par reduction in mimo broadcast ofdm systems using tone...

PAR Reduction in MIMO Broadcast OFDM

Systems using Tone Reservation

Maja Taseska

Jacobs University Bremen

Electrical and Computer Engineering

A thesis submitted for the degree of

Bachelor of Science

May, 2010

Supervisor: Prof. Dr.-Ing Werner Henkel

mailto:[email protected]

http://www.jacobs-university.de

mailto:[email protected]

Abstract

The combination of multicarrier modulation (OFDM, DMT) and Mul-

tiple-Input Multiple-Output (MIMO) transmission significantly improves

the quality and efficiency of modern communication systems. The peak-

to-average ratio problem in multicarrier modulation requires additional

transmitter-sided processing which may become considerably complex in

the case of MIMO transmission. This problem is particularly emphasized

in MIMO broadcast scenarios (MIMO BC), where the necessary precoding

block at the transmitter side severely affects the complexity of the PAR

reduction algorithms. Many of the most important PAR reduction schemes

that have been developed for SISO (Single-Input Single-Output) and MIMO

systems, become too complex and inadequate for practical implementations

when precoding is performed at the transmitter. Herein, on the one hand,

we will refer to Selected Mapping (SLM) and Selected Sorting (SLS) PAR

reduction schemes, which operate by generating multiple signal represen-

tations in the frequency domain. On the other hand we will analyze the

performance of the Tone Reservation Method (TR), originally proposed by

Jose Tellado. We show that the Tone Reservation method, which operates

entirely in time domain, significantly improves the PAR in MIMO scenarios

as well, while retaining low complexity.

Contents

1 Introduction 1

2 The Principles of OFDM 4

2.1 Introduction: Single-Carrier Modulation

vs OFDM? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 OFDM Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Discrete Fourier Transform (DFT) . . . . . . . . . . . . . . . . 5

2.2.2 The Cyclic Extension . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Peak-to-Average Power Ratio in OFDM . . . . . . . . . . . . . . . . . 10

3 Multilple-Input Multiple-Output Transmission (MIMO) 14

3.1 Overview of MIMO systems . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.1 MIMO Channel Model . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 SVD-based Two-Sided Processing in a Single-User (Point-to-Point) Sce-

nario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.3 MIMO Multi-User Scenario . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3.1 QR Decomposition for MAC (upstream) Interference Cancellation 18

3.3.2 Precoding for the Broadcast Channel . . . . . . . . . . . . . . . 19

3.4 MIMO-OFDM System . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 PAR Reduction in SISO and MIMO Systems 23

4.1 Overview of the Existing Methods . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Selected Mapping (SLM) . . . . . . . . . . . . . . . . . . . . . . 24

4.1.2 Partial Transmit Sequences (PTS) . . . . . . . . . . . . . . . . . 26

4.1.3 Tone Reservation . . . . . . . . . . . . . . . . . . . . . . . . . . 27

ii

CONTENTS

5 PAR Reduction by the Tone Reservation Method in a MIMO Broad-

cast Scenario 31

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 The System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.3 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.3.1 The PAR Reduction . . . . . . . . . . . . . . . . . . . . . . . . 34

5.3.2 Comments on Complexity . . . . . . . . . . . . . . . . . . . . . 36

6 Conclusions 38

iii

Chapter 1

Introduction

Designing wireless digital communication systems that efficiently exploit the spatial

domain of the transmission medium provides a significant improvement of the spectral

efficiency. Multiple-Input Multiple-Output (MIMO) systems employ spatial multiplex-

ing by using multiple transmit and receive antennas and provide significant capacity

gains over single path systems. The combination of the recent advances in MIMO

systems with the numerous well analyzed advantages of multicarrier modulation, par-

ticularly Orthogonal Frequency Division Multiplexing (OFDM), gave rise to the MIMO

OFDM systems which are the most prominent systems in modern wireless communi-

cations technology. OFDM as the most popular technique for digital transmission over

frequency selective channels, is a crucial part of the MIMO systems, because it trans-

forms the wideband frequency selective MIMO channel into a number of parallel flat

fading subchannels. This allows for less complex signal processing, needed for reliable

support of multiple users.

While OFDM transmission over mobile communication channels alleviates the dis-

tortions induced by the channel, and constitutes a crucial block of the MIMO systems,

current research efforts are focused on dealing with the inherent drawbacks of the

multi-carrier modulated systems, one of them being the high Peak-to-Average Power

ratio (PAR). This problem is expressed more than in single carrier systems due to the

fact that the independent subcarriers of the OFDM are added together increasing the

probability of forming large peaks which significantly exceed the average power value.

This in turn, leads to non-linear distortion at the power amplifier and out-of-band

1

radiation. There are two approaches suggested to mitigate the problem that the PAR

of an OFDM system can cause: either developing techniques to reduce the PAR, or

improving the amplification stage of the transmitter. Regarding the first solution, sev-

eral distortionless peak-power reduction techniques have been developed which offer

satisfactory improvement at the cost of additional signal processing at the transmitter.

There are well developed schemes for PAR reduction in single antenna systems (SISO),

and the same are being modified and adjusted for MIMO systems. The Selected Map-

ping scheme (SLM), proposed by Fischer, Huber and Bauml, and its modification for

MIMO broadcast channels, the Selected Sorting (SLS), proposed by Fischer and Siegl

in [4], operate in frequency domain by allowing for a selection of an optimal signal

representation. The Tone Reservation method (TR) is based on subtraction of time

domain Dirac-like signals from the original multicarrier symbol and operates fully in

time domain.

In this work, we concentrate on the PAR reduction problem in MIMO OFDM sys-

tems, particularly in the MIMO broadcast channel case (downlink). Highest complexity

of the PAR algorithms can be observed in the case of a broadcast channel, where due

to the physical separation of the users at the receiving end, joint processing is only

possible at the transmitter. Non-linear precoding is necessary to reduce spatial inter-

ference among the multiple users. SLM and SLS have been applied on this transmitter

structure of the BC scenario in [4], and PAR improvement is observed at very large

computational cost. In our work, we simulate the MIMO BC transmitter structure

and we apply Tellado’s Tone Reservation Method (TR) after the precoding block. The

benchmarks presented in the results section are the complexity of TR versus the com-

plexity of SLM, and the increase in complexity of TR when used in the MIMO BC

scenario as compared to the TR employed in SISO. Regarding the performance of the

TR algorithm in MIMO systems, we compare two different approaches of the algorithm

application, based on the selection of the reduced peak. We show that the Tone Reser-

vation Method is applicable in MIMO OFDM systems as well, and offers very good

performance at lower complexity than the other PAR reduction methods.

In Chapter 2, the principles of multicarrier modulation (OFDM) are revised and the

PAR problem is formalized, Chapter 3 is an overview of the variants of MIMO systems

2

(Point-to-Point, Multiple Access - MAC, Broadcast - BC) and a detailed description

of the precoding procedure in the downlink case is given. In Chapter 4, we look more

closely at the theory and the ongoing research into the PAR reduction methods men-

tioned above, and their performance and complexity. In Chapter 5, our transmitter

model is described, where we apply the Tone Reservation Method for PAR Reduc-

tion. The results of the simulation are presented along with complexity observations.

Chapter 6 presents the conclusions of the work.

3

Chapter 2

The Principles of OFDM

2.1 Introduction: Single-Carrier Modulation

vs OFDM?

Modern communication services are in continuous demand for higher data rates.

The reliability of the data transmitted at high rates highly depends on the digital

transmission infrastructure. In single-carrier systems, as higher rates are used, the

symbol duration becomes shorter and thus the system becomes more susceptible to

information loss from impulse noise, and Intersymbol Interference (ISI) becomes more

pronounced.

OFDM extends the concept of single-carrier modulation by using multiple subcarriers

within the same channel. The total data rate is then divided among the subcarriers.

The serial data stream is passed through a S/P converter and is split into data blocks of

size N , where N is the number of parallel channels (subcarriers) of the system. Each

data channel is used to modulate a separate carrier at different frequencies. Since

every N th symbol is applied to a carrier, this procedure has an interleaving effect on

the symbols. At the receiver, the OFDM signal is demultiplexed and the N separate

carriers are demodulated. Then, the baseband signals are deinterleaved using a P/S

converter to obtain the original data. In practice, OFDM is implemented using the

Discrete Fourier Transform (DFT). The sinusoids of DFT form an orthogonal basis set

and a signal in the vector space of the DFT can be represented as a linear combination

4

2.2 OFDM Structure

of the orthogonal sinusoids. The Inverse Discrete Fourier Transform (IDFT) is used at

the OFDM transmitter to map the input signal onto a set of orthogonal subcarriers,

and similarly, the DFT is used again at the OFDM receiver to recover the received

subcarriers. The orthogonality of the subcarriers offers higher spectral efficiency than

the conventional Frequency Division Multiplexed (FDM) systems (more about FDM

in [10, 11]).

In single-carrier modulation, the data stream is applied to a single carrier, and

the modulated signal occupies the whole bandwidth available for the system. This

means that impulse noise, or deep fades of the channel are likely to cause bursts of

transmission errors. In contrast, in multi-carrier system, during N symbol periods

each of the subcarriers transmits only one symbol, each with N -times longer duration.

Therefore, an identical channel fade would only affect a fraction of the duration of

each symbol. In this case, the system is more likely to recover the partially distorted

N symbols, and is thus less susceptible to impulse noise and channel fades. Moreover,

as the symbol duration is increased, the ISI decreases.

The usage of a cyclic prefix in the multi-carrier system implementation offers further

enhancement of the performance of OFDM, namely, a very effective ISI mitigation and

a possibility of simple equalization techniques. The length of the cyclic prefix, Lcp

should be at least LC− 1 samples long, in order to achieve complete removal of the ISI

(LC is the length of the impulse response of the channel). The cyclic prefix of length

Lcp, is realized by appending the last Lcp time domain samples of the previous OFDM

symbol to the new symbol. The next section explains the usage of the cyclic prefix, as

well the overall structure of the OFDM in more detail .

2.2 OFDM Structure

2.2.1 Discrete Fourier Transform (DFT)

The key components of a modern OFDM system implementation are the IDFT

and the DFT at the transmitter and the receiver respectively. For a system with N

5

2.2 OFDM Structure

subcarriers, these components have to implement the following operations:

dn =1√N

N−1∑k=0

DkWknN , n = 0, 1, ..., N − 1 , (2.1)

Dn =1√N

N−1∑k=0

dkW−knN , n = 0, 1, ..., N − 1 , (2.2)

where WN is the N th root of unity:

WN = ej2π/N . (2.3)

The DFT essentially correlates the input with each of the sinusoidal basis functions.

If the input signal has energy at a certain frequency, there will be a peak in the

correlation of the input with the sinusoid at that frequency. Since the basis functions of

the DFT are orthogonal (uncorrelated), the correlation performed in the DFT domain

for a given subcarrier only sees the energy for that subcarrier. The energies from the

other subcarriers do not contribute.

The formation of a single OFDM symbol is as follows: the IDFT block takes a

vector of length N , which contains frequency domain complex data symbols drawn

from a particular constellation (usually the standard PSK or QAM):

D = [D0, D1, . . . , DN−1] (2.4)

In the IDFT block, this vector is converted to a time domain vector by using Eq. (2.1):

d = [d0, d1, . . . , dN−1] (2.5)

The IDFT correlates the frequency domain input with its orthogonal basis functions

and maps it onto the sinusoidal subcarriers. The IDFT output signal is actually a

summation of all sinusoidal basis functions. The output vector of N time-domain

samples makes up a single OFDM symbol. At the receiver, a DFT is used to perform the

reverse operation according to Eq. (2.2), i.e, to transform the time domain vector back

into frequency domain. The IDFT and the DFT blocks are practically implemented

by using efficient FFT algorithms.

6

2.2 OFDM Structure

2.2.2 The Cyclic Extension

The method of introducing a cyclic prefix prior to transmission in ODFM allows for

easy ISI removal and enables efficient channel equalization. In order to understand the

principle, consider a linear time-invariant channel (LTI) with a finite number of taps

represented by a vector of length L:

h = [h0, h1, . . . , hL−1] , (2.6)

and recall the important property of continuous time and discrete time Fourier trans-

forms (FT and DFT):

FT{d(x) ? h(x)} = FT{d(x)} × FT{h(x)} , (2.7)

DFT{d[n]⊗ h[n]} = DFT{d[n]} ×DFT{h[n]} , (2.8)

where ?, ⊗, and × denote linear convolution, circular convolution, and multiplication,

respectively.

In DFT domain, the duality between convolution in time domain and multiplication

in frequency domain does not hold for the linear convolution, but for the circular one.

However, in the transmission process, the OFDM signal is linearly convolved with the

channel impulse response. The cyclic prefix is realized in such a way that for every

block of length N denoted by

d = [d0, d1, . . . , dN−1] , (2.9)

we form an input block of length N + L− 1:

d = [dN−L+1, dN−L+2, . . . , dN−1, d0, d1, . . . , dN−1] . (2.10)

This creates pseudo-periodicity of the input which enables the linear convolution to be

equivalent to a circular one. The beneficial result is that the effect of the channel in

frequency domain becomes multiplicative, according to Eq. (2.8). This means that the

spectrum of the OFDM signal (i.e. the frequency domain constellation points) will be

multiplied with the frequency response of the channel. Each subcarrier symbol will be

multiplied by a complex factor corresponding to the channel response at the subcarrier’s

frequency. The effects of the channel can be undone by designing a frequency-domain

7

2.2 OFDM Structure

equalizer which is much simpler than time domain equalizers. It takes place after the

FFT block of the receiver, and compensates for the phase and amplitude distortion of

the channel by performing a single complex multiplication per subcarrier. In addition,

the ISI is easily removed, because it extends only into the first L− 1 received symbols,

which are anyhow discarded at the receiver as they correspond to the cyclic prefix. The

relevant data is ISI - free. The simplification of dealing with ISI comes at the cost of

reduced power and data rate efficiency. Namely, part of these two resources has to be

allocated for the cyclic prefix and therefore cannot be used for data transmission. The

fraction of data rate and energy that is lost is LL+N

. The overhead can be reduced by

increasing the number of carriers N as much as possible. However, due to the time-

varying nature of the channels in practical implementations, the maximum number of

carriers that can be used is limited.

INPUT BITSTREAM QAM/PSK

MAPPING

IDFT

P/S

+

CYCLICPREFIX

CHANNEL

S/P

REMOVEPREFIX

DFT+

D1

D2

DN dN

d2

d1 d1

d2

dN

dN-L+1 yN-L+1

y2

yN

y1 y

1

y2

yN

Y1

YN

Y2

Figure 2.1: OFDM Block Diagram

Looking into the OFDM system in terms of matrices gives useful insights regarding

the relation between frequency selective channel and MIMO channel discussed in the

next chapter. In order to further clarify the concept behind OFDM, we will rehearse

the standard approach of a matrix description of the circular convolution and the DFT,

used in the standard Signals and Systems textbooks. The starting point for the analysis

is the duality between circular convolution and DFT, expressed in Eq. (2.8). Let d

be a time-domain vector of length N . If the channel impulse response has a length L

(L ≤ N), for convenience, we apply zero-padding of length N − L in order to be able

to use the channel impulse response in the matrix operations. Thus we obtain vector

8

2.2 OFDM Structure

h of length N

h = [h0, h1, . . . , hL−1, 0, . . . , 0]

If we take T to be the Toeplitz matrix of the channel (circular convolution matrix),

and y the signal after the channel (at the receiver), then the convolution of the input

with the noiseless channel can be represented by the linear operation:

y = Td ; (2.11)

Now, Eq. (2.8) can be equivalently written as:

WTd = ΛWd , (2.12)

where W is the DFT matrix,

1 1 1 · · · 11 W W 2 · · · WN−1

1 W 2 · · · · · · W 2(N−1)

.... . . . . .

......

. . . . . ....

1 WN−1 · · · · · · W (N−1)(N−1)

(2.13)

and Λ is a diagonal matrix with diagonal entries equal to the DFT coefficients of the

channel. Now, by simply reordering Eq. (2.12), we obtain:

T = W−1ΛW , (2.14)

which means that the Toeplitz matrix is diagonalized in the coordinate system defined

by the columns of the DFT matrix and the eigenvalues of T are the DFT coefficients

of the channel impulse response h. Finally, for noiseless channel, we have the following

input-output relation:

y = Td = WΛW−1d . (2.15)

When the channel is Gaussian, with AWGN (Additive Gaussian White Noise), the

input-output relation is:

y = Td + n = WΛW−1d + n . (2.16)

where both n and n are AWGN vectors.

9

2.3 Peak-to-Average Power Ratio in OFDM

Utilizing this matrix analysis in the context of the OFDM structure, where D is

a frequency-domain input vector, T is the Toeplitz matrix of the channel and Y is

the frequency domain vector on the receiver side after the DFT block, we obtain the

following equation:

Y = W−1TWD

⇒ Y = ΛD (2.17)

Under the assumption that a cyclic prefix is used in the process, the last equations show

that the OFDM decomposes the ISI channel into N orthogonal flat fading subchannels.

This is accomplished by using the IDFT and DFT rotations as pre and post-processing

matrices. Knowledge of the channel frequency response matrix H is not necessary

neither at the transmitter nor at the receiver.


The major drawback of the multicarrier modulation is the possibility of extreme

amplitude excursions of the signal in time domain. The basic problem is that depending

on the transmitted symbol sequence, the subcarriers can constructively interfere to yield

amplitudes that scale with the number of subcarriers N . Thus, the peak power scales

as N2, whereas the average scales only as N . This creates the possibility that the

PAR reaches values as high as N . For example, if N = 256, this means that the PAR

can reach up to 24 dB, imposing the requirement that the power amplifier stays in a

linear region over a range of 24 dB higher than operating power required on average.

However, the PAR value of N occurs with extremely low probabilities of order 10−5.

Thus, the best way to describe the PAR is by assigning probability distribution to

it. For that purpose, the statistical model of the transmitted time-domain samples

d0, d1, . . . , dN should be analyzed.

As described in the previous sections, the time domain OFDM signal is obtained by

taking the IDFT of the vector D of frequency-domain symbols. Thus, it represents

a sum of complex exponential functions, whose amplitudes and phases depend on the

data contained in D. The symbols from the most commonly used PSK and QAM con-

stellations satisfy the following statistical properties: E{Dk} = 0 and E{|Dk|2} = 1,

10


where E{♦} denotes the expected value. Assuming that the different symbols Dk are

independent, the time domain samples dk represent linear combination of N indepen-

dent identically distributed random variables (IDFT is a linear map). By the central

limit theorem, dk can be approximated by Normal Gaussian complex, i.e, Rayleigh dis-

tributed random variables. If the variance is normalized to σ = 1√2, then the cumulative

distribution function (cdf) of a Rayleigh random variable is:

P{x ≤ x0} = 1− e−x0 (2.18)

Moreover, due to the row orhtogonality of the IDFT matrix, the random variables dk

are uncorrelated to each other, which in the case of Gaussian random variables implies

that they are independent. We have now presented all the statistical analysis necessary

to derive the PAR in an OFDM system. The expression that defines the PAR is:

PAR =maxk{|dk|2}E{|dk|2}

. (2.19)

From the analysis of the distributions of dk, it follows that the random variables Pk ≡|dk|2 can be modeled as Rayleigh random variables, where by convenience, they can be

normalized so that E{Pk} = 1. Taking into account this normalization in Eq. (2.19),

we obtain PAR = maxk{Pk}. Thus, the cumulative distribution function (cdf) of the

PAR can be computed as follows:

P [PAR > γ] = 1− P [maxk{Pk} ≤ γ]

= 1− P [P0 ≤ γ, P1 ≤ γ, . . . , PN−1 ≤ γ]

= 1− (1− e−γ)N (2.20)

Here, we used the independence of dk. It is clear from the expression that the PAR

problem becomes more severe as the number of subcarriers, N , increases. However,

the above mentioned PAR value N for a system with N subcarriers is very pessimistic

for properly randomized data. The PAR in practical systems is usually always below

15 dB, which is still considered undesirably high for the system even though it occurs

with very low probabilities.

The problem of high peaks in the amplitude is most severe at the transmitter output.

In order to transmit these peaks without clipping, the range of the D/A converter must

accommodate the peaks, and more importantly, the power amplifier must remain linear

11


over an amplitude range that includes the peak amplitudes. This implies a significant

back-off of the average operating power with respect to the maximum operating power

available, leading to poor power efficiency of the amplifier. In order to avoid the

large back-offs, saturation or clipping of the amplifier must be allowed. However, this

additional non-linear distortion leads to increased bit-error rates (BER) and causes

spectral widening of the signal and adjacent channel interference.

Linear

Region

NonlinearRegion

Vin

Vout

Figure 2.2: Power Amplifier Response

Therefore, a different approach and additional transmitter signal processing are

required to reduce the PAR in an OFDM system. The most popular and the most

efficient techniques for PAR reduction are analyzed in detail in Chapter 4.

12


Figure 2.3: Probability of the peak amplitude per OFDM symbol being above the

threshold value

Figure 2.4: Probability of the instanteneous signal amplitude of the time-domain

OFDM signal being above the threshold value (N = 128)

13

Chapter 3

Multilple-Input Multiple-Output

Transmission (MIMO)

3.1 Overview of MIMO systems

Multiple-Input Multiple-Output (MIMO) systems operate with multiple antennas

at the transmitter and the receiver. The multiple antennas provide additional spatial

dimensions for communication and can be used to increase data rates through mul-

tiplexing as well as improve performance through diversity. Multiplexing is realized

by exploiting the structure of the channel gain matrix to obtain independent signaling

paths that can be used to send independent data, thus increasing channel capacity.

The multiple antennas can also be used to obtain diversity, by sending the same sym-

bol via many antennas. The performance gain of the MIMO systems comes with an

increased cost of the multiple antennas used and their space and power requirements,

as well as increased signal processing complexity.

MIMO communication systems have gained a lot of popularity mostly because of

their usage in single-user point-to-point scenarios. However, recent research has proven

that multi-user systems can exploit the multiple spatial dimensions offered by the mul-

tiple antennas even more efficiently than the single-user systems. In addition to the

multiplexing and diversity gains as in the single-user case, multiple antennas allow the

base station to simultaneously transmit and receive data from the different users. In

a multi-path environment, the channel affects the different input signals differently,

14

3.1 Overview of MIMO systems

and therefore, the output signal needs to be decomposed in order to obtain the orig-

inal signals corresponding to the different transmitter antennas. The most common

mathematical tools used for MIMO channel decomposition are the Singular Value De-

composition for the point-to-point (single-user) scenarios where two-sided processing is

applicable, and the QR decomposition for the multi-user scenarios, where due to spatial

separation of either the transmitters or the receivers, only one-sided signal processing

is possible. These techniques will be explained in the subsequent sections.

3.1.1 MIMO Channel Model

In the most general model, the MIMO channel is considered to cause temporal (ISI)

and spatial (multiuser) interference. Suppose that the impulse response has length L.

Then the channel can be represented by the matrix polynomial:

H(z) =L−1∑k=0

Hkz−k , (3.1)

where each matrix Hk is an Nr×Nt matrix, containing the complex channel coefficients

of delay k between all pairs of antennas. For reliable data transmission over this kind

of channel, the following is required:

• Temporal interference (ISI) is eliminated by employing multi-carrier modulation

(OFDM) in the system. The available bandwith is thus partitioned into many

orthogonal bands which can be modeled as narrowband channels by Nr×Nt chan-

nel coefficient matrices. Denote the channel matrix of subcarrier n as H(n). Then

the narrowband transmission for that subcarrier can be modelled by the equation

y(n) = H(n)x(n) + n , (3.2)

where H(n) is a Nr×Nt matrix of the channel gains H(n)ij from the ith transmit

to the jth receive antenna at subcarrier n, x(n) is Nt×1 input vector, y is Nr×1

output vector, and n is the noise vector. The noise samples are assumed to be

white with variance N2

.

• Spatial interferences which occur in each of the narrow-band subchannels are

eliminated by additional transmitter and/or receiver side processing, reviewed in

the next sections.

15

3.2 SVD-based Two-Sided Processing in a Single-User (Point-to-Point)Scenario

3.2 SVD-based Two-Sided Processing in a Single-

User (Point-to-Point) Scenario

The multiplexing gain of a MIMO channel results from the fact that it can be decom-

posed into R parallel independent subchannels where R = rank{H}. By multiplexing

independent data onto these independent channels, an R-fold increase in data rate can

be obtained, in comparison to a system with just one antenna at the transmitter and

receiver.

The parallel decomposition of the channel is obtained by defining a transformation on

the channel input and output x and y through transmitter pre-processing and receiver

post-processing. The underlying concept is the singular value decomposition of the

channel matrix,

H = UΣVH, (3.3)

where U and V are Nr×Nr and Nt×Nt unitary matrices, respectively, and Σ is a

Nr×Nt diagonal matrix, with entries the singular values σi of the channel matrix H,

i = 1, 2, . . . , R , i.e, the square roots of the eigenvalues of HHT = HTH.

In this analysis we assume the channel to be narrowband, so that it can be modeled

by the matrix of complex channel coefficients H. When the SVD of the channel matrix

is computed, the pre- and post-processing are realized by the linear transformations:

x = Vx (3.4)

y = UHy (3.5)

The diagonalization of the channel matrix is clearly explained through the following

equations:

y = UH (Hx + n)

= UH(UΣVHx + n

)= UH

(UΣVHVx + n

)= UHUΣVHVx + UHn

= Σx + n (3.6)

Since unitary linear transformations do not change the average power of white Gaussian

noise, the input noise n and the output noise vector n follow the same distribution.

16

3.3 MIMO Multi-User Scenario

Η = UΣVHV UHx x~ y~ y

Figure 3.1: SVD Channel Decomposition

In this way, the presented method of transmitter pre- and receiver post-processing,

transforms the channel into R parallel subchannels with input vector x, output vector

y and channel gains σi, i = 1, 2, . . . , R. The parallel subchannels are then used for

data multiplexing. It can be noticed that the SVD decomposition of a MIMO channel

is very similar to the OFDM transmission explained in Section 2.2.2. Both transform

the channel into parallel subchannels by unitary matrix operations at the transmitter

and the receiver. However, in the SVD decompositon, the channel matrix has to be

known on both sides.


In order to support multiple users, the signal space dimensions of a multi-user system

must be allocated to the different users. There are multiple ways to distribute the

resources among the different users, resulting in a capacity region for the system, which

determines at which rates can the users communicate with arbitrarily small probabiltiy

of error. In this work, the relevant aspect of the multi-user scenarios is the transmitter

or receiver-side signal processing, necessary for achieving cancellation of the spatial

interference. The spatial separation of the users at the receiver side for MAC or at the

transmitter side for BC scenarios, do not allow for a double-sided processing as the

one based on SVD. In the MAC case, only post-processing and in the BC case only

pre-processing can be performed. The QR channel decomposition leads to a spatial

decision feedback equalizer at the MAC receiver, or a spatial Tomlinson-Harashima

precoder (THP) at the BC transmitter.

17


3.3.1 QR Decomposition for MAC (upstream) Interference

Cancellation

In this section, a very short outline of this scenario is given. The underlying concept

is to consider the QR decomposition of the channel matrix:

H = QR , (3.7)

where Q is a unitary and R an upper triangular matrix. The post-processing of the

received signal y is done by first multiplying with the matrix QH, where the superscriptH denotes Hermitian transpose. We assume the channel to be narrowband

and the channel matrix to be a U×U matrix, i.e, there are U users each equipped

with a single antenna, and U antennas at the base station.

y = QH QRx︸︷︷︸y

+n = Rx + n (3.8)

I - [diag(R)]-1R

decisiondiag(R)-1QH

yy~

x

Figure 3.2: Spatial DFE based on QR decomposition

Due to the fact that R is a triangular matrix, the estimates xk of the transmitted

symbols xk can be easily obtained:

xk =1

Rk,k

· yk −U∑

j=k+1

Rk,j

Rk,k

· xj (3.9)

18


3.3.2 Precoding for the Broadcast Channel

The relevant scenario for this work is the one involving communication from a base

station to a number to geographicaly separated users. Due to the geographical separa-

tion, joint processing at the receiver is not feasible. Therefore, signal processing based

on the QR decomposition is performed at the transmitter. The concept is similar to

the one presented for the MAC scenario. For the model we consider again the coeffi-

cient matrix H as described in the previous section. The first step is to obtain the QR

decompostion of the Hermitian transpose of matrix H.

HH = QR (3.10)

Denote as x the input to the channel, obtained from the information vector x by

appropriate preprocessing, which is now going to be analyzed. Then the following

equation holds:

y = RHQHx + n . (3.11)

Q is a unitary matrix, so multiplication by Q simply rotates the signal space without

causing an energy increase. Therefore, we can define x′ such that x = Qx′. Now, Eq.

(3.11) becomes:

y = RHx′ + n (3.12)

For the information vector x, reception without interference can be achieved if the

following equation holds:

diag(RH)x = RHx′ , i.e., (3.13)

r11x1r22x2

...rUUxU

=

r11 0 · · · 0r21 r22 · · · 0...

.... . .

...rU1 rU2 · · · rUU

·x′1x′2...x′U

Due to the triangular structure of the matrix RH, the equation leads to the following

precoding operation:

19

3.4 MIMO-OFDM System

x′1 = x1

x′2 = x2 −r21r22

x′1

...

x′U = xU −rU,U−1rUU

x′U−1 − · · ·rU,1rUU

x′1

This computation may lead to very large energy increase of x′ and thus modulo opera-

tion has to be included in order to bound the energy of the transmitted signal. Finally,

the precoding operation can be described as follows:

x′1 = x1

x′2 = ΓM2

[x2 −

r21r22

x′1

]...

x′U = ΓMU

[xU −

rU,U−1rUU

x′U−1 − · · ·rU,1rUU

x′1

] (3.14)

ΓMi[x] is defined as:

ΓMi[x] = x−Mid

⌊x+ Mid

2

Mid

⌋, (3.15)

where Mi is the PAM constellation size for user i, d is the constellation point spacing,

and x is a real number. If M -QAM is employed, as it will be done in the relevant

simulations for this work, x is a complex number and then ΓMi[x] is defined as:

Γ√Mi[Re(x)] + jΓ√Mi

[Re(x)] (3.16)


We have seen in the previous sections that many of the explained methods for pro-

cessing the MIMO channel matrix were based on the assumption of a narrowband

channel. A way to remove this constraint is by employing multi-carrier modulation

20


I - [diag(RH)]-1RH

modulo Qx x' x~

Figure 3.3: Precoder based on QR decomposition (spatial THP)

(OFDM or DMT) and then perform the described signal processing techniques to each

subcarrier separately. The key requirement of the overall system is that the cyclic

prefix is at least as long as the channel impulse response, as described in Section 2.2.2

and that the IDFT blocks of all users are synchronized. Then the model described

with Eq. (3.2) holds for each subcarrier and each can be pre- and/or post-processed.

In the context of this work, the relevant scenario is combining the precoding proce-

dure described in the previous section, with OFDM modulation. The way to do this

is to regroup the U information vectors

Xu = [X(1)u , X(2)

u , . . . , X(N)u ] , (3.17)

where X(n)u is the data of user u at subcarrier n, and to form N new vectors

D(n) = [D(n)1 , D

(n)2 , . . . , D

(n)U ] , (3.18)

where D(n)u denotes the data of user u at subcarrier n, and N is the number of sub-

carriers. After having formed the vectors D(n) for each subcarrier, then the precoding

can be applied on each subcarrier, separately.

21


PRECODER 1

PRECODER 2

PRECODER N

ANTENNA 1

ANTENNA 2

ANTENNA U

CHANNEL

RECEIVER 1

RECEIVER 2

RECEIVER U

.

.

.

.

.

.

.

.

.

D(1)

D(2)

D(N)

Figure 3.4: MIMO BC OFDM System

22

Chapter 4

PAR Reduction in SISO and

MIMO Systems

4.1 Overview of the Existing Methods

The simplest technique to reduce the PAR is to clip the signal such that the peak

amplitude is reduced to the desired maximum level. Since large peaks occur with low

probability (see Eq. (2.20)), in some cases clipping could be an effective technique

for PAR reduction. However, clipping is a nonlinear process which may cause inband

distortion and out-of-band noise. These severely degrade the system performance, and

cause interference to other systems. The clipping technique can be improved by using

a different pulse shaping function instead of a rectangular one. In any case, in order

to reduce out-of-band radiation, the window should be as narrowband as possible.

This implies a longer window in time domain, which adversely affects the bit-error rate

(BER). By making an acceptable trade-off, in some cases clipping and signal windowing

can be effective techniques of PAR reduction.

However, new distortionless schemes have been developed which introduce nonlinear

predistortion of the transmit signal to prevent signal peaks prior to amplification. In

this chapter, we will give an overview of the newest and most efficient distortionless

PAR reduction methods. Some of them are based on generating a set of OFDM signals

by multiplying the data vector in frequency domain with different phase vectors and

choosing the optimal one for transmission. Another group of peak amplitude reduction

23


techniques is based on modifying the time domain signal, wherever its amplitude ex-

ceeds the desired peak limit. The straightforward clipping and windowing mentioned

previously belong to this group as well. The method that we will put emphasis on is

the Tone Reservation method (TR), which is one of the most efficient PAR reduction

techniques, has very low complexity and operates entirely in time domain. All of the

mentioned techniques can be modified for usage both in SISO and MIMO systems.

In this work, we will particularly analyze the performance of the Tone Reservation

method in MIMO broadcast scenario and compare its performance to the other popu-

lar techniques when used in similar scenarios.

4.1.1 Selected Mapping (SLM)

This method benefits from the multicarrier signal sensitivity to phase shifts. The

original vector of frequency domain symbols, D, of length N , where N is the number of

subcarriers, is multiplied by M different phase vectors P(m), m = 1, 2, . . . ,M creating

M different representations of the original block. The phase vectors are of the following

type:

P(m) = [P(m)1 , P

(m)2 , . . . , P

(m)N ] (4.1)

where P(m)µ = ejφ

(m)µ , φ

(m)µ ∈ [0, 2π), µ = 1, 2, . . . , N ,

and each of the new M signal representations is:

D(m)

= D�P(m), (4.2)

where � denotes element-wise multiplication. The M different signal representations

are then converted to time domain and the block with minimum PAR is transmitted.

Ideally, M representations are independent mappings of the original vector D. In that

case the probability in Eq. (2.20) decreases exponentially with M :

P{PAR > γ} = [1− (1− e−γ)N ]M (4.3)

The reduced PAR comes with the cost of increased computational complexity in the

OFDM IDFT block, because for every transmitted OFDM frame, M IFFTs have to

be performed instead of one. At the receiver, the signal must be post-processed in

order to obtain the correct estimate D of the original block of symbols D. Thus, the

receiver needs to know the vector P(m) that was chosen at the transmitter to minimize

24


the PAR. Therefore, the index m has to be transmitted with high reliability, which

requires additional information of dlog2(M)e and additional measures to ensure reliable

transmission of m. Eventually, the decoding process at the transmitter performs the

following operation:

D = DFT{d(m)} �P∗(m) (4.4)

where � is element-wise multiplication, and ∗ denotes conjugation.

For SISO systems, the application of the SLM method is straightforward according

to the theory given above. This technique can easily be extended to MIMO systems.

• SLM in MIMO Single-User Scenario (Point-to-Point)

The described SLM technique can be applied easily in MIMO single user sce-

narios as independent implementation of the SISO SLM for every transmitted

antenna. The complexity and the additional information for the phase vectors

used now increase linearly with the number of antennas since MNT IDFTs and

NT dlog2(M)e bits of side information are necessary (NT is the number of transmit

antennas).

• SLM in MIMO Multi-User Uplink (MAC)

In this case, every user is assumed to be equipped with a single transmit antenna,

and again SISO SLM is applied on every transmit antenna.

• SLM in MIMO Multi-User Downlink (BC)

This is scenario is of biggest interest in current research, because the precoding

block at the transmitter described in Section 3.3.2 leads to very high complexity of

the system when SLM is employed. The SLM technique, when used as described,

requires receiver side channel equalization, because it is impossible to determine

the impact of the phase vector P(m)µ to the µth transmit antenna if precoding

is performed at the transmitter, since the precoding permutes the order of the

antennas (users) differently for different subcarriers. However, in BC scenarios,

joint processing at the receiver is not applicable. Another alternative is to perform

the phase rotation after the precoding block, which is also not applicable, since

the rotation would distroy the channel equalization of the precoding block. The

only possibility to apply SLM in this case is if one optimal phase vector P (u) is

25


chosen and used for each of the transmitting antennas. This will, of course, lead

to reduced efficiency in the PAR reduction, and moreover the complexity of the

precoding procedure becomes very high for practical implementation. Therefore,

the SLM method is not optimal in the case of a broadcast channel. A modification

of SLM, the Selected Sorting (SLS), proposed by Fischer and Siegl in [?], is

designed particularly for the broadcast channel. It operates in the same way as

the SLM method, but in order to reduce complexity, only selected subcarriers are

multiplied by the phase vectors.

4.1.2 Partial Transmit Sequences (PTS)

Partial Transmit Sequences represents a PAR technique derived from SLM. The

difference is that instead of rotating all subcarries by different phases as in the case

of SLM, the subcarriers are grouped and rotated block by block. The first step of the

algorithm is a subdivision of the frequency domain OFDM block D of length N into NB

disjoint blocks of length B = NNB

. Then, NB vectors, denoted as Xi, i = 1, 2, . . . , NB,

each of length N are formed. The vector Xi corresponds to the ith symbol block, where

the entries that correspond to the subcarrier indices belonging to other block, are set

to zero. In this way, we have:

X =

NB∑i=1

Xi ; (4.5)

Next, each Xi vector is rotated by constant rotation factor:

ri = ejφ φi ∈ [0, 2π) , (4.6)

to obtain the modified vector:

X =

NB∑i=1

ri ·Xi =

NB∑i=1

ejφ ·Xi ; (4.7)

The vector X is converted to time domain by the IFFT block of the OFDM:

x = IFFT{NB∑i=1

ejφ ·Xi} =

NB∑i=1

ejφ · IFFT{Xi} =

NB∑i=1

ejφ · xi ; (4.8)

The last step of the algorithm is picking optimal set of phase rotations,

{ropt1 , ropt2 , . . . , roptNB} that minimizes PAR.

26


An important advantage of PTS over SLM is the fact that the PAR optimization

can be done in time domain after the IFFT block, so no additional IFFTs are required.

This is possible due to the linearity of the IFFT, as shown in Eq. (4.8). Research

has proven that PTS has a slightly better performance than SLM. PTS can also be

modified for the MIMO scenarios, in the same way as SLM, described in the previous

section.

Data SourceX

S / P

+

Partition

intoBlocks

X1

X2

XNB

IFFT

IFFT

IFFT

x1

x2

xNB

r1 r2 rNB

OPTIMIZATION

+ x

Figure 4.1: Partial Transmit Sequences diagram

4.1.3 Tone Reservation

The Tone Reservation method is based on iteratively adding a data-dependent, time

domain signal to the original symbol to reduce its peaks. The transmitter compute the

signal by time-domain efficient methods, which results in reduced complexity of the

system as compared to the previously mentioned methods. Tone Reservation belongs

to larger class of PAR reducion methods known as Additive Methods, or Average Power

Increasing Methods, due to the fact that by adding a signal to reduce the peaks, the

average power of the transmitted symbols is increased. Instead of using signal clipping

for PAR reduction, which leads to non-linear distortion of the signal, the idea of the

Tone Reservation method makes use of the finding that the clipped portion of the

27


signal can be represented as:

x− xclip =∑i

βi · (δ → mi) , (4.9)

where βi are the values that exceed the threshold xclip, and mi are the clip locations.

→ m denotes a cyclic shift by m and δ is the Dirac vector. However, the generation

of a Dirac function requires the whole DFT frame which would occupy the whole

transmission capacity of the system. Instead, the Tone Reservation method generates

Dirac-like signals p using only some reserved carriers (frequency bins) and subtracts

them iteratively from the signal formed by the remaining carriers. This leads to the

following approximation of Eq. (4.9):∑i

βi · (δ → mi) ≈∑i

γi · (p→ mi) , (4.10)

where γi are weighting coefficients.

The frequency bins used for the Dirac-like function generation can either be chosen

at random or can be fixed. If they are chosen at random, after certain number of trials,

a set of bins is found that shows a sufficient peak compared to the sidelobes in the

time domain vector. The other option is to have the reserved bins fixed beforehand,

either by using the carriers that are not used for data transmission or where the SNR

is not satisfactory. After selecting the bins and storing the corresponding time domain

vector, the tone reservation algorithm is applied according to the following steps:

1. Initialize X to be the DFT-domain information vector when the reserved bins

are set to zero.

2. Initialize the time domain solution x(0) to x, obtained as the IFFT of X

3. Find the value x(i)m and location m for which |x(i)m | = maxk|x(i)k |.

4. If |x(i)m | < xtarget or i > imax then stop and transmit x(i), otherwise

5. Update the time-domain vector:

x(i+1) = x(i) − α ·(x(i)m − sign

(x(i)m)· xtarget

)· (p→ m) (4.11)

i := i+ 1 and go to Step 3.

28


The term α ·(x(i)m − sign(x

(i)m ) · xtarget

)is a possible realization of the weighting coef-

ficients γi in Eq. (4.10). α represents the step size and is chosen dependent on xtarget

which denotes the desired maximum value after the algorithm. The iterative applica-

tion of the algorithm tries to force the exceeding values to below the threshold xtarget.

The diagram of the Tone Reservation method is shown in the next figure. c(t) is the

time domain Delta-like function that is generated using the reserved carriers and is

added to the original signal.

CHANNEL

D0

D1

DN

C1

CN

C2

IDFT

IDFT

d(t)

c(t)

d(t)+c(t)

Figure 4.2: Tone Reservation diagram

Despite its efficiency in terms of computational complexity, the Tone Reservation

method also has several disadvantages that should be pointed out. The first and most

obvious drawback is the inevitable decrease in data rate due to the reserved carriers.

In some systems, it is possible to overcome this drawback by using carriers with very

low SNR which are anyway not used for data transmission. However, it may happen

that the generated signal by those predefined carriers does not show a significant peak

29


with respect to its sidelobes, so the problem of choosing optimal carriers for reservation

is still a challenge. On the one hand, the choice of carriers certainly affects the amount

of PAR reduction, but on the other hand, searching over all possible sets significantly

increases the complexity for practical implementation. There is a trade-off between

the number of reserved carriers and the performance of the method. The more carriers

are reserved, the larger is the PAR reduction and less power is needed for the power

reduction signal, at the expense of the other system resource, namely the data rate.

30

Chapter 5

PAR Reduction by the Tone

Reservation Method in a MIMO

Broadcast Scenario

5.1 Motivation

In the previous chapter some of the recent and most efficient PAR reduction tech-

niques have been presented. For SISO systems, these techniques have been well ana-

lyzed and optimized, and current research has been dealing with applying these tech-

niques to MIMO systems. Increasing the number of antennas at the transmitter, in-

evitably leads to an increase in the complexity of any PAR reduction technique, because

the PAR reduction algorithms need to be applied to all antennas, separately. In the

scenarios where joint signal processing can be performed at the receiver side, such as

in the single-user point-to-point and in the multi-user MAC (multipoint-to-point) sys-

tems, the complexity increase is related only to the increase in number of transmit

antennas. However, in the multi-user BC (point-to-multipoint) systems, the precoding

block at the transmitter additionaly affects the complexity, usually to an extent that

is undesirable for practical implementations.

The SLM technique, presented in Section 4.1.1 has many shortcomings when used

in broadcast scenarios. Moreover, even in the other MIMO systems, single-user, or

multi-user MAC, the multiple IFFT operations at each antenna, already impose very

31

5.2 The System Model

large computational complexity. In addition, we saw that for the BC case, there is the

restriction that the same vector has to be used for all the antennas, due to the presence

of a precoding block. Eventually, it is clear that this leads to a less efficient PAR

reduction procedure, where the complexity is severely increased, and moreover, the

amount of PAR reduction is reduced. In this chapter, we will analyze the performance

of the Tone Reservation method in MIMO systems, particularly in the MIMO broadcast

system, and we will see that this method offers a high efficiency even for this critical

scenario. The main reason is that the complexity of the algorithm is independent of the

precoding block, and the only PAR reduction complexity increase of any MIMO system

over a SISO system arises due to the larger number of antennas. In addition, the fact

that TR operates entirely in time domain, saves lot of computational complexity that

is lost with the IFFT operations when SLM algorithm is used.

5.2 The System Model

In practical situations, the most general MIMO channel model can be described by

Eq. (3.1). Therefore, as explained in the previous chapters, an OFDM modulation

is necessary to decompose the wideband MIMO channel into multiple narrowband

channels. In the MIMO broadcast scenario, a precoding block is employed at each

subcarrier as shown in Fig. 3.4. The Tone Reservation algorithm described in Section

4.1.3, is applied to the time domain signals at each transmitter, just after the IFFT

block of the OFDM. In the MIMO system, there are multiple OFDM transmitters

and TR is separately applied at each of them, before transmission. We will consider

the transmitter model discussed in Section 3.4, whereby we add a processing block

that performs the PAR reduction using TR algorithm. The full transmitter model is

presented in Fig. 5.1.

32

5.3 Simulation and Results

PRECODER 1

PRECODER 2

PRECODER N

ANTENNA 1

IFFT

ANTENNA 2

IFFT

ANTENNA U

IFFT

CHANNEL...

.

.

.

D(1)

D(2)

D(N)

d1

d2

dU

TR

Algorithm

TR

Algorithm

TR

Algorithm

d1^

d2

dU

Figure 5.1: Transmitter Model


For the simulation of the transmitter model in the previous figure, we used the

following parameters:

• NT = NR = U = 4, where U is the number of users at the receiver side;

• The length of the channel impulse response is L = 5, and the channel coefficients

are complex i.i.d. Gaussian distributed with variance 1L

;

• The randomly generated information symbols are drawn from a 16-QAM con-

stellation;

• Number of carriers of the system: N = 128;

The channel is assumed to be time-varying, and therefore the Q and R matrices used

in the precoding procedure described in (3.3.2) have to constantly be recomputed after

several OFDM blocks. After precoding each subcarrier and applying the IFFT on the

frequency domain vectors of all antennas, as shown in the previous figure, we are ready

to apply the Tone Reservation method following the algorithm in Section 4.1.3.

33


5.3.1 The PAR Reduction

In order to assess the performance of the algorithm in the presented scenario, we

analyzed the sample Complementary Cumulative Distribution Function (CCDF) of

the PAR of the time domain signal before, and after the peak reduction. There are

two factors which generally affect the amount of peak reduction:

1. The number of iterations, i.e., how many times the Delta-like function is sub-

tracted from the original signal in order to reduce the critical peaks.

2. The percentage of reserved carriers, used for generation of the Delta-like function.

The more carriers are reserved, the more emphasized is the peak in the Delta-

function with respect to the rest of the samples. This contributes to faster and

more substantial PAR reduction of the data signal, at the expense of the data

rate. In order to present the effect of the number of subcarriers, we performed

a simulation both with 5 % and 10 % reserved carriers, and compared the PAR

reduction after 4 iterations of the algorithm.

These two factors are the common factors for evaluating the performance of the Tone

Reservation algorithm. Here, we present another, third aspect of its performance,

which is specific only for the case when the TR algorithm is used in a MIMO scenario.

3. The selection of a peak for reduction. Namely, at the output of the IFFT block

of all the NT transmitters of the system, we have a signal with high peak-to-

average ratio. As described in the Tone Reservation algorithm in Section 4.1.3, a

search should be performed to find the maximum peak, and then cyclically shift

the Delta-like function to the appropriate position. Once we set the maximum

number of iterations for the system to I, there are two ways to apply the searching

for the worst peak:

• An equal number of iterations is performed at each antenna, i.e., b INTc

• The total number of iterations is used to search all the antennas jointly, thus

selecting the worst peak among all the antennas.

The latter searching method is more efficient, because it performs the peak reduc-

tion exactly on those samples, where the PAR is most critical, and in general, the

34


critical samples are not equally distributed among all the antennas. Therefore,

giving equal treatment to all the antennas, is not the most efficient approach. The

increased efficiency when performing a joint search, comes at the cost of a more

time-consuming search, as it has to be performed on a longer array, containing

the samples from all transmit antennas.

The results of the simulation of the overal model, and the influence of the three factors

mentioned above, are shown in the next two plots.

Figure 5.2: Comparison between the two search methods, for a different number of

iterations. The number of iterations shown on the plot is the average number of

iterations performed per antenna. For example, if the number is 4, this means total of

16 iterations are applied in the system, with either of the search methods.

35


Figure 5.3: Effect of the number of reserved carriers

5.3.2 Comments on Complexity

The application of the PAR algorithms in MIMO systems is a very popular field in

current research, and the main objective is to reduce the complexity to a level that

is not problematic for practical applications. In the previous chapter, we explained

the drawback of the SLM method when it is applied in a MIMO scenario, namely

the severe increase in computational complexity due to the multiple frequency-time

domain transformations at each antenna. Moreover, in the scenario of interest, i.e., the

broadcast scenario, the complexity and the performance efficiency of the SLM method

is additionally affected by the precoder.

Having multiple antennas at the transmitter, would inevitably lead to increased

complexity of any PAR reduction method, as the algorithms have to be applied on

all transmit antennas. However, performing the TR iterations on the time domain

signal requires much lower computational cost than an IFFT operation. This already

36


offers and advantage of the TR method over SLM, where all the modified data vectors

have to be transformed in time domain by IFFT, and then the best one is selected for

transmission. However, one more important observation of this work is the fact that

the Tone Reservation method, does not suffer from additional complexity due to the

precoder block in a broadcast system. The PAR reduction is done on the time-domain

signal at each transmitter, and the effects of any additional transmitter pre-processing

are irrelevant at this stage.

37

Chapter 6

Conclusions

In this work we analyzed the performance of the Tone Reservation method for PAR

reduction in MIMO broadcast systems. In the first chapters, the importance of the

OFDM and MIMO systems was presented, along with the PAR problem of the multicar-

rier modulation. Chapter 4 gave an overview of the newest PAR reduction techniques

and their application in SISO and MIMO scenarios. The fact that the Tone Reser-

vation method has already proven to be extremely efficient in single-antenna systems,

was a motivation to try to apply this method to the case of multiple antennas, and

moreover, in broadcast systems, where the precoding block usually severely affects the

PAR reduction algorithms. We simulated the system model, where all of the system

components (precoding block, OFDM, and PAR reduction block) were included. The

simulations analyzed the statistical behavior of the time-domain signal taking all an-

tennas into account, before and after the Tone Reservation algorithm was applied. In

order to characterize the performance, we compared the PAR according to:

1. Number of iterations applied (4,8,12);

2. Searching method for the peaks (Search performed at each antenna individually,

versus, overall search on all antennas);

3. Number of reserved carriers for generation of the Delta-like signal. (5 % versus

10 %).

The amount of PAR reduction depends on all of these factors. For optimum results,

a trade-off between computational efficiency, data rate, and PAR reduction has to be

38

found. Obviously, the more iterations are applied, the more peaks are reduced and

there is a larger reduction in the PAR. However, it should be noted that regardless

of the number of iterations, the reduction power of the algorithm is limited by the

generated Delta-like function. Namely, due to the fact that this signal is generated

only by a small percentage of subcarriers, its addition to the original data signal in-

troduces undesired amplitude variations not only at the critical peak, but at the other

time-domain samples, as well. Reserving more subcarriers for the Delta-like signal,

increases its PAR reduction potential, but at the same time leads to higher losses of

data rate. Eventually, analyzing the specific factor for MIMO scenarios, namely, the

peak-search method, we conclude that performing a joint search over all antennas leads

to significantly larger PAR reduction than performing the search on each antenna, sep-

arately. The reason for this is that the first search ensures that the overall most critical

peaks will be treated first by the algorithm iterations, whereas the latter one performs

equal number of iterations on all antennas. As it may happen that the large peaks are

concentrated only at a small number of antennas, performing all the iterations only on

these antennas, would lead to much better PAR results than trying to reduce peaks at

the antennas which do not have a critical PAR ratio at all.

Finally, it is obvious that there are many aspects to be optimized and improved in

each of the PAR reduction algorithms, including the Tone Reservation, when they are

applied in MIMO scenarios. This problem represents an open and challenging field in

current research which deserves further work and effort for improvement. Designing

power- and data rate-efficient MIMO OFDM transmitters greatly contributes to the

development of modern wireless communication systems.

39

APPENDIX

The appendix for this work is not public.

Bibliography

[1] J. Tellado, “Peak-to-Average Power Reduction for Multicarrier Modulation,” Ph.D.

thesis, Stanford University, Sept. 1999.

[2] J. Tellado, J.M. Cioffi, “PAR Reduction in Multicarrier Transmission Systems,”

Delayed Contribution ITU-T 4/15, D.150 (WP 1/15), Geneva, February 9-20, 1988

[3] “Wireline Communications”, An online book project of the wireline communications

course at Jacobs University Bremen , edited by W. Henkel

http://trsys.faculty.jacobs-university.de

[4] W. Henkel, V. Zrno , “PAR reduction revisited: an extension to Tellado’s method,”

6th International OFDM Workshop, Hamburg, 2001 2

[5] C. Siegl, R.F.H. Fischer , “Peak-to-Average Ratio Reduction in Multi-User OFDM,”

ISIT2007, Nice, France, June 24-June 29, 2007

[6] G. Ginis, J.M. Cioffi, “A Multi-user Precoding Scheme achieving Crosstalk Can-

cellation with Application to DSL Systems,” Department of Electrical Engineering,

Stanford University

[7] R.J. Baxley, “Analyzing Selected Mapping for Peak-to-Average Power Reduction

in OFDM,” Master Thesis, Georgia Institute of Technology, May 2005.

[8] R. W. Bauml, R.F.H. Fischer, J.B. Huber, “Reducing the Peak-to-Average Power

Ratio of Multicarrier Modulation Systems by Selected Mapping, ” Electron. Lett., vol.

32, pp. 2056-2057, Oct. 1996.

[9] A. Goldsmith, Wireless Communications, Cambridge University Press, 2007

[10] U. Madhow, Fundamentals of Digital Communication, Cambridge University Press,

2008 5

41

http://trsys.faculty.jacobs-university.de

BIBLIOGRAPHY

[11] D. Tse, P. Viswanath, Fundamentals of Wireless Communication, Cambridge Uni-

versity Press, 2005 5

[12] A.R.S. Bahai, B.R. Saltzberg, Multi-Carrier Digital Communications: Theory and

Applications of OFDM, Kluwer Academic Publishers, 2002

[13] R. Prasad, OFDM for Wireless Communication Systems, 2004

[14] L. Hanzo, M. Muenster, B.J. Choi, T. Keller, OFDM and MC-CDMA for Broad-

band Multi-User Communications, WLANs and Broadcasting, Wiley, 2003

42

par reduction in mimo broadcast ofdm systems using tone...

Documents