -0.3151 14.9551

DCT 20 6.0880 5.2343 12.0230 4.1559 1.5699 3.5081 -4.7831 -0.1611 16.4111

DCT 30 9.0769 5.6015 9.0341 4.7366 1.4726 2.9274 -5.5879 -0.0509 17.2159

DCT 40 8.2610 3.9953 9.8500 4.8099 1.1459 2.8541 -6.2800 -0.1580 17.908

DCT 50 8.7019 4.4531 9.4091 5.0522 1.1882 2.6118 -6.7103 -0.0683 18.3383

DCT 60 8.8635 4.4852 9.2475 5.1270 1.1230 2.537 -7.2375 -0.0885 18.8655

DCT 70 8.3644 4.4771 9.7466 5.1100 1.0920 2.554 -7.4138 -0.0418 19.0418


A. 45

DCT 80 9.4263 7.8654 8.6847 5.3400 1.0060 2.324 -7.4558 0.3902 19.0838

DCT 90 9.7652 4.5126 8.3458 5.3276 0.9636 2.3364 -8.0063 -0.0343 19.6343

DCT 100 9.4938 3.7970 8.6172 5.3421 0.7981 2.3219 -8.2689 0.1031 19.8969

DCT 120 8.9284 2.9316 9.1826 5.3968 0.7328 2.2672 -8.6476 0.0044 20.2756

DCT 140 10.3110 4.9379 7.8000 5.5709 0.9669 2.0931 -8.9126 -0.0406 20.5406

DCT 160 9.7695 3.8668 8.3415 5.5726 0.7746 2.0914 -8.9073 -0.0603 20.5353

DCT 180 10.1595 4.6343 7.9515 5.6011 0.8171 2.0629 -9.0835 0.3385 20.7115

DCT 200 10.2138 4.3451 7.8972 5.5775 0.6735 2.0865 -9.6093 -0.2373 21.2373

DCT 220 9.3948 2.5560 8.7162 5.6864 0.6674 1.9776 -9.1604 0.4501 20.7884

DCT 240 10.3095 8.1129 7.8015 5.6913 0.8773 1.9727 -9.7126 0.2594 21.3406

DCT 250 9.0514 3.3762 9.0596 5.7476 0.8066 1.9164 -9.8144 0.1576 21.4424

DCT 255 10.4596 4.0920 7.6514 5.7487 0.7647 1.9153 -9.8207 0.0193 21.4487

DCT 260 9.5897 3.4650 8.5213 5.6988 0.7008 1.9652 -10.0232 0.0488 21.6512

DCT 280 10.6507 4.1288 7.4603 5.8074 0.7434 1.8566 -10.0021 0.2699 21.6301

DCT 300 9.8587 3.6550 8.2523 5.7534 0.6994 1.9106 -10.2307 0.1413 21.8587

DCT 320 10.7427 5.0020 7.3683 5.8207 0.8567 1.8433 -10.4537 0.0433 22.0817

DCT 500 10.1111 3.1952 7.9999 5.9902 0.6262 1.6738 -10.8432 0.4488 22.4712

DCT 700 10.8348 3.5456 7.2762 6.0670 0.5730 1.597 -11.7915 0.0805 23.4195

DCT 1000 11.1737 5.9580 6.9373 6.1554 0.7714 1.5086 -12.0486 0.0874 23.6766

DST 5 5.4486 5.4169 12.0163 3.0512 1.9442 4.4718 -2.3432 -0.6349 13.9712

DST 10 6.8133 5.8954 10.6516 3.7000 1.9470 3.823 -3.5551 -0.5431 15.1831

DST 20 6.7980 6.5904 10.6669 4.1850 1.7400 3.338 -5.0565 -0.4345 16.6845

DST 30 8.3677 5.5384 9.0972 4.6087 1.4857 2.9143 -6.3507 -0.8137 17.9787

DST 40 7.6474 4.0278 9.8175 4.7031 1.1801 2.8199 -6.8122 -0.6902 18.4402

DST 50 9.2676 5.6649 8.1973 5.0013 1.2783 2.5217 -6.9744 -0.3324 18.6024

DST 60 9.0030 5.2708 8.4619 5.0522 1.1892 2.4708 -7.7744 -0.6254 19.4024

DST 70 9.2551 6.0139 8.2098 5.1230 1.2460 2.4 -8.1161 -0.7441 19.7441

DST 80 9.2642 8.3494 8.2007 5.2023 1.0093 2.3207 -8.4165 -0.5705 20.0445


A. 46

DST 90 9.4738 4.8673 7.9911 5.2485 1.0255 2.2745 -8.5426 -0.5706 20.1706

DST 100 9.3820 4.3313 8.0829 5.3318 0.9288 2.1912 -8.6108 -0.2388 20.2388

DST 120 9.7971 4.4464 7.6678 5.3680 0.8450 2.155 -9.0312 -0.3792 20.6592

DST 140 9.7060 4.9790 7.7589 5.4550 0.9920 2.068 -9.3167 -0.4447 20.9447

DST 160 10.0042 4.7476 7.4607 5.4857 0.8287 2.0373 -9.6425 -0.7955 21.2705

DST 180 9.8794 5.0003 7.5855 5.5538 0.9108 1.9692 -9.7710 -0.3490 21.399

DST 200 10.1197 4.8971 7.3452 5.5763 0.8133 1.9467 -10.0101 -0.6381 21.6381

DST 220 9.5615 3.3688 7.9034 5.6042 0.7262 1.9188 -10.1189 -0.5084 21.7469

DST 240 10.2651 8.7146 7.1998 5.6338 0.9608 1.8892 -10.4585 -0.4865 22.0865

DST 250 9.6395 4.6104 7.8254 5.6467 0.8467 1.8763 -10.3842 -0.4122 22.0122

DST 255 9.5708 3.8493 7.8941 5.6370 0.7940 1.886 -10.4538 -0.6138 22.0818

DST 260 8.1344 2.6558 9.3305 5.5533 0.6963 1.9697 -10.7142 -0.6422 22.3422

DST 280 9.7061 3.8303 7.7588 5.6830 0.7600 1.84 -10.7494 -0.4774 22.3774

DST 300 8.2172 2.6596 9.2477 5.6017 0.6887 1.9213 -10.8662 -0.4942 22.4942

DST 320 9.1446 4.0500 8.3203 5.6361 0.8131 1.8869 -10.8145 -0.3175 22.4425

DST 500 10.3526 4.0828 7.1123 5.8364 0.6134 1.6866 -11.6766 -0.3846 23.3046

DST 700 8.6398 1.9967 8.8251 5.8487 0.4957 1.6743 -11.6894 0.1826 23.3174

DST 1000 10.6545 6.0849 6.8104 6.0095 0.7665 1.5135 -12.5300 -0.3940 24.158

DHT 5 1.9728 12.4473 4.9859 0.7742 3.8402 2.5758 -0.7449 1.0094 12.3269

DHT 10 2.3552 11.9435 4.6035 0.9217 3.3417 2.4283 -1.2059 1.8521 12.7879

DHT 20 2.6599 12.9585 4.2988 1.0222 2.7502 2.3278 -2.1132 2.5548 13.6952

DHT 30 2.8047 10.4816 4.1540 1.0703 2.1203 2.2797 -2.4565 3.1265 14.0385

DHT 40 2.8756 9.7622 4.0831 1.0526 1.7026 2.2974 -2.7943 3.3737 14.3763

DHT 50 2.9382 9.8417 4.0205 1.0762 1.5262 2.2738 -3.3194 3.3686 14.9014

DHT 60 2.9855 9.7595 3.9732 1.0940 1.4040 2.256 -3.5662 3.6288 15.1482

DHT 70 3.0230 10.2880 3.9357 1.1081 1.4041 2.2419 -3.7728 3.6452 15.3548

DHT 80 3.0564 12.6478 3.9023 1.0898 1.0698 2.2602 -3.7046 4.1874 15.2866

DHT 90 3.0824 8.9821 3.8763 1.0982 1.0482 2.2518 -3.7978 4.2202 15.3798


A. 47

DHT 100 3.0929 8.5484 3.8658 1.1892 0.9592 2.1608 -3.7443 4.6737 15.3263

DHT 120 3.1412 8.2967 3.8175 1.1173 0.7673 2.2327 -4.0950 4.6030 15.677

DHT 140 3.1703 8.9495 3.7884 1.1267 0.8367 2.2233 -4.2397 4.6783 15.8217

DHT 160 3.1929 8.4425 3.7658 1.2065 0.7225 2.1435 -4.2496 4.6434 15.8316

DHT 180 3.2132 8.8403 3.7455 1.2134 0.7434 2.1366 -4.3751 5.0929 15.9571

DHT 200 3.2306 8.5142 3.7281 1.2193 0.6293 2.1307 -4.4275 4.9905 16.0095

DHT 220 3.2458 7.5593 3.7129 1.2244 0.5194 2.1256 -4.5322 5.1243 16.1142

DHT 240 3.2399 12.1956 3.7188 1.1561 0.6561 2.1939 -4.8847 5.1333 16.4667

DHT 250 3.2541 8.7312 3.7046 1.2272 0.6002 2.1228 -4.9731 5.0449 16.5551

DHT 255 3.2490 8.0337 3.7097 1.1591 0.4891 2.1909 -5.2350 4.6510 16.817

DHT 260 3.2519 8.2795 3.7068 1.1600 0.4760 2.19 -5.2350 4.8830 16.817

DHT 280 3.2708 7.9012 3.6879 1.2337 0.4837 2.1163 -5.0874 5.2306 16.6694

DHT 300 3.2715 8.2201 3.6872 1.2504 0.5104 2.0996 -5.0789 5.3391 16.6609

DHT 320 3.2898 8.7014 3.6689 1.2412 0.5912 2.1088 -5.3548 5.1882 16.9368

DHT 500 3.3475 7.5839 3.6112 1.2639 0.2139 2.0861 -5.7347 5.6033 17.3167

DHT 700 3.3767 7.2398 3.5820 1.2848 0.1048 2.0652 -5.7148 6.2032 17.2968

DHT 1000 3.4307 9.3673 3.5280 1.3010 0.2310 2.049 -6.0038 6.1782 17.5858

Table A.32 (Pre-coding + RCT) Results and compared with the results of each of (Pre-coding) and (RCT)

Pre-coding R X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

WHT .9 3.3459 2.3713 19.4918 1.3932 1.0386 8.5114 -0.1006 -0.0541 11.7306

WHT .8 5.1820 3.4754 17.6557 2.1560 0.9329 7.7486 -0.2737 0.0833 11. 9037

WHT .7 6.1784 1.4698 16.6593 2.6885 0.8419 7.2161 -1.0924 -0.3087 12.7224

WHT .6 8.2743 0.7913 14.5634 3.5811 0.3590 6.3235 -1.9051 -0.1351 13.5351

WHT .5 10.1461 1.2348 12.6916 4.3915 0.3219 5.5131 -3.0635 -0.2435 14.6935

WHT .4 13.2863 1.9778 9.5514 5.6906 0.6075 4.214 -4.9407 -0.4257 16.5707

WHT .3 15.4834 1.1986 7.3543 6.6295 0.4349 3.2751 -7.1992 -0.3042 18.8292


A. 48

WHT .2 17.7489 0.9000 5.0888 7.6444 0.3143 2.2602 -10.7564 -0.1364 22.3864

WHT .1 20.1831 0.2180 2.6546 8.7120 0.0754 1.1926 -17.0412 -0.3712 28.6712

DCT .9 2.6159 6.3680 15.4951 0.8902 2.7762 6.7738 -0.0147 0.0338 11.6427

DCT .8 4.0365 7.0566 14.0745 1.5129 2.5304 6.1511 -0.3815 -0.0225 12.0095

DCT .7 5.5106 5.5287 12.6004 2.1557 2.5497 5.5083 -1.0401 -0.2544 12.6681

DCT .6 6.0660 3.3097 12.0450 2.5025 1.5210 5.1615 -1.8911 -0.1191 13.5191

DCT .5 7.8308 3.6462 10.2802 3.2498 1.4208 4.4142 -3.0857 -0.2637 14.7137

DCT .4 9.6726 3.0908 8.4384 4.0337 1.1912 3.6303 -4.7014 -0.1844 16.3294

DCT .3 11.9638 2.4057 6.1472 4.9843 1.0303 2.6797 -6.6401 0.2569 18.2681

DCT .2 13.6369 1.5147 4.4741 5.7340 0.6445 1.93 -10.2162 0.4058 21.8442

DCT .1 15.8025 0.5641 2.3085 6.6729 0.2769 .9911 -16.5112 0.1608 28.1392

DST .9 1.7433 6.1415 15.7216 0.7555 2.7825 6.7675 -0.0393 0.0092 11.6673

DST .8 3.1975 6.8637 14.2674 1.3692 2.5277 6.1538 -0.3827 -0.0237 12.0107

DST .7 4.7059 5.3701 12.7590 2.0010 2.5360 5.522 -0.9126 -0.1269 12.5406

DST .6 6.2744 4.1642 11.1905 2.6500 1.8095 4.873 -1.8332 -0.0612 13.4612

DST .5 7.9097 4.3712 9.5552 3.3369 1.6489 4.1861 -2.9499 -0.1279 14.5779

DST .4 9.1755 3.2398 8.2894 3.9292 1.2277 3.5938 -4.6498 -0.1328 16.2778

DST .3 11.0822 2.1702 6.3827 4.7562 0.9432 2.7668 -7.0794 -0.1824 18.7074

DST .2 13.0856 1.6095 4.3793 5.6273 0.6788 1.8957 -10.5597 0.0623 22.1877

DST .1 15.2166 0.6243 2.2483 6.5535 0.2985 .9695 -16.8057 -0.1337 28.4337

DHT .9 0.3560 15.2604 6.6027 0.1052 6.3052 3.2448 0.0718 0.1663 11.5102

DHT .8 0.7383 14.9107 6.2204 0.2788 5.6103 3.0712 -0.2792 0.1258 11.8612

DHT .7 1.1257 12.2961 5.8330 0.5084 5.2164 2.8416 -0.6722 0.1595 12.2542

DHT .6 1.5174 9.9134 5.4413 0.6703 4.0028 2.6797 -1.1254 0.6926 12.7074

DHT .5 1.9183 8.8860 5.0404 0.7518 3.2368 2.5982 -1.5191 1.3489 13.1011

DHT .4 2.3380 6.9085 4.6207 0.8374 2.3089 2.5126 -2.5360 2.0270 14.118

DHT .3 2.7565 4.3507 4.2022 1.0510 1.4110 2.299 -3.4451 3.4979 15.0271

DHT .2 3.1938 2.2239 3.7649 1.0568 0.2813 2.2932 -5.8003 4.8677 17.3823

DHT .1 3.6504 -0.4357 3.3083 1.2249 -0.8571 2.1251 -9.5778 7.1402 21.1598


A. 49

Table A.33(Pre-coding +AEXP) Results and compared with the results of each of (Pre-coding) and (AEXP companding).

Precoding AEXP d X X1 PAPR Y Y1 CCDF OF

PAPR

Z Z1 SNR

(BER= )

WHT 2 10.7804 1.0238 12.0573 4.9111 0.1598 4.9935 -4.3415 -1.2415 15.9715

WHT 1.9 11.3253 1.5116 11.5124 5.0575 0.2929 4.8471 -4.5962 -1.5262 16.2262

WHT 1.8 11.7057 1.0663 11.1320 5.2111 0.2250 4.6935 -5.3536 -2.1256 16.9836

WHT 1.7 12.2595 0.6391 10.5782 5.4254 0.2908 4.4792 -4.7950 -1.9750 16.425

WHT 1.6 12.3909 0.2494 10.4468 5.5148 0.1952 4.3898 -4.3508 -1.7808 15.9808

WHT 1.5 12.9152 0.1439 9.9225 5.6605 0.1139 4.2441 -5.2656 -2.5818 16.8956

WHT 1.4 13.4360 0.2128 9.4017 5.8436 0.0855 4.061 -4.0073 -1.0688 15.6373

WHT 1.3 14.2743 0.4181 8.5634 6.0031 0.0785 3.9015 -4.0536 -1.3836 15.6836

WHT 1.2 14.8045 0.4168 8.0332 6.2983 0.1997 3.6063 -4.1271 -0.4271 15.7571

WHT 1.1 15.2924 0.4070 7.5453 6.4313 0.1262 3.4733 -4.4413 -1.2213 16.0713

WHT 1 15.8627 0.4024 6.9750 6.6393 0.1087 3.2653 -4.2999 -0.6299 15.9299

WHT .9 16.4192 0.3315 6.4185 6.8279 0.1153 3.0767 -4.2078 0.4622 15.8378

WHT .8 17.0353 0.2782 5.8024 7.1031 0.1185 2.8015 -5.5993 2.0157 17.2293

WHT .7 17.4907 0.1783 5.3470 7.2846 0.0170 2.62 -6.5697 11.8003 18.1997

WHT .6 18.2253 0.1768 4.6124 7.6039 0.0393 2.3007 -9.5950 8.7750 21.225

WHT .5 18.8350 0.1317 4.0027 7.9506 0.1535 1.954 -18.3700 0 >30

WHT .4 19.5373 0.1035 3.3004 8.2612 0.0966 1.6434 -18.3700 0 >30

WHT .3 20.2553 0.0694 2.5824 8.5546 -0.0228 1.35 -18.3700 0 >>30

WHT .2 21.0167 0.0148 1.8210 8.9118 -0.0503 .9928 -18.3700 0 >>30

WHT .1 21.8494 -0.0193 0.9883 9.3313 -0.0617 .5733 -18.3700 0 >>>30

DCT 2 6.7007 1.6708 11.4103 2.9261 0.4154 4.7379 -3.9507 -0.8487 15.5787

DCT 1.9 7.1704 2.0834 10.9406 3.0746 0.5506 4.5894 -3.9175 -0.8455 15.5455

DCT 1.8 7.6401 1.7274 10.4709 3.2279 0.4824 4.4361 -3.8548 -0.6248 15.4828

DCT 1.7 8.2400 1.3463 9.8710 3.4745 0.5805 4.1895 -3.1582 -0.3362 14.7862


A. 50

DCT 1.6 8.5743 1.1595 9.5367 3.6095 0.5305 4.0545 -3.2536 -0.6816 14.8816

DCT 1.5 9.0476 1.0030 9.0634 3.7703 0.4643 3.8937 -2.9112 -0.2254 14.5392

DCT 1.4 9.5246 1.0281 8.5864 3.9270 0.4095 3.737 -2.7500 0.1905 14.378

DCT 1.3 10.0065 0.8770 8.1045 4.0957 0.4117 3.5683 -2.6924 -0.0204 14.3204

DCT 1.2 10.4319 0.7709 7.6791 4.5142 0.6562 3.1498 -2.4536 1.2484 14.0816

DCT 1.1 10.9577 0.7990 7.1533 4.3928 0.3283 3.2712 -2.4922 0.7298 14.1202

DCT 1 11.4605 0.7269 6.6505 4.5868 0.2968 3.0772 -3.0850 0.5870 14.713

DCT .9 11.9813 0.6203 6.1297 4.8018 0.3298 2.8622 -3.7744 0.8976 15.4024

DCT .8 12.5214 0.4910 5.5896 5.0146 0.2706 2.6494 -5.2869 2.3301 16.9149

DCT .7 13.1121 0.5264 4.9989 5.3103 0.2833 2.3537 -6.8777 11.4943 18.5057

DCT .6 13.6710 0.3492 4.4400 5.5512 0.2272 2.1128 -18.3720 0 >30

DCT .5 14.2768 0.3002 3.8342 5.8225 0.2660 1.8415 -18.3720 0 >30

DCT .4 14.9229 0.2158 3.1881 6.1132 0.1892 1.5508 -18.3720 0 >30

DCT .3 15.6162 0.1570 2.4948 6.4346 0.0978 1.2294 -18.3720 0 >>30

DCT .2 16.3628 0.0876 1.7482 6.8036 0.0821 .8604 -18.3720 0 >>30

DCT .1 17.1819 0.0399 0.9291 7.1850 0.0326 .479 -18.3720 0 >>>30

DST 2 5.7603 1.3765 11.7046 2.8291 0.4594 4.6939 -3.4924 -0.3904 15.1204

DST 1.9 6.2487 1.8078 11.2162 2.9692 0.5862 4.5538 -2.9424 0.1296 14.5704

DST 1.8 6.7376 1.4710 10.7273 3.1148 0.5103 4.4082 -1.7942 1.4358 13.4222

DST 1.7 7.4201 1.1725 10.0448 3.3569 0.6039 4.1661 -2.1349 0.6871 13.7629

DST 1.6 7.9659 1.1972 9.4990 3.4962 0.5582 4.0268 -1.5678 1.0042 13.1958

DST 1.5 8.2102 0.8117 9.2547 3.5812 0.4162 3.9418 -1.8026 0.8832 13.4306

DST 1.4 8.6838 0.8334 8.7811 3.7896 0.4131 3.7334 -2.0358 0.9047 13.6638

DST 1.3 9.2006 0.7172 8.2643 3.9663 0.4233 3.5567 -2.4312 0.2408 14.0592

DST 1.2 9.6836 0.6687 7.7813 4.1170 0.4000 3.406 -3.5155 0.1865 15.1435

DST 1.1 10.2053 0.6927 7.2596 4.3442 0.4207 3.1788 -4.6118 -1.3898 16.2398

DST 1 10.6868 0.5993 6.7781 4.5007 0.3517 3.0223 -17.9720 -14.3000 29.6

DST .9 11.3528 0.6379 6.1121 4.7532 0.4222 2.7698 -18.3720 -13.7000 >30

DST .8 11.9005 0.5162 5.5644 4.9746 0.3716 2.5484 -18.3720 -10.7550 >30


A. 51

DST .7 12.4544 0.5148 5.0105 5.2316 0.3456 2.2914 -18.3720 0 >30

DST .6 13.0419 0.3662 4.4230 5.4339 0.2509 2.0891 -18.3720 0 >>30

DST .5 13.6703 0.3398 3.7946 5.6985 0.2830 1.8245 -18.3720 0 >>30

DST .4 14.3023 0.2413 3.1626 5.9822 0.1992 1.5408 -18.3720 0 >>30

DST .3 15.0261 0.2130 2.4388 6.3203 0.1245 1.2027 -18.3720 0 >>30

DST .2 15.7624 0.1333 1.7025 6.6784 0.0979 .8446 -18.3720 0 >>>30

DST .1 16.5640 0.0681 0.9009 7.0644 0.0530 .4586 -18.3720 0 >>>30

DHT 2 -0.6937 5.4287 7.6524 0.1988 2.0021 3.1512 -1.9173 1.2307 13.4993

DHT 1.9 -0.4865 5.5788 7.4452 0.2950 2.0850 3.055 -1.5431 1.5749 13.1251

DHT 1.8 -0.3933 4.8463 7.3520 0.3265 1.8950 3.0235 -1.5752 1.7008 13.1572

DHT 1.7 -0.2155 4.0431 7.1742 0.3825 1.8025 2.9675 -1.0598 1.8082 12.6418

DHT 1.6 0.0648 3.8023 6.8939 0.5006 1.7356 2.8494 -0.8207 1.7973 12.4027

DHT 1.5 0.3050 3.4127 6.6537 0.5780 1.5860 2.772 -0.8038 1.9280 12.3858

DHT 1.4 0.2838 2.9396 6.6749 0.6027 1.3992 2.7473 -1.0845 1.9020 12.6665

DHT 1.3 0.5355 2.5583 6.4232 0.6680 1.2980 2.682 -1.0560 1.6620 12.638

DHT 1.2 0.8006 2.2919 6.1581 0.7343 1.1903 2.6157 -0.8581 2.8899 12.4401

DHT 1.1 1.1528 2.1464 5.8059 0.7608 1.0103 2.5892 -0.3896 2.8784 11.9716

DHT 1 1.5750 1.9937 5.3837 0.9006 0.9246 2.4494 -0.5745 3.1435 12.1565

DHT .9 1.8437 1.6350 5.1150 0.9706 0.8126 2.3794 -0.6731 4.0449 12.2551

DHT .8 2.1033 1.2252 4.8554 1.0415 0.6115 2.3085 -0.8873 6.7757 12.4693

DHT .7 2.3210 0.8876 4.6377 1.0391 0.3261 2.3109 -1.4496 6.9684 13.0316

DHT .6 2.5943 0.4248 4.3644 1.1280 0.1180 2.222 -1.7265 16.6915 13.3085

DHT .5 2.8462 0.0219 4.1125 1.1485 -0.0940 2.2015 -2.3714 16.0466 13.9534

DHT .4 3.1118 -0.4430 3.8469 1.2126 -0.3974 2.1374 -3.0057 15.4123 14.5877

DHT .3 3.3695 -0.9374 3.5892 1.2777 -0.7451 2.0723 -4.1479 14.2701 15.7299

DHT .2 3.6525 -1.4704 3.3062 1.3818 -1.0257 1.9682 -18.4180 0 17.8183 -30

DHT .1 3.8711 -2.1186 3.0876 1.3298 -1.5086 2.0202 -18.4180 0 >>30


A. 52

Table A.34 (Pre-coding +cos) Results and compared with the results of each of (Pre-coding) and (cos companding).

Precoding Cos

X X1 PAPR Y Y1 CCDF of

PAPR

Z Z1 SNR

(BER= )

WHT 1 7.3889 0.1118 15.4488 3.0367 -0.0179 6.8679 -0.0936 0.0764 11.7236

WHT .9 8.6546 0.2175 14.1831 3.5637 0.0751 6.3409 -0.3785 0.2165 12.0085

WHT .8 9.9471 0.2189 12.8906 4.2290 0.1894 5.6756 -0.8393 0.1107 12.4693

WHT .7 11.3571 0.1886 11.4806 4.7941 0.1710 5.1105 -1.5116 0.2424 13.1416

WHT .6 12.8150 0.2831 10.0227 5.4976 0.1960 4.407 -2.6935 0.2150 14.3235

WHT .5 14.2197 0.1510 8.6180 6.0483 0.0987 3.8563 -3.9494 0.2806 15.5794

WHT .4 15.7226 0.0911 7.1151 6.6971 0.0715 3.2075 -6.0413 -0.1113 17.6713

WHT .3 17.3259 0.0641 5.5118 7.4495 0.1449 2.4551 -9.0243 0.6457 20.6543

WHT .2 19.0411 3.7966 8.1231 1.7815 -18.3700 >30

WHT .1 20.8281 2.0096 8.9201 .9845 -18.3700 >>30

DCT 1 4.5103 1.9599 13.6007 1.7336 0.9196 5.9304 -0.0147 0.1573 11.6427

DCT .9 5.5939 1.8835 12.5171 2.2109 0.9629 5.4531 -0.3795 0.2175 12.0075

DCT .8 6.7188 1.7173 11.3922 2.6944 0.8954 4.9696 -1.0026 -0.0506 12.6306

DCT .7 7.8898 1.4480 10.2212 3.1956 0.8131 4.4684 -1.5364 0.2196 13.1644

DCT .6 9.1125 1.3073 8.9985 3.7198 0.6588 3.9442 -2.5236 0.3869 14.1516

DCT .5 10.0773 0.7353 8.0337 4.1538 0.4448 3.5102 -3.9940 0.2380 15.622

DCT .4 11.4935 0.5887 6.6175 4.7682 0.3832 2.8958 -5.9277 0.0043 17.5557

DCT .3 12.9874 0.4523 5.1236 5.4192 0.3552 2.2448 -11.3055 -1.6335 22.9335

DCT .2 14.5824 3.5286 6.0790 1.585 -18.3720 >30

DCT .1 16.2663 1.8447 6.8477 .8163 -18.3720 >>30

DST 1 3.4964 1.5921 13.9685 1.4699 0.7969 6.0531 -0.5719 -0.3999 12.1999

DST .9 5.0333 1.9690 12.4316 2.0815 0.9745 5.4415 -0.6383 -0.0413 12.2663

DST .8 6.0390 1.6836 11.4259 2.5870 0.9290 4.936 -1.3140 -0.3620 12.942

DST .7 7.0375 1.2418 10.4274 3.0940 0.8525 4.429 -2.1014 -0.3454 13.7294

DST .6 8.3999 1.2408 9.0650 3.5805 0.6605 3.9425 -3.0753 -0.1648 14.7033


A. 53

DST .5 9.6431 0.9472 7.8218 4.1717 0.6037 3.3513 -4.0155 0.2165 15.6435

DST .4 11.1108 0.8521 6.3541 4.6548 0.4108 2.8682 -6.3518 -0.4198 17.9798

DST .3 12.4423 0.5533 5.0226 5.3188 0.3958 2.2042 -8.6413 1.0307 20.2693

DST .2 13.9884 3.4765 5.9826 1.5404 -17.8447 29.4727

DST .1 15.6603 1.8046 6.7346 .7884 -18.3720 >>>30

DHT 1 -0.8692 7.7327 7.8279 0.2092 3.7092 3.1408 -0.1667 0.0513 11.7487

DHT .9 -0.3893 7.0526 7.3480 0.3310 3.3970 3.019 -0.3262 0.3168 11.9082

DHT .8 0.0275 6.1783 6.9312 0.4508 2.9658 2.8992 -0.5046 0.4934 12.0866

DHT .7 0.5943 5.3048 6.3644 0.5863 2.5178 2.7637 -0.9311 0.8709 12.5131

DHT .6 1.2213 4.5684 5.7374 0.6807 1.9337 2.6693 -1.1133 1.8432 12.6953

DHT .5 1.6803 3.4906 5.2784 0.7570 1.3620 2.593 -2.2444 2.0336 13.8264

DHT .4 2.1705 2.4180 4.7882 0.8625 0.7915 2.4875 -2.6397 3.3383 14.2217

DHT .3 2.6262 1.2434 4.3325 1.0832 0.3332 2.2668 -4.0128 5.7052 15.5948

DHT .2 3.1364 3.8223 1.2095 2.1405 -18.4180 30

DHT .1 3.6225 3.3362 1.3423 2.0077 -18.4180 >>30

Table A.35 (Pre-coding +NERF) Results and compared with the results of each of (Pre-coding) and (NERF companding).

Precoding NERF X X1 PAPR Y Y1 CCDF of

PAPR

Z Z1 SNR

(BER= )

WHT 12.7937 0.3968 10.0440 5.6684 0.2838 4.2362 -2.1831 -0.4681 13.8131

DCT 9.0407 1.3705 9.0703 3.7171 0.5731 3.9469 -1.7823 -0.0653 13.4103

DST 8.4569 1.4328 9.0080 3.6090 0.6060 3.914 non-NaN.

DHT 1.1145 4.5966 5.8442 0.6177 1.7877 2.7323 non-NaN

Table A.36 (Pre-coding + tanhR) Results and compared with the results of each of (Pre-coding) and (tanhR companding).

Precoding k y X X1 PAPR Y Y1 CCDF of Z Z1 SNR


A. 54

PAPR (BER= )

WHT 5 1 -0.1807 -6.3055 23.0184 0.2938 -2.2032 9.6108 0.1112 0.6106 11.5188

WHT 5 .8 3.7573 -5.2332 19.0804 2.2354 -1.6111 7.6692 -0.6251 0.4161 12.2551

WHT 5 .5 10.6855 -3.2479 12.1522 4.7480 -1.2705 5.1566 -3.0382 0.9195 14.6682

WHT 5 .2 18.3499 -0.9432 4.4878 7.9541 -0.4230 1.9505 -10.9634 2.1297 22.5934

WHT 10 1 0.0656 -13.8409 22.7721 0.3874 -5.3082 9.5172 0.1112 3.1298 11.5188

WHT 10 .8 4.2484 -10.4413 18.5893 2.1231 -4.0236 7.7815 -0.4102 2.8919 12.0402

WHT 10 .5 11.1658 -5.2722 11.6719 4.9280 -2.1485 4.9766 -3.1630 2.8359 14.793

WHT 10 .2 18.6064 -1.1811 4.2313 8.0609 -0.4986 1.8437 -11.2273 3.1586 22.8573

WHT 15 1 0.3235 -17.2828 22.5142 0.4487 -6.8572 9.4559 -0.0924 17.8776 11.7224

WHT 15 .8 4.9015 -12.6933 17.9362 2.3646 -5.0135 7.54 -0.6197 10.2159 12.2497

WHT 15 .5 10.8964 -6.9994 11.9413 5.1420 -2.5305 4.7626 -3.2753 6.5694 14.9053

WHT 15 .2 18.7511 -1.3429 4.0866 8.1767 -0.5252 1.7279 -11.4066 3.5210 23.0366

WHT 20 1 0.0789 -19.3807 22.7588 0.3788 -7.7030 9.5258 -0.2736 17.6964 11.9036

WHT 20 .8 5.7588 -13.4839 17.0789 2.6213 -5.4421 7.2833 -0.7083 17.2617 12.3383

WHT 20 .5 12.0332 -7.0284 10.8045 5.1936 -2.9215 4.711 -3.4580 14.5120 15.088

WHT 20 .2 19.1134 -1.1533 3.7243 8.2956 -0.4775 1.609 -11.7669 5.5423 23.3969

WHT 30 1 2.9821 19.8556 1.4998 8.4048 0.1416 11.4884

WHT 30 .8 6.1658 16.6719 2.7222 7.1824 -0.6555 12.2855

WHT 30 .5 12.9867 9.8510 5.6462 4.2584 -3.4154 15.0454

WHT 30 .2 18.9871 3.8506 8.2285 1.6761 -12.3505 23.9805

WHT 40 1 4.2670 18.5707 1.8924 8.0122 0.0095 11.6205

WHT 40 .8 7.5802 15.2575 3.4669 6.4377 -0.6978 12.3278

WHT 40 .5 13.8960 8.9417 5.9673 3.9373 -3.9522 15.5822

WHT 40 .2 19.3685 3.4692 8.3837 1.5209 -12.5536 24.1836

DCT 5 1 4.9948 3.5967 13.1162 1.8882 1.6318 5.7758 -0.2843 0.2171 11.9123

DCT 5 .8 7.4490 3.1852 10.6620 2.9477 1.3418 4.7163 -1.2305 -0.1873 12.8585

DCT 5 .5 11.3493 2.1426 6.7617 4.6478 0.8699 3.0162 -4.0362 -0.0765 15.6642

DCT 5 .2 15.4567 0.8903 2.6543 6.4640 0.3275 1.2 -12.4393 0.6558 24.0673


A. 55

DCT 10 1 10.0154 0.8356 8.0956 4.0445 0.5895 3.6195 -2.1407 0.8799 13.7687

DCT 10 .8 10.8270 0.8640 7.2840 4.4516 0.5455 3.2124 -2.6784 0.6257 14.3064

DCT 10 .5 12.7260 1.0147 5.3850 5.3042 0.4683 2.3598 -6.2448 -0.2439 17.8728

DCT 10 .2 15.6243 0.5635 2.4867 6.5815 0.2626 1.0825 -13.8424 0.5455 25.4704

DCT 15 1 13.2826 0.4030 4.8284 5.2743 0.2090 2.3897 -7.3080 10.6640 18.936

DCT 15 .8 13.3063 0.4382 4.8047 5.3592 0.2217 2.3048 -7.1678 3.6698 18.7958

DCT 15 .5 13.9101 0.7410 4.2009 5.7489 0.3170 1.9151 -8.8231 1.0236 20.4511

DCT 15 .2 15.8469 0.4796 2.2641 6.6630 0.2017 1.001 -14.9256 0.0040 26.5536

DCT 20 1 14.9735 0.2406 3.1375 5.9965 0.1553 1.6675 -18.3720 -0.4000 >30

DCT 20 .8 14.7766 0.2606 3.3344 5.9706 0.1478 1.6934 -18.3720 -0.4000 >30

DCT 20 .5 14.7322 0.3973 3.3788 6.1017 0.2272 1.5623 -13.4514 4.5206 25.0794

DCT 20 .2 16.0073 0.4673 2.1037 6.7400 0.2075 .924 -16.1143 1.1969 27.7423

DCT 30 1 16.4535 1.6575 6.6685 .9955 -18.3720 >>30

DCT 30 .8 16.2397 1.8713 6.5711 1.0929 -18.3720 >>30

DCT 30 .5 15.7594 2.3516 6.5149 1.1491 -18.3720 >30

DCT 30 .2 16.2183 1.8927 6.8333 .8307 -17.5814 29.2094

DCT 40 1 17.0593 1.0517 6.9437 .7203 -18.3720 >>30

DCT 40 .8 16.9063 1.2047 6.9363 .7277 -18.3720 >>30

DCT 40 .5 16.3777 1.7333 6.7772 .8868 -18.3720 >>30

DCT 40 .2 16.3828 1.7282 6.8898 .7742 -18.3720 >30

DST .5 1 2.7791 14.6858 1.2087 6.3143 -18.3720 >30

DST .5 .8 5.6384 11.8265 2.4243 5.0987 -18.3720 >>30

DST .5 .5 10.1178 7.3471 4.3530 3.17 -18.3720 >>>30

DST .5 .2 14.6923 2.7726 6.3053 1.2177 -18.3720 >>>30

DST 1 1 7.7531 9.7118 3.1540 4.369 -18.3720 >30

DST 1 .8 9.1948 8.2701 3.7915 3.7315 -18.3720 >30

DST 1 .5 11.8357 5.6292 4.9741 2.5489 -18.3720 >>>30

DST 1 .2 15.0636 2.4013 6.4409 1.0821 -18.3720 >>>30

DST 5 1 16.5353 15.7833 0.9296 6.8695 6.7541 .6535 -18.3720 -17.8706 >>>30


A. 56

DST 5 .8 16.3710 12.7533 1.0939 6.8535 5.3886 .6695 -18.3720 -17.3288 >>>30

DST 5 .5 15.8264 7.2658 1.6385 6.6730 3.0361 .85 -18.3720 -14.4123 >>30

DST 5 .2 15.8076 1.8873 1.6573 6.7862 0.7907 .7368 -18.3720 -5.2769 >30

DST 10 1 17.1973 8.6636 0.2676 7.2727 3.9587 .2503 -18.3720 -15.3514 >>>30

DST 10 .8 17.1359 7.8190 0.3290 7.2537 3.4886 .2693 -18.3720 -15.0679 >>>30

DST 10 .5 16.8045 5.7393 0.6604 7.1194 2.4245 .4036 -18.3720 -12.3711 >>>30

DST 10 .2 16.1937 1.7790 1.2712 6.9533 0.7754 .5697 -18.3720 -3.9841 >>30

DST 15 1 17.3352 5.1017 0.1297 7.3498 2.4255 .1732 -18.3720 -0.4000 >>>30

DST 15 .8 17.3044 5.0824 0.1605 7.3447 2.3482 .1783 -18.3720 -7.5344 >>>30

DST 15 .5 17.1086 4.5856 0.3563 7.2769 1.9860 .2461 -18.3720 -8.5253 >>>30

DST 15 .2 16.3513 1.6301 1.1136 7.0354 0.7151 .4876 -18.3720 -3.4424 >>30

DST 20 1 17.3829 3.2961 0.0820 7.4045 1.7043 .1185 -18.3720 -0.4000 >>>30

DST 20 .8 17.3648 3.4949 0.1001 7.3987 1.7169 .1243 -18.3720 -0.4000 >>>30

DST 20 .5 17.2385 3.5497 0.2264 7.3456 1.6121 .1774 -18.3720 -0.4000 >>>30

DST 20 .2 16.4904 1.5965 0.9745 7.0845 0.6930 .4385 -18.3720 -1.0608 >>30

DST 30 1 17.4153 0.0496 7.4273 .0957 -18.3720 >>>30

DST 30 .8 17.4074 0.0575 7.4277 .0953 -18.3720 >>>30

DST 30 .5 17.3451 0.1198 7.3967 .1263 -18.3720 >>>30

DST 30 .2 16.6910 0.7739 7.1582 .3648 -18.3720 >>>30

DST 40 1 17.4283 0.0366 7.4550 .068 -18.3720 >>>30

DST 40 .8 17.4237 0.0412 7.4562 .0668 -18.3720 >>>30

DST 40 .5 17.3863 0.0786 7.4420 .081 -18.3720 >>>30

DST 40 .2 16.7987 0.6662 7.2140 .309 -18.3720 >>>30

DHT 5 1 4.0819 13.8361 2.8768 1.4368 5.4944 1.9132 -1.4034 -0.8560 12.9854

DHT 5 .8 4.0969 10.9854 2.8618 1.4315 4.1396 1.9185 -0.9347 0.1545 12.5167

DHT 5 .5 3.9269 5.8725 3.0318 1.4552 1.9913 1.8948 -1.3122 2.6935 12.8942

DHT 5 .2 3.8097 0.3956 3.1490 1.3754 -0.4471 1.9746 -5.7323 7.4088 17.3143

DHT 10 1 4.1283 6.1008 2.8304 1.4791 2.3381 1.8709 -1.5015 1.5651 13.0835

DHT 10 .8 4.1282 5.3175 2.8305 1.4791 1.8870 1.8709 -1.0322 2.3179 12.6142


A. 57

DHT 10 .5 4.0944 3.5354 2.8643 1.4683 0.9464 1.8817 -1.1244 4.9225 12.7064

DHT 10 .2 3.8978 -0.0107 3.0609 1.4054 -0.5995 1.9446 -5.7624 8.6715 17.3444

DHT 15 1 4.1283 2.4010 2.8304 1.4791 0.7278 1.8709 -1.5015 16.5165 13.0835

DHT 15 .8 4.1283 2.4125 2.8304 1.4791 0.6556 1.8709 -1.0322 9.8514 12.6142

DHT 15 .5 4.1193 2.1025 2.8394 1.4762 0.3583 1.8738 -1.0754 8.8173 12.6574

DHT 15 .2 3.9387 -0.2763 3.0200 1.4185 -0.7288 1.9315 -5.7564 9.2192 17.3384

DHT 20 1 4.0983 0.5177 2.8604 1.5718 0.0446 1.7782 -1.3866 16.6314 12.9686

DHT 20 .8 4.0983 0.7346 2.8604 1.5619 0.0531 1.7881 -0.6494 17.3686 12.2314

DHT 20 .5 4.0954 0.9128 2.8633 1.5609 0.0004 1.7891 -1.2818 16.7362 12.8638

DHT 20 .2 3.9368 -0.4509 3.0219 1.5058 -0.7127 1.8442 -5.5015 11.8557 17.0835

DHT 30 1 4.1032 2.8555 1.5600 1.79 -1.3244 12.9064

DHT 30 .8 4.1264 2.8323 1.5207 1.8293 -0.8384 12.4204

DHT 30 .5 4.1028 2.8559 1.4790 1.871 -1.0972 12.6792

DHT 30 .2 3.9758 2.9829 1.4365 1.9135 -5.7159 17.2979

DHT 40 1 4.1264 2.8323 1.5207 1.8293 -1.4180 13

DHT 40 .8 4.1024 2.8563 1.5619 1.7881 -0.8458 12.4278

DHT 40 .5 4.1263 2.8324 1.5207 1.8293 -1.2255 12.8075

DHT 40 .2 4.0197 2.9390 1.4816 1.8684 -5.5194 17.1014

Table A.37 (Pre-coding +logR) Results and compared with the results of each of (Pre-coding) and (logR companding).

Precoding k y X X1 PAPR Y Y1 CCDF of

PAPR

Z Z1 SNR

(BER= )

WHT 5 1 0.4263 22.4114 0.5366 9.368 0.0852 11.5448

WHT 5 .8 4.6287 18.2090 2.2852 7.6194 -0.4767 12.1067

WHT 5 .5 11.9168 10.9209 5.1927 4.7119 -3.2966 14.9266

WHT 5 .2 18.3833 4.4544 7.9769 1.9277 -11.4059 23.0359

WHT 10 1 1.0356 -5.0601 21.8021 0.7807 -1.8094 9.1239 0.0587 0.9287 11.5713


A. 58

WHT 10 .8 5.2030 -2.9008 17.6347 2.5204 -1.0067 7.3842 -0.5679 1.0021 12.1979

WHT 10 .5 11.7711 -1.3242 11.0666 5.1833 -0.4813 4.7213 -3.3075 0.8125 14.9375

WHT 10 .2 18.4830 -0.2614 4.3547 8.0193 -0.0853 1.8853 -11.4272 0.5928 23.0572

WHT 20 1 1.9256 -6.4950 20.9121 1.4846 -2.1290 8.42 -0.2059 1.7801 11.8359

WHT 20 .8 5.9356 -4.0630 16.9021 3.0211 -1.3535 6.8835 -0.7820 1.4260 12.412

WHT 20 .5 12.1732 -1.7503 10.6645 5.4977 -0.4789 4.4069 -3.4292 1.3740 15.0592

WHT 20 .2 18.5564 -0.0850 4.2813 8.1109 0.0983 1.7937 -11.4118 0.8582 23.0418

WHT 30 1 2.8015 -6.3259 20.0362 1.4262 -2.7184 8.4784 -0.2365 2.8718 11.8665

WHT 30 .8 6.5904 -4.7333 16.2473 3.0154 -1.8692 6.8892 -0.8880 2.1820 12.518

WHT 30 .5 12.4999 -1.9038 10.3378 5.5439 -0.7607 4.3607 -3.4371 2.0157 15.0671

WHT 30 .2 18.6257 -0.4202 4.2120 8.1529 -0.0657 1.7517 -11.8373 0.3127 23.4673

WHT 40 1 4.7516 -5.6271 18.0861 2.1352 -2.3524 7.7694 0.0341 4.0906 11.5959

WHT 40 .8 8.0905 -3.9205 14.7472 3.5446 -1.6600 6.36 -0.8033 3.5667 12.4333

WHT 40 .5 13.3591 -1.1251 9.4786 5.7738 -0.5108 4.1308 -3.5819 2.4001 15.2119

WHT 40 .2 18.9242 -0.0447 3.9135 8.2183 -0.0078 1.6863 -11.7032 0.5536 23.3332

WHT 50 1 4.2553 -6.9289 18.5824 2.0956 -2.7290 7.809 -0.1544 4.7956 11.7844

WHT 50 .8 7.6258 -4.9426 15.2119 3.4779 -1.9347 6.4267 -0.9347 3.6953 12.5647

WHT 50 .5 12.9890 -2.0679 9.8487 5.6726 -0.8200 4.232 -3.9124 1.8816 15.5424

WHT 50 .2 18.7335 -0.3517 4.1042 8.1261 -0.1105 1.7785 -11.5775 1.1925 23.2075

WHT 70 1 5.1623 -7.3198 17.6754 2.6547 -2.6579 7.2499 -0.4487 7.9213 12.0787

WHT 70 .8 8.2282 -4.8379 14.6095 3.8796 -1.7250 6.025 -1.1765 4.9185 12.8065

WHT 70 .5 13.2299 -1.9328 9.6078 5.9123 -0.6263 3.9923 -4.1543 2.3437 15.7843

WHT 70 .2 18.7585 -0.1039 4.0792 8.1934 -0.0062 1.7112 -11.6655 0.5845 23.2955

WHT 90 1 5.9939 -7.1447 16.8438 2.7271 -2.7575 7.1775 -0.6104 17.7596 12.2404

WHT 90 .8 8.7922 -4.9729 14.0455 3.9383 -2.0163 5.9663 -1.3420 7.0280 12.972

WHT 90 .5 13.4752 -2.2699 9.3625 5.9566 -0.8240 3.948 -4.0355 3.3159 15.6655

WHT 90 .2 18.8058 -0.3343 4.0319 8.2281 -0.0765 1.6765 -12.0414 0.6886 23.6714

DCT 5 1 4.0783 14.0327 1.5751 6.0889 -0.4586 12.0866


A. 59

DCT 5 .8 5.8776 12.2334 2.4715 5.1925 -0.8859 12.5139

DCT 5 .5 10.1251 7.9859 4.2709 3.3931 -4.2414 15.8694

DCT 5 .2 14.7651 3.3459 6.2418 1.4222 -11.6890 23.317

DCT 10 1 5.2700 3.9010 12.8410 2.0919 1.7424 5.5721 -0.7392 0.1328 12.3672

DCT 10 .8 7.5462 4.1691 10.5648 3.0042 1.7177 4.6598 -1.4409 0.1311 13.0689

DCT 10 .5 10.9668 2.5982 7.1442 4.5075 1.0835 3.1565 -4.2249 -0.1029 15.8529

DCT 10 .2 14.9984 0.9807 3.1126 6.2912 0.4272 1.3728 -11.9552 0.0668 23.5832

DCT 20 1 6.4419 2.7480 11.6691 2.7293 1.3563 4.9347 -1.6394 0.3486 13.2674

DCT 20 .8 8.8030 3.5311 9.3080 3.5713 1.4373 4.0927 -2.3940 -0.1840 14.022

DCT 20 .5 11.5302 2.3334 6.5808 4.7449 1.0089 2.9191 -5.0332 -0.2280 16.6612

DCT 20 .2 15.1147 1.2000 2.9963 6.3263 0.5543 1.3377 -12.1126 0.1594 23.7406

DCT 30 1 8.4747 4.0740 9.6363 3.3960 1.4920 4.268 -2.4175 0.6928 14.0455

DCT 30 .8 9.6024 3.0054 8.5086 3.8900 1.2460 3.774 -2.9251 0.1469 14.5531

DCT 30 .5 11.9149 2.2379 6.1961 4.9083 0.8443 2.7557 -5.0536 0.4012 16.6816

DCT 30 .2 14.9918 0.6726 3.1192 6.3167 0.3387 1.3473 -12.0250 0.1270 23.653

DCT 40 1 8.4754 2.8234 9.6356 3.4950 1.2480 4.169 -3.5951 0.4634 15.2231

DCT 40 .8 9.4797 2.1954 8.6313 3.9296 0.9656 3.7344 -3.7052 0.6668 15.3332

DCT 40 .5 11.6821 1.9246 6.4289 4.8823 0.8383 2.7817 -5.4937 0.4903 17.1217

DCT 40 .2 15.0467 0.8045 3.0643 6.3449 0.3594 1.3191 -12.5112 -0.2524 24.1392

DCT 50 1 9.6001 3.1426 8.5109 3.8828 1.2988 3.7812 -3.6787 1.2733 15.3067

DCT 50 .8 9.8944 2.0527 8.2166 4.0978 0.9258 3.5662 -3.9254 0.7066 15.5534

DCT 50 .5 11.8863 1.5561 6.2247 4.9704 0.7184 2.6936 -6.2297 -0.4337 17.8577

DCT 50 .2 15.0848 0.7263 3.0262 6.3745 0.3785 1.2895 -12.2911 0.4809 23.9191

DCT 70 1 9.7691 2.0137 8.3419 4.0838 1.0118 3.5802 -5.0376 3.3344 16.6656

DCT 70 .8 10.4531 2.1137 7.6579 4.3864 1.0224 3.2776 -4.8662 1.2308 16.4942

DCT 70 .5 12.1597 1.7237 5.9513 5.1309 0.8329 2.5331 -6.3668 0.1332 17.9948

DCT 70 .2 15.2596 1.1239 2.8514 6.4068 0.4478 1.2572 -13.0394 -0.7874 24.6674

DCT 90 1 10.5879 2.1760 7.5231 4.3312 1.0872 3.3328 -6.3649 12.0071 17.9929

DCT 90 .8 11.1396 2.1012 6.9714 4.5770 0.8630 3.087 -5.9901 2.3819 17.6181


A. 60

DCT 90 .5 12.5958 1.5774 5.5152 5.2242 0.6842 2.4398 -6.7500 0.6034 18.378

DCT 90 .2 15.3170 0.9036 2.7940 6.4156 0.3516 1.2484 -12.6807 0.0513 24.3087

DST 5 1 8.5202 8.9447 3.6111 3.9119 -2.6156 14.2436

DST 5 .8 9.3524 8.1125 4.0014 3.5216 -3.4388 15.0668

DST 5 .5 11.4249 6.0400 4.8296 2.6934 -5.9240 17.552

DST 5 .2 14.5077 2.9572 6.2632 1.2598 -13.2083 24.8363

DST 10 1 9.8718 9.1489 7.5931 4.1997 3.9912 3.3233 -5.1839 -4.3119 16.8119

DST 10 .8 10.0125 7.2815 7.4524 4.3860 3.2405 3.137 -5.3699 -3.7979 16.9979

DST 10 .5 11.7969 4.0744 5.6680 5.0592 1.7762 2.4638 -7.6083 -3.4863 19.2363

DST 10 .2 14.6244 1.2528 2.8405 6.3016 0.5786 1.2214 -13.0385 -1.0165 24.6665

DST 15 1 10.4876 6.9773 4.5158 3.0072 -7.2293 18.8573

DST 15 .8 10.8296 6.6353 4.6424 2.8806 -6.8672 18.4952

DST 15 .5 12.3484 5.1165 5.2379 2.2851 -8.0515 19.6795

DST 15 .2 14.6892 2.7757 6.2822 1.2408 -13.6972 25.3252

DST 20 1 10.9548 7.9070 6.5101 4.6516 3.4196 2.8714 -8.9870 -6.9990 20.615

DST 20 .8 11.2190 6.5932 6.2459 4.8452 2.8522 2.6778 -8.1981 -5.9881 19.8261

DST 20 .5 12.3938 3.8431 5.0711 5.2302 1.6352 2.2928 -8.7103 -3.9051 20.3383

DST 20 .2 14.6429 1.3743 2.8220 6.3268 0.6958 1.1962 -13.8951 -1.6231 25.5231

DST 30 1 11.6232 7.8686 5.8417 4.9818 3.2188 2.5412 -15.0476 -11.9373 26.6756

DST 30 .8 11.7005 5.7496 5.7644 5.0121 2.5091 2.5109 -11.0759 -8.0039 22.7039

DST 30 .5 12.5336 3.5027 4.9313 5.3564 1.4334 2.1666 -9.2805 -3.8257 20.9085

DST 30 .2 14.7947 1.1216 2.6702 6.3350 0.4980 1.188 -13.9518 -1.7998 25.5798

DST 40 1 11.9575 6.9516 5.5074 5.0939 2.9879 2.4291 -18.3720 -14.3135 >30

DST 40 .8 12.1610 5.5228 5.3039 5.1844 2.3614 2.3386 -13.6810 -9.3090 25.309

DST 40 .5 12.9059 3.7945 4.5590 5.5148 1.6118 2.0082 -9.9412 -3.9572 21.5692

DST 40 .2 14.8824 1.2863 2.5825 6.3855 0.5410 1.1375 -13.8786 -1.6198 25.5066

DST 50 1 11.9685 6.1571 5.4964 5.1368 2.6938 2.3862 -18.3720 -13.4200 >30

DST 50 .8 12.1433 4.9477 5.3216 5.2135 2.1825 2.3095 -17.2375 -12.6055 28.8655

DST 50 .5 12.8274 3.1433 4.6375 5.5130 1.4020 2.01 -10.5717 -4.7757 22.1997


A. 61

DST 50 .2 14.8421 1.1297 2.6228 6.3588 0.5038 1.1642 -13.7840 -1.0120 25.412

DST 70 1 12.3682 5.2589 5.0967 5.2927 2.3617 2.2303 -18.3720 -10.0000 >30

DST 70 .8 12.5000 4.8067 4.9649 5.3530 2.1300 2.17 -18.3720 -12.2750 >30

DST 70 .5 13.0636 3.2737 4.4013 5.6055 1.4485 1.9175 -11.9018 -5.4018 23.5298

DST 70 .2 14.7955 1.3059 2.6694 6.4405 0.6225 1.0825 -14.2162 -1.9642 25.8442

DST 90 1 12.8291 5.0633 4.6358 5.4648 2.3618 2.0582 -18.3720 0 >30

DST 90 .8 12.9326 4.5403 4.5323 5.5119 1.9389 2.0111 -18.3720 -10.0000 >30

DST 90 .5 13.4032 3.0309 4.0617 5.7247 1.3257 1.7983 -11.9831 -4.6297 23.6111

DST 90 .2 14.9895 1.2222 2.4754 6.4377 0.5147 1.0853 -14.6467 -1.9147 26.2747

DHT 5 1 2.1510 4.8077 0.8595 2.4905 -0.3995 11.9815

DHT 5 .8 2.3800 4.5787 0.9299 2.4201 -0.4631 12.0451

DHT 5 .5 2.8097 4.1490 1.0939 2.2561 -1.8985 13.4805

DHT 5 .2 3.4943 3.4644 1.3200 2.03 -6.1537 17.7357

DHT 10 1 2.5046 12.2879 4.4541 0.9697 4.9342 2.3803 -0.8641 0.0539 12.4461

DHT 10 .8 2.6241 10.3993 4.3346 1.0064 4.0339 2.3436 -0.5591 1.0589 12.1411

DHT 10 .5 2.9457 5.7294 4.0130 1.1058 1.9958 2.2442 -1.5443 2.6237 13.1263

DHT 10 .2 3.5365 0.6711 3.4222 1.2908 -0.2592 2.0592 -6.0508 6.0172 17.6328

DHT 20 1 2.7535 10.2119 4.2052 1.1068 4.0478 2.2432 -0.5381 1.4959 12.1201

DHT 20 .8 2.8192 8.6996 4.1395 1.1329 3.3129 2.2171 -0.4235 1.8325 12.0055

DHT 20 .5 3.0364 4.9919 3.9223 1.2124 1.7904 2.1376 -1.4934 3.3578 13.0754

DHT 20 .2 3.5319 0.7695 3.4268 1.3662 -0.0918 1.9838 -5.6755 6.6425 17.2575

DHT 30 1 2.8996 9.6512 4.0591 1.0915 3.5015 2.2585 -1.0229 2.1334 12.6049

DHT 30 .8 2.9317 7.4870 4.0270 1.0909 2.7609 2.2591 -0.5152 2.6028 12.0972

DHT 30 .5 3.1032 4.5785 3.8555 1.1471 1.3971 2.2029 -1.3132 4.1876 12.8952

DHT 30 .2 3.5497 0.3828 3.4090 1.2944 -0.3696 2.0556 -5.8967 6.3013 17.4787

DHT 40 1 2.9635 8.4638 3.9952 1.1013 3.1683 2.2487 -1.0149 3.0896 12.5969

DHT 40 .8 3.0145 6.8825 3.9442 1.1159 2.4659 2.2341 -0.4055 4.0125 11.9875

DHT 40 .5 3.1604 4.5552 3.7983 1.1686 1.4386 2.1814 -1.4070 4.6230 12.989

DHT 40 .2 3.5777 0.4878 3.3810 1.3201 -0.3514 2.0299 -5.5958 6.7090 17.1778


A. 62

DHT 50 1 3.0338 7.7286 3.9249 1.1228 2.8528 2.2272 -0.9711 4.0269 12.5531

DHT 50 .8 3.0455 6.3561 3.9132 1.1738 2.3158 2.1762 -0.5152 4.1628 12.0972

DHT 50 .5 3.1724 3.9945 3.7863 1.2150 1.2770 2.135 -1.3591 4.4829 12.9411

DHT 50 .2 3.5636 0.3574 3.3951 1.3427 -0.3393 2.0073 -6.0558 6.7622 17.6378

DHT 70 1 3.0908 6.4877 3.8679 1.1885 2.4305 2.1615 -0.8581 7.5599 12.4401

DHT 70 .8 3.1424 5.9553 3.8163 1.1830 2.1330 2.167 -0.6713 5.4717 12.2533

DHT 70 .5 3.2468 3.9631 3.7119 1.2180 1.2340 2.132 -1.2287 5.3173 12.8107

DHT 70 .2 3.6069 0.6235 3.3518 1.3394 -0.3056 2.0106 -5.5747 6.7233 17.1567

DHT 90 1 3.1589 5.8993 3.7998 1.1722 2.2422 2.1778 -1.1746 17.2434 12.7566

DHT 90 .8 3.1763 5.2902 3.7824 1.1776 1.7776 2.1724 -0.6321 7.7859 12.2141

DHT 90 .5 3.2661 3.4000 3.6926 1.2057 0.9797 2.1443 -1.3063 6.0931 12.8883

DHT 90 .2 3.6028 0.3417 3.3559 1.3118 -0.4382 2.0382 -5.9216 6.8564 17.5036

Appendix B MATLAB Code

B. 1

Appendix B

MATLAB Code

&& OFDM CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------

M = 4; % QPSK signal constellatio

fft_size = 128; % have 128 data point

fspacing=15000;

fs=15000*128;

Ts = 1/fs; % Sampling period of channel

Fd = 0; % Max Doppler frequency shift

nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------

% Generate 1 x 128 vector of random data points

source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source

cp_length = .25*fft_size; % length of cyclic prefix

sp = reshape(source , 128 , nsym-2);% s/p

s = size (sp);

tx = [];

for i=2:nsym-1

% QPSK modulation (mapping)

qpsk_mod = pskmod(sp(:,i-1), M);

% making s/p

serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p

% to know the start and end of cp

cp_start = fft_size-cp_length;

cp_end = fft_size;

% make ifft to each block and add CP

ifft_data_matrix = ifft(serial_to_paralle,fft_size);

% Compute and append Cyclic Prefix

for j=1:cp_length,


B. 2

cp(j,1) = ifft_data_matrix(j+cp_start,1);

end

% Append the CP to the existing block to create the actual OFDM block

ifft_data = vertcat(cp,ifft_data_matrix);

% s/p for transmission

[rows_ifft_data, cols_ifft_data]=size(ifft_data);

length_ofdm_data = rows_ifft_data*cols_ifft_data;

pilot = zeros(1,length_ofdm_data);

ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s

Signal_Power = abs(ofdm_signal.^2);

Peak_Power = max(Signal_Power,[],2);

Mean_Power = mean(Signal_Power,2);

PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power);

tx = [tx ofdm_signal];

end

[cdf0, PAPR0] = ecdf(PAPR_Orignal);

%--------------------------------------------------------------------------

semilogy(PAPR0,1-cdf0,'-b')

legend('Orignal')

xlabel('PAPR0 [dB]');

ylabel('CCDF (Pr[PAPR>PAPR0])');

tt =[ pilot tx pilot];

x_abs=abs(tt);

papr=10*log(max(x_abs.^2)/mean(x_abs.^2))

% ------------

% CHANNEL

% ------------

% Create Rayleigh fading channel object.

% Frequency selective channel with 4 taps

tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; % Path delays

pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains

h = rayleighchan(Ts, Fd,tau,pdb);

h.StoreHistory = 0;

h.StorePathGains = 1;

h.ResetBeforeFiltering = 1;

% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


B. 3

for ii=1:length(snr)

rx= [];

rx_signal_matrix = reshape(tt,length(ifft_data),nsym);

[~, c] = size(rx_signal_matrix);

for j = 2: nsym-1

hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel

a = h.PathGains;

AM = h.channelFilter.alphaMatrix;

g = a*AM; % Channel coefficients

G(j,:) = fft(g,fft_size); % DFT of channel coefficients

% Add AWGN no

rx_signal = awgn(hx,snr(ii),'measured');

d=size(rx_signal);

% Convert Data back to "parallel" form to perform FFT

con=reshape( rx_signal , length(ifft_data),1);

% Remove CP

con(1:cp_length,:)=[];

% Perform FFT

% FFT

fft_data_matrix = fft(con,fft_size);

% p/s

rx_serial_data = reshape(fft_data_matrix, 1,fft_size);

fftrx = rx_serial_data./G(j,:);

% Demodulate the data

qpsk_dem_data = pskdemod(fftrx,M);

rx =[rx qpsk_dem_data];

end

figure(2)

[no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation

end

semilogy(snr,ratio,'--*r','linewidth',1);

hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')

title('Bit error probability curve for qpsk using OFDM');


B. 4

&& RCF CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR = 3;

ITERATE_NUM = 4;

K = 128; % SIZE OF OFDM Symbol

IF = 2; % Interpolation factor

(Oversamplingfactor)

fft_size = K*IF; % SIZE OF FFT

mm = 193 %when IF =1.125 =81 ;when IF =1.25 =97 ;when IF

=1.5 =129; when IF = 2 =193; when IF =3 =321; when IF = 4=449

% ------------------

% TRANSMITTER

% ------------------


source = randsrc(1, K*(nsym-2), 0:M-1); %the data source

cp_length = .25*K; % length of cyclic prefix

sp = reshape(source , K , nsym-2);% s/p

s = size (sp);

tx = [];

PAPR_Orignal = zeros(1,nsym);

PAPR_RCF = zeros(ITERATE_NUM,nsym);

for i=2:nsym-1



% making s/p

serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p

xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)];

ifft_data_matrix = ifft(xy,fft_size);


pilot = zeros(1,length(ifft_data_matrix));


B. 5

ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s





for nIter=1:ITERATE_NUM

% Clipping

x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power);

x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp);

ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp;

% Filtering

XX = fft(ofdm_signal,[],2);

XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K);

ofdm_signal = ifft(XX,[],2);

% PAPR Compute




PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power);

end


serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p



cp_end = fft_size;


for j=1:cp_length,

cp(j,1) =serial_to_paralle2(j+cp_start,1);

end


ifft_data_cp = vertcat(cp,serial_to_paralle2);

ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s

pilot = zeros(1,length(ifft_data_cp));

tx = [tx ofdm];

end


[cdf1, PAPR1] = ecdf(PAPR_RCF(1,:));




%--------------------------------------------------------------------------


B. 6

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'-

c',PAPR4,1-cdf4,'-m')

legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four

clip and filter')




x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path power

gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];

rx_signal_matrix =reshape(tt,length( ifft_data_cp),nsym);


for j = 2: nsym-1


a = h.PathGains;



G(j,:) = fft(g,K); % DFT of channel coefficients

% Add AWGN no

rx_signal = awgn(hx ,snr(ii),'measured');


con=reshape( rx_signal, length( ifft_data_cp),1);

% Remove CP


B. 7


% Perform FFT

% FFT


du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(mm:fft_size)];

% p/s

rx_serial_data = reshape(du, 1,K);





end

figure(2)


end

ofdm_signal = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('SNR');

ylabel('BER')


&& RCF I=1 CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR = 2;

ITERATE_NUM = 4;

% ------------------


B. 8

% TRANSMITTER

% ------------------


source = randsrc(1, fft_size*(nsym-2), 0:M-1); %the data source


sp = reshape(source , fft_size , nsym-2);% s/p

s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p

serial_to_paralle = reshape(qpsk_mod, fft_size , 1);% s/p



cp_end = fft_size;


ifft_data_matrix = ifft(serial_to_paralle ,fft_size);









% Clipping




% PAPR Compute





end




B. 9



cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end

figure(1)






%--------------------------------------------------------------------------




clip and filter')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;


B. 10



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no




% Remove CP


% Perform FFT

% FFT


% p/s






end

figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')


B. 11

grid on

xlabel('snr');

ylabel('BER')


&& RCF( I =pilot =76 in this case )CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR = 4;

ITERATE_NUM = 4;


% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p








B. 12






% Clipping




% Filtering




% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end

figure(1)






%--------------------------------------------------------------------------


B. 13




clip and filter')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no





B. 14

% Remove CP


% Perform FFT

% FFT


du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(91:fft_size)];

% p/s






end

% r = rx(1,(K+1:length(rx)-K));

figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& RFC CODE :

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR = 3;


B. 15

ITERATE_NUM = 4;


% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p












% Filtering




% Clipping




% PAPR Compute





B. 16


end





cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end






figure(1)

%--------------------------------------------------------------------------


legend('I =pilot ','1.125','I =1.25 ','I =1.5 ','I= 2','I =3','I =4')




t = size (tx);

Q = size(tt);

figure(2)

plot(real(tt)); xlabel('Time'); ylabel('Amplitude');

title('OFDM Signal');grid on;

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------





B. 17

pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no




% Remove CP


% Perform FFT

% FFT



% p/s






end

figure(2)


end



B. 18

v = size(rx);

semilogy(snr,ratio,'--*b','linewidth',1);

hold on;

axis([0 30 10^-4 1])

legend('I =pilot ','1.125','I =1.25 ','I =1.5 ','I= 2','I =3','I =4')

grid on

xlabel('snr');

ylabel('BER')


&& A_ law CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------



%figure(1)

cp_length = .25*128 ; % length of cyclic prefix


s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p



B. 19



cp_end = fft_size;




for j=1:cp_length,


end







ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/s





A = 90; % Parameter for A-law compander

V = max(abs(ofdm_signal));

compsig = compand(ofdm_signal,A,V,'A/compressor');

Signal_Power = abs(compsig.^2);



PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power);

tx = [tx compsig];

end

figure(1)


[cdf1, PAPR1] = ecdf(PAPR_Orignal1);

%--------------------------------------------------------------------------

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


B. 20

papr=10*log(max(x_abs.^2)/mean(x_abs.^2));

disp('PAPR of original signal in dB');

disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


V = max(abs(rx_signal));

compsig = compand(rx_signal,A, V,'A/expander');


con=reshape( compsig, length(ifft_data),1);

% Remove CP


% Perform FFT

% FFT


% p/s


B. 21






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& CODE

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];


B. 22

for i=2:nsym-1



%scatterplot(qpsk_mod);

%title('MODULATED TRANSMITTED DATA');

% making s/p




cp_end = fft_size;




for j=1:cp_length,


end








ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/





Mu = 700; % Parameter for mu-law compander


x = compand(ofdm_signal,Mu,V,'mu/compressor');

Signal_Power = abs(x.^2);




tx = [tx x];

end

figure(1)



%--------------------------------------------------------------------------


B. 23





x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no



xx = compand(rx_signal,Mu,V,'mu/expander');


con=reshape(xx , length(ifft_data),1);

% Remove CP


B. 24


% Perform FFT

% FFT


% p/s






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& Rooting CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

y = .5;

% ------------------

% TRANSMITTER

% ------------------



B. 25




s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p




cp_end = fft_size;




for j=1:cp_length,


end












a =abs (ofdm_signal );

b =a.^y;

x= complex(b.*cos(angle(ofdm_signal )),b.*sin(angle(ofdm_signal )));





tx = [tx x];

end



%--------------------------------------------------------------------------


B. 26





t = size (tx);

Q = size(tt);

x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


aa =abs (rx_signal);

bb =aa.^(1/y);

xx= complex(bb.*cos(angle(rx_signal)),bb.*sin(angle(rx_signal)));



B. 27


% Remove CP


% Perform FFT

% FFT


% p/s






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& NERF CODE :

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------


B. 28

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p




cp_end = fft_size;




for j=1:cp_length,


end












b =erf(((abs(ofdm_signal)))./(sqrt(2).*std(ofdm_signal)));

f= (((2).*std(ofdm_signal).*b));

h= sign(ofdm_signal).*f;

Signal_Power = abs(h.^2);





B. 29

tx = [tx h];

end


t = size (tx);



%--------------------------------------------------------------------------




x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


B. 30


a =abs(rx_signal);

dd=erfinv((a)./(2).*std(rx_signal));

v =sqrt(2).*std(rx_signal);

s =(v.*dd);

ff=abs(s);

rr =sign(rx_signal).*ff;


con=reshape( rr , length(ifft_data),1);

% Remove CP


% Perform FFT

% FFT


% p/s






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& AEXP CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



B. 31


fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p




cp_end = fft_size;




for j=1:cp_length,


end













B. 32

d =1.4;

a =var(abs(ofdm_signal))+ mean(abs(ofdm_signal));

b =exp(-((abs(ofdm_signal)).^2)./var(ofdm_signal));

c =(1-b).^2;

e =(c).^(d/2);

E =( a./mean(e)).^(d/2);

f= (E.*(1-b)).^(d/2);

h= sign(ofdm_signal).*f;





tx = [tx h];

end


t = size (tx);



%--------------------------------------------------------------------------




x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];


B. 33

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


a =abs(rx_signal);

c =a.^(2/d);

aa =var(abs(rx_signal))+ mean(abs(rx_signal));

b =exp(-((abs(rx_signal)).^2)./var(rx_signal));

cc =(1-b).^2;

e =(cc).^(d/2);

E =( aa./mean(e)).^(d/2);

dd=log(1-(c./E));

v =var(rx_signal);

s =sqrt(-v.*dd);

ff=abs(s);



con=reshape( rr , length(ifft_data),1);

% Remove CP


% Perform FFT

% FFT


% p/s






end

figure(2)


end

x = size (source);


B. 34

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& COS CODE :

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p




cp_end = fft_size;



B. 35



for j=1:cp_length,


end












a =abs (ofdm_signal );

b =sqrt(a);

x= complex(b.*cos(angle(ofdm_signal )),b.*sin(angle(ofdm_signal )));





tx = [tx x];

end



%--------------------------------------------------------------------------





t = size (tx);

Q = size(tt);

x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




B. 36


pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


aa =abs (rx_signal);

bb =aa.^2;

xx= complex(bb.*cos(angle(rx_signal)),bb.*sin(angle(rx_signal)));



% Remove CP


% Perform FFT

% FFT


% p/s






end

figure(2)


B. 37


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


$$ tanhR CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 10;

snr = 0:0.8:30;

k=5;

k1 =1;

y = 1;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p

serial_to_paralle = reshape(qpsk_mod, fft_size, 1);% s/p


B. 38



cp_end = fft_size;




for j=1:cp_length,


end












x = k1.*tanh((((abs(ofdm_signal).*k).^(y)))).* sign(ofdm_signal);





tx = [tx x];

end

figure(1)



%--------------------------------------------------------------------------





x_abs=abs(tt);



disp(papr);

% ------------


B. 39

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


xx = abs((atanh(abs(rx_signal)./(k))).^(1/y))./((k1).^(1/y)) .* sign(rx_signal);



% Remove CP


% Perform FFT

% FFT


% p/s

rx_serial_data = reshape( fft_data_matrix, 1,fft_size);






B. 40

end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& logR CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



%scatterplot(qpsk_mod);

%title('MODULATED TRANSMITTED DATA');

% making s/p



B. 41



cp_end = fft_size;




for j=1:cp_length,


end













k=10;

k1 =1;

y = .6;

x = k1.*log((((abs(ofdm_signal).*k).^(y))+1)).* sign(ofdm_signal);





tx = [tx x];

end

figure(1)



%--------------------------------------------------------------------------





t = size (tx);

Q = size(tt);


B. 42

x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no

rx_signal = awgn(hx ,snr(ii),'measured');h;

xx = abs((exp(abs(rx_signal)./k)-1).^(1/y))./((k1).^(1/y)) .* sign(rx_signal);



% Remove CP


% Perform FFT

% FFT


% p/s


B. 43






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& dht function CODE:

function X=dht(x)

N=length(x);

X=zeros(size(x));

i=sqrt(-1);

for k=0:N-1

for n=0:N-1

X(k+1)=X(k+1)+x(n+1)*(cos(2*pi*k*n/N)+sin(2*pi*k*n/N));

end

end

&&idht function CODE:

function x=idht(X)

N=length(X);

x=zeros(size(X));

i=sqrt(-1);

for k=0:N-1

for n=0:N-1,

x(k+1)=x(k+1)+X(n+1)*(cos(2*pi*k*n/N)+sin(2*pi*k*n/N));

end

end

x=x/N;


B. 44

&& precoding CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];

for i=2:nsym-1



% making s/p

serial_to_paralle = reshape(qpsk_mod, fft_size, 1);% s/p

ddg = dht(serial_to_paralle);

% ddg = fft(serial_to_paralle); % for DFT precoding

% ddg = dct(serial_to_paralle); % for DCT precoding

% ddg = dst(serial_to_paralle); % for DST precoding

% ddg = fwht(serial_to_paralle); % for WHT precoding



cp_end = fft_size;


ifft_data_matrix = ifft(ddg,fft_size);


for j=1:cp_length,


end


B. 45












tx = [tx ofdm_signal];

end


%--------------------------------------------------------------------------


legend('Orignal')




x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



B. 46


for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


d=size(rx_signal);


con=reshape( rx_signal , length(ifft_data),1);

% Remove CP


% Perform FFT

% FFT


dg = idht(fft_data_matrix);

% dg = ifft(fft_data_matrix); % for DFT precoding

% ddg = idct(serial_to_paralle); % for DCT precoding

% ddg = idst(serial_to_paralle); % for DST precoding

% ddg = ifwht(serial_to_paralle); % for WHT precoding

% p/s

rx_serial_data = reshape(dg, 1,fft_size);





end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')



B. 47

hybrid

&& precoding + RCF CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR =4;

ITERATE_NUM = 4;


IF = 2; % Interpolation factor (Oversampling factor)


mm=193; %when IF =1.125 =81 ;when IF =1.25 =97 ;when IF =1.5 =129;

when IF = 2 =193; when IF =3 =321; when IF = 4=449

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p







You can use

another type of

precoding

compnding


B. 48

xy = [ddg(1:K/2) ; zeros(fft_size-K,1); ddg(K/2+1:K)];










% Clipping




% Filtering




% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end

figure(1)



B. 49





%--------------------------------------------------------------------------




clip and filter')




t = size (tx);

Q = size(tt);

figure(2)

plot(real(tt)); xlabel('Time'); ylabel('Amplitude');

title('OFDM Signal');grid on;

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;


B. 50




% Add AWGN no




% Remove CP


% Perform FFT

% FFT


du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(mm:fft_size)];

dg = idht(du);

% ddg = ifft(du); % for DFT precoding

% ddg = idct(du); % for DCT precoding

% ddg = idst(du); % for DST precoding

% ddg = ifwht(du); % for WHT precoding

% p/s

rx_serial_data = reshape(dg, 1,K);





end


figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


You can use

another type of

precoding

compnding


B. 51

&& precoding +RCF I =1 CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 10;

snr = 0:0.8:30;

CR = 2;

ITERATE_NUM = 4;

% ------------------

% TRANSMITTER

% ------------------


source = randsrc(1, fft_size*(nsym-2), 0:M-1); %the data source


sp = reshape(source , fft_size , nsym-2);% s/p

s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p

serial_to_paralle = reshape(qpsk_mod, fft_size , 1);% s/p








cp_end = fft_size;




B. 52









% Clipping




% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end

figure(1)






%--------------------------------------------------------------------------


B. 53




clip and filter')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no





B. 54

% Remove CP


% Perform FFT

% FFT




% dg = idct(fft_data_matrix); % for DCT precoding

% dg = idst(fft_data_matrix); % for DST precoding

% dg = ifwht(fft_data_matrix); % for WHT precoding

% p/s






end


figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& precoding + RCF( I =pilot =76 in this case )CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;




B. 55

nsym = 1000;

snr = 0:0.8:30;

CR = 1.5;

ITERATE_NUM = 4;


% ------------------

% TRANSMITTER

% ------------------



%figure(1)



s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p







xy = [ddg(1:K/2) ; zeros(fft_size-K,1); ddg(K/2+1:K)];










% Clipping




% Filtering



B. 56



% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





tx = [tx ofdm];

end

figure(7)






%--------------------------------------------------------------------------




clip and filter')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);



B. 57

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no




% Remove CP


% Perform FFT

% FFT



dg = idht(du);

% ddg = ifft(du); % for DFT precoding

% ddg = idct(du); % for DCT precoding

% ddg = idst(du); % for DST precoding


B. 58

% ddg = ifwht(du); % for WHT precoding

% p/s

rx_serial_data = reshape(dg, 1,K);





end

figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& precoding + companding code:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 100;

snr = 0:0.8:30;

% ------------------

% TRANSMITTER

% ------------------



cp_length = .25*128 ; % length of cyclic prefix


s = size (sp);


B. 59

tx = [];

for i=2:nsym-1



% making s/p









cp_end = fft_size;




for j=1:cp_length,


end







ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/s





A = 10; % Parameter for A-law compander


compsig = compand(ofdm_signal,A,V,'A/compressor');

Signal_Power = abs(compsig.^2);




tx = [tx compsig];

You can use

another type of

compnding

compnding

You can use

another type of

precoding

compnding


B. 60

end

figure(1)



%--------------------------------------------------------------------------





t = size (tx);

Q = size(tt);

x_abs=abs(tt);



disp(papr);

% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no


B. 61



compsig = compand(rx_signal,A, V,'A/expander');


con=reshape( compsig, length(ifft_data),1);

% Remove CP


% Perform FFT

% FFT




% dg = idct(fft_data_matrix); % for DCT precoding

% dg = idst(fft_data_matrix); % for DST precoding

% dg = ifwht(fft_data_matrix); % for WHT precoding

% p/s






end

figure(2)


end

x = size (source);

v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& RCF + companding CODE:

clear all

clc

close

% ---------------

% Parameters

You can use

another type of

compnding

compnding

You can use

another type of

precoding

compnding


B. 62

% ---------------



fspacing=15000;

fs=15000*128;



nsym = 1000;

snr = 0:0.8:30;

CR = 4;

ITERATE_NUM = 4;


IF = 2; % Interpolation factor

(Oversampling factor)


% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p












% Clipping



B. 63



% Filtering




% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





k= 5;

k1 =1;

y = 1;

x = k1.*tanh((((abs(ofdm).*k).^(y)))).* sign(ofdm);





tx = [tx x];

end

figure(1)






You can use

another type of

compnding

compnding


B. 64


%--------------------------------------------------------------------------


c',PAPR4,1-cdf4,'-m',PAPR5,1-cdf5,'-k')


clip and filter','tanh')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no



B. 65

xx = abs((atanh(abs(rx_signal)./(k))).^(1/y))./((k1).^(1/y)) .* sign(rx_signal);


con=reshape(xx , length(ifft_data_cp),1);

% Remove CP


% Perform FFT

% FFT


du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix((fft_size/2+K/2)+1:fft_size)];

% p/s






end

figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


&& RFC + companding CODE:

clear all

clc

close

% ---------------

% Parameters

% ---------------



fspacing=15000;

fs=15000*128;



You can use

another type of

compnding

compnding


B. 66

nsym = 1000;

snr = 0:0.8:30;

CR = 3;

ITERATE_NUM = 4;


IF = 2; % Interpolation factor (Oversampling

factor)


d =.8;

% ------------------

% TRANSMITTER

% ------------------





s = size (sp);

tx = [];



for i=2:nsym-1



% making s/p












% Filtering




% Clipping




B. 67


% PAPR Compute





end





cp_end = fft_size;


for j=1:cp_length,


end





a =var(abs(ofdm))+ mean(abs(ofdm));

bb =cos(-((abs(ofdm)))./std(ofdm));

b =exp(-((abs(ofdm)))./std(ofdm));

c =(1-b).^2;

e =(c).^(d/2);

E1 =( a./mean(e)).^(d/2);

f= (E1.*(1-bb)).^(d/2);

h= sign(ofdm).*f;





tx = [tx h];

end

figure(1)







You can use

another type of

compnding

compnding


B. 68

%--------------------------------------------------------------------------


c',PAPR4,1-cdf4,'-m',PAPR5,1-cdf5,'-k')


clip and filter','COS')




t = size (tx);

Q = size(tt);

x_abs=abs(tt);


% ------------

% CHANNEL

% ------------




pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; % Avg path

power gains


h.StoreHistory = 0;



% ---------------

% RECEIVER

% ------------

no_of_error=[];

ratio=[];


rx= [];



for j = 2: nsym-1


a = h.PathGains;




% Add AWGN no



B. 69

a =abs(rx_signal);

c =a.^(2/d);

aa =var(abs(rx_signal))+ mean(abs(rx_signal));

b =exp(-((abs(rx_signal)).^2)./var(rx_signal));

cc =(1-b).^2;

e =(cc).^(d/2);

E =( aa./mean(e)).^(d/2);

dd=acos(1-(c)./E);

v =std(rx_signal);

s =(-v.*dd);

ff=abs(s);



con=reshape( rr , length(ifft_data_cp),1);

% Remove CP


% Perform FFT

% FFT


du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix((fft_size/2+K/2)+1:fft_size)];

% p/s






end


figure(2)


end


v = size(rx);


hold on;

axis([0 30 10^-4 1])

legend('simulated')

grid on

xlabel('snr');

ylabel('BER')


You can use

another type of

compnding

compnding

B. 70

الخالصة

انزي أصثخ ,ض يرؼذد انالم فؼال ( ظاو ذشيOFDMيعاػفح ذمغى انرشدد انرؼايذ )

يؤخشا شؼثا غثا ف كم ي انظى االذصاالخ انغهكح انلعهكح نهثااخ انعائػ انرؼذدج

خػ اشرشان سل غش يراشمك اعرخذاي ف صى انظى انؼشفح يصم OFDMمم.

(ADSLاإلرشد، انرهفض انشل / انثس اإلراػ، انشثكح ) ( انذهح انلعهكحLANs ،)

(.LTE) يششع يرطس غم األيذ

، يا ؤدي إنى OFDM انؼة انشئغ ل (PAPR) انؼذلإنى انمذسج انؼظىاسذفاع غثح

أكصش PAPR.يشكهح الOFDMاخفاض انكفاءج ف اعرلن انمذسج تانران ؼشلم ذفز

سج أيش تانغ األح خاصح ف انذطح انرمهح أل أح ف اإلسعال أل كفاءج يعخى انمذ

انثطاسح نذا غالح يذذدج.

زا االيش رطهة OFDMذذذز رجح نرمهثاخ كثشج ف إشاسج انذايهح نم PAPRاسذفاع

(. يعخاخ انمذسج انؼانح ػذيا ذك HPAدسجح ػانح ي انخطح ف يعخى انمذسج انؼان )

٪ ي دجى االسعال صؼتح ثخ غانح انص، ظخح تغثح ذصم انىخطح تشكم كثش ذص

انرصغ.

،نمذ ذى الرشاح ػذج ذماخ ف ز األغشدح، اال لذ ذى الرشح ػهى PAPRي أجم انذذ ي

( تانماسح يغ انطشمح انرفشج ذكشاس انمص RFCذكشاس ذصفح طاق انرشدد لص )

، I ف األداء خصصا ػذيا كد RCFأفعم RFC(. RCFذصفح طاق انرشدد ) ػهى انشغى ي أ نذى فظ انرؼمذ انركهفح.

يؼذل خطأ انشاسج اا أعا ذذغ PAPRانطشمح انمرشدح نغد فمػ ذؼم ػهى ذذغ

(BER)أفعم رجح ف ز انطشمح نم .BER ػذ I = 4 CR = 4 دس ف دس ،

6.تمذاس ) ( ) BERػذيا (SNR) غغثح اإلشاسج إنى انعججذذغد

66.تمذاس ) PAPR( نم CCDFدغثم(، ذذغ انذانح انكهح نهرصغ انرشاك)

دغثم(. 66.تمذاس ) PAPRدغثم(، ذذغ

I = 4 ف BERتذس ال ذرذس PAPR CCDF of PAPRأفعم ذذغ ادذج ف

CR = 1.75 يمذاسانرذغ ف .PAPR ( =8.681 ،)دغثمCCDF of PAPR (=

دغثم(. .يمذاس = ) ( )SNR at BERدغثم(، ذرذغ 8.0187

يماسرا يغ لا companding( لذ ذى الرشاح عرح أاع جذذج ي RFCتاإلظافح إنى )

μ- لاA compandings -ز األعانة انمرشدح نا أداء أفعم ي لا . كمμ

( (AEXPيمرشح انطهك األع companding، أفعم compandings A لا

. يمذاس .= d ػذيا ذك PAPR CCDF of PAPRافعم يمذاسذذغ ف

دغثمCCDF of PAPR (7.2405 ،) =دغثمPAPR ( =6.1 ،)انرذغ ف

دغثم(. 8.-ذرذسيمذاس= ) ( )SNR at BERا ت

( ف ز االغشدح ي شى يماسرا يغ precodingذى اعرخذاو خغح أاع ي لثم انرشيض)

ذذم فس PAPR BERف ذمهم precodingتؼعا انثؼط. أفعم ع ي

.WHT)ذذم انش اداياسد )تا اعء ع يماسح يغ انثمح DFT)(انرمطغ

B. 71

. ز انطشق :PAPRكا ذى الرشاح أستؼح أاع جذذج ي ذماخ جح نهذذ ي

. RCFيغ precodings WHT) ،ذذم جة انراو انرمطغ (DCT) ، ذذم جة

((.DHTانرمطغ) اسذه ذذم(، DST) انرمطغ

. . RCF يغcompandings ع ال)نجغ ااcompandings انمرشدح، انما- μ

(.A compandings-انما

. .RFC يغcompandings نجغ ااع ال(compandings انمرشدح، انما- μ

( A compandings-انما

)نجغ compandings(.، يغ precodings WHT) ،DCT ،DST ،DHT . أخشا

(A compandings-انما μ -انمرشدح، انما compandingsااع ال

PAPR ،CCDF of( ألا ذؼم ػهى ذذغ كم ي AEXPيغ RFCأفعم دانح )

PAPR ،BER افعم لح نرذغ .PAPR CCDF of PAPR, ػذ d =.

CR = 4 يمذاسانرذغ ف .PAPR = (.1 )دغثم CCDF of PAPR,(= 8.7178

دغثم(. .) =( )SNR at BER ثم( دغ

DHT ( يغ ظم ذاو انجزسيtanhR نا رائج جذج دس ذرذغ كم ي )PAPR

CCDF of PAPR تا ال ذرذسBER .يمذاس كثش

. k = ،y =.8 DHT ػذ ,PAPR CCDF of PAPRأفعم لح نرذغ

دغثمCCDF of PAPR(= 8.9691 )دغثمPAPR = (.66 ،)يمذاسانرذغ ف

دغثم(. 88.-) ذرذس يمذاس=( )SNR at BER تا

ذى يذاكاج كم انطشق تاعرخذاو ياذلب.

نظرة عامة : تقلل نسبة القدرة العظمى الى المعدل مضاعفة تقسم التردد المتعامد باستخدام ف نظام

طرق بعض الطرق الجددة

)مع ماتالب كود(

تأليف

زنب سعد هادي الهاشم

الت خرجة كلة الهندسة قسم االكترونك واالتصا

جامعة بغداد

1436 2015

ظمى الى المعدل نظرة عامة: تقلل نسبة القدرة العمضاعفة تقسم التردد المتعامد باستخدام ف نظام

)مع ماتالب كود( بعض الطرق الجددة

زنب سعد هادي الهاشم