technia international journal of computing science and ... · ofdm transmitter is simulated using...

TECHNIA – International Journal of Computing Science and Communication Technologies, VOL. 3, NO. 2, Jan. 2011. (ISSN 0974-3375)

Studies on Performance of Pulse Shaped

OFDM Signal

1D. K. Sharma, 2A. Mishra, 3Rajiv Saxena

1Ujjain Engineering College, Ujjain, MP 2Madhav Institute of Technology & Science, Gwalior, MP 3Jaypee Institute of Engineering & Technology, Guna, MP

[email protected], [email protected], [email protected],

Abstract- The Orthogonal Frequency Division Multiplexing

(OFDM) transmission system is one of the optimum versions

of the multi-carrier transmission scheme. The OFDM is

referred in the literature as Multi-carrier, Multi-tone and

Fourier Transform based modulation scheme. The OFDM is

a promising candidate for achieving high data rate

transmission in mobile environment.

In this paper, different power spectral density (PSD)

curves of OFDM signal with various pulse shapes are

presented. The final pulse shaped OFDM waveform is then

analyzed for frequency domain response and the PSD in

each case and it’s also an analyzed on the basis of its

modulation index which equally varies with the used window

function during transmission. The simulation results are

presented in a tabular manner enabling to analyze and

establish the superiority, at a glance, of a specific window

function applied (pulse shaped). The OFDM signals with the

pulse shapes, like Rectangular, Blackman, Gaussian,

Hamming and Hanning are tried. The effect of some of these

time waveforms on the OFDM system performance in terms

of power spectral density (PSD) & modulation index has

been investigated.

Keywords- MI, PSD, FFT, ISI, ICI, OFDM.

I. INTRODUCTION

The concept of using parallel data transmission and

frequency multiplexing was published in the mid of

1960s. After more than thirty years of research and

development, OFDM has been widely implemented in

high speed digital communications. Due to recent

advances of digital signal Processing (DSP) and Very

Large Scale Integrated circuit (VLSI) technologies, the

initial obstacles of OFDM implementation such as

massive complex computation and high speed memory do

not exist anymore [1-3].

The use of Fast Fourier Transform (FFT) algorithms

eliminates arrays of sinusoidal generators and coherent

demodulation required in parallel data systems and makes

the implementation of the technology cost effective [4-5].

The OFDM concept is based on spreading the data to

be transmitted over a large number of carriers, each being

modulated at a low rate. The carriers are made orthogonal

to each other by appropriately choosing the frequency

spacing between them [6-7].

In contrast to conventional Frequency Division

Multiplexing, the spectral overlapping among sub-carriers

are allowed in OFDM since orthogonality will ensure the

sub-carrier separation at the receiver, providing better

spectral efficiency and the use of steep band-pass filter

was eliminated.[8-11]

The orthoganality of sub channels in OFDM can be

maintained and individual sub channels can be completely

separated by the FFT at the receiver when there are no

inter symbol interference (ISI) and inter carrier

interference (ICI) introduced by the transmission channel

distortion.[12-14]

One way to prevent ISI is to create a cyclically

extended guard interval, where each OFDM symbol is

preceded by a periodic extension of the signal itself.

When the guard interval is longer than the channel

impulse response or multi-path delay, the ISI can be

eliminated [15-16].

This paper is organized as follows: in section 2, the

OFDM transmitter is simulated using Matlab. In section 3,

the OFDM transmitter model along with various pulse

shapes are included with their mathematical expressions.

The simulation of various pulse shapes is presented. In

section 4, pulse shaped OFDM waveform is then analyzed

for MI & frequency domain response and the PSD in each

case. In section 5, the simulation results are shown. The

conclusive remarks are given in last section.

II. OFDM TRANSMITTER

A brief description of the model is provided in Figure-

1. The incoming serial data is first converted from serial

to parallel and grouped into x bits, each to form a complex

number. The complex numbers are modulated in a base-

band fashion by the IFFT and converted back to serial

data for transmission [17-18].

Fig.1: OFDM Transmitter.

A guard interval is inserted between symbols to

avoid inter symbol interference (ISI) caused by multi-path

distortion. The discrete symbols are converted to analog

and low-pass filtered for RF up-conversion [19].

III. FFT IMPLEMENTATION

The selection of FFT also plays an important role in

design of an OFDM system because of the size of the FFT

is to be taken as balance between the protection against

Doppler shift, multipath and design complexity. The

610


OFDM spectrum is centered on cf , i.e., sub-carrier 1 is

7.61/2 MHz to the left of the carrier and sub-carrier 1,705

is 7.61/2 MHz to the right. One simple way to achieve the

centering is to use a 2N-IFFT and T/2 as the elementary

period. The OFDM symbol duration, U , is specified

considering a 2,048-IFFT (N=2,048); therefore, we shall

use a 4,096-IFFT. A block diagram of the generation of

one OFDM symbol is as shown in Figure-2, where the

variables indicated are used in the source code. The next

task to consider is the appropriate simulation period. T is

defined as the elementary period for a base-band signal,

but since we are simulating a band-pass signal, we have to

relate it to a time-period, ,/1 sR that considers at least

twice the carrier frequency. For simplicity, we use an

integer relation, sR =40/T. This relation gives a carrier

frequency close to 90 MHz, which is in the range of a

VHF channel five, a common TV channel in any city [20].

Fig.2: Simulation of OFDM Transmitter.

The complex envelope of the OFDM signal,

consisting of N carriers is given [22] by,

)1.....(..1

0

2

,

k

N

n

tT

jn

kntotal ekttgaS

Where g(t) is rectangular pulse of duration T and T is

OFDM symbol duration.

A. Pulse shaping

There are two effects caused by frequency offset; (I)

reduction of signal amplitude in the output of the filters

matched to each of the carriers and (II) introduction of ICI

from the other carriers which are now no longer

orthogonal to the filter. The time-domain pulse shaping

method can also reduce ICI through multiplying the

transmitted time-domain signals by a well-designed pulse

shaping function proposed a pulse shaping method to

reduce ICI by cyclically extending by v samples the time

domain signal associated with each symbol [23-24]. The

whole of the resulting signal is then shaped with the pulse

function. It is important to note that DFT transform in the

receiver is N point whereas that in the transmitter is N/2

point. If v< N/2, then the signal corresponding to each

symbol is zero padded at the receiver to give length N [25,

26, 27].

The simplest way to reduce the Peal to Average Power

Ration (PAPR) is to clip the signal, but this significantly

increases the out of band radiation. A different approach

is to multiply large signal peak with a Gaussian pulse

shaped proposed. But, in fact any pulse shaping function

can be used, provided it has good spectral properties.

Since the OFDM signal is multiplied with several of these

pulse functions the resulting spectrum is a convolution of

the original OFDM spectrum with the spectrum of the

applied pulse function. So, ideally the pulse should be as

narrow band as possible. On the other hand, the pulse

function should not be too long in the time domain,

because that implies that many signal samples are

affected, which increases the bit error ratio. Examples of

suitable pulse functions are the Cosine, Kaiser and

Hamming window [28-29].

B. Methodology

Among the existing methods, Pulse shaping exhibits

good properties such as very simple to implement,

independent of number of carriers, no affect in coding rate

and large reduction in PAPR. Then, we propose a method

to be used with pulse shaping to reduce the PAPR further.

OFDM signal is multiplied by the pulse shaping function

when the signal peak exceeds the clipping level. Unlike

the clipping, the OFDM signal within the pulse width is

modified. This results in a smoothed OFDM signal.

Consider the OFDM system shown in Figure-2, IFFT

output, exhibit PAPR and is multiplied by a pulse shaping

function to reduce PAPR. This will cause signal distortion

[30-31].

Let, the modulated data be nx where n=0, 1, 2, …., N-1.

OFDM signal can be expressed as,

)3.......(

)2(..........

0

2

N

k

N

nkj

kn

nn

ex

IFFTx

The OFDM signal after multiplication by pulse shaping

function can be evaluated as,

.2/,...1,0;_

)4.......(..........,

2/

Mjlevelclipxif

otherwisex

gxz

n

n

JMjn

n

C Mathematical Model of OFDM:

The expression for an OFDM symbol at

t = ts is given as:

)5.......(..........,0)(

,5.0

2exp2

2

2/Re)(

Tsttsttts

Tsttsts

ttT

ic

fj

sN

sN

is

Nidts

Where, id = complex modulation symbols, S =

number of sub carriers, T = symbol duration, cf = carrier

frequency.

The above equation (5) can also be expressed as:

)6....()(.Re)(0

67

0

,,,,

2max

min

m l

klm

k

kk

klm

tfjtCets c

W

here

611


SS

mLk

j

klm mltml

otherwise

etSS

U

16868;

7..................;0

{68

'2

,,

Where: k = carrier number; l = OFDM symbol number; m

= transmission frame number; K = number of transmitted

carriers; S = symbol duration; U = inverse of the

carrier spacing; Δ = duration of the guard interval; cf =

central frequency of the radio frequency (RF) signal; k′ =

carrier index relative to the center frequency,

2/minmax

' and kimC ,, = complex

symbol for the carrier K of the data symbol frame from

number 0, 1, 2, 3….67 in frame number m.

D. OFDM Pulse Shaping

The ISI can be reduced by increasing the symbol

duration or by introducing the guard interval between the

OFDM symbols, the expected multipath delay spread can

be taken care of by increasing the guard interval and in

this way the ISI can be completely eliminated but addition

of guard interval to an OFDM system accounts for a

decrease in bandwidth efficiency and an increase power

requirement which are not the desirable features for

spectrally efficient wireless communication system [32-

33].

One way to solve this problem is to adopt a

proper prototype pulse shape function well localized in

time and frequency domain so that the combined ISI / ICI

can be combated efficiently without utilizing any cyclic

prefix [34-35].

At the stage of modulating the OFDM signal by

applying the pulse g(t) an exhaustive analysis has been

done by varying the pulse shape g(n) as follows:

Case1: Rectangular Pulse

g(n) = 1, for n = 0,1,2,3...,M.

0 , otherwise.

Case2: Blackman pulse

1/4cos0801/2cos50420 nn

Case3: Gaussian pulse

g(n) = 2

0,/22

1exp

2

nforn

Case 4: Hamming pulse

g(n) = 1/2cos460540 n

Case5: Hanning pulse

g(n) = 1/2cos12

1n

The above mentioned five pulse shapes have been

simulated and representing in time and frequency domain

as shown in Figure-3 and Figure-4.

10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Samples

Am

plit

ude

Time domain

w indow #1

w indow #2

w indow #3

w indow #4

w indow #5

Fig.3: Various pulse shapes in time domain.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9-140

-120

-100

-80

-60

-40

-20

0

20

40

Normalized Frequency ( rad/sample)

Magnitu

de (

dB

)

Representation of Different pulse shapes in frequency domain

rectw in#1

blackman#2

gaussw in#3

hamming#4

hanning#5

Fig.4: Various pulse shapes in frequency domain.

IV. PULSE SHAPED OFDM SIGNAL

In OFDM systems, the information bit stream (bit rate

bR =

b

1) is first modulated in base band signal using

M-ary quadrature amplitude modulation (M-QAM) with

symbol duration being defined as S = 2logb , and is

then divided into N parallel symbol streams which are

then multiplied by pulse shape function g(t) or g(n) [2, 8,

28].

The transmitted signal in the analytic form can be

represented as

)()( ,, tgatSn

nmnm

(8)

Where )1......,2,1,0,(, Nmna nm represents

the base band modulated information symbol conveyed

with the sub-carrier of index m during the symbol time of

index n, and )(, tg nm represent the pulse shape of index

(m, n) in the synthesis basis which is derived by the time-

frequency translated version of the pulse shaped function

g(t) as

,)( 2

, nTtgetg mFtj

nm

Where, nm, (9)

612


Where ,1j F represents the inter-carrier

frequency spacing and T is the OFDM symbol duration,

hence )(, tg nm forms an infinite set of pulse spaced at

multiples of T and frequency shifted by multiple of F.

Consequently the density of OFDM lattice is TF/1 .

Transmitter Model

We consider an OFDM system with a total of N

orthogonal sub-carriers. Each sub-carrier is modulated

with a low rate sequence of symbols and uses a pulse

shape of the same duration as the OFDM symbol duration

T [27]. The transmitter block diagram is shown in Figure-

5.

Fig.5: Block diagram of the OFDM scheme using time-

limited waveforms.

The equivalent low-pass representation of the

transmitted signal is given

by

1

0

2

, .1,)(m

tfj

km tetpSt m

………. (10)

where kkkoK sssS ,1,1, ,.....,,

uncorrelated complex base-band modulated signals related

to the used modulation scheme and

m

fm ,

m=0,1,2,……N-1 is the carrier frequency of sub-carrier

m.

)(tpm is a pulse shape of duration T used at sub-carrier

m with .)(2dttpm

This pulse is defined according to [10] to be

t

sm

itptwwhere

mtmtwtp

.)(

)11_(__________.1......,

......2,1,0,2/)(

is a

periodic function with a period T and p(t) is a time limited

waveform of duration T and

.2/2/,1

.,0

t

otherwiset

S =T/N; is the symbol duration of the base-band

modulation signal. We assume that the total number of

harmonics for the pulse )(tpm shape is L+U+1. Using

(11), the pulse shape of sub-carrier m becomes

,2/)(22

teectpU

L

T

tij

N

mij

im

L, U<N, M= 0, 1, 2….., N-1…(12)

The discrete representation of this set of time waveforms

is given in [28-30] and is written in a vector form as:

mp [ N

mLj

Lec2

, ,.....,

)1(2

1N

Lmj

L ec

N

Umj

U ec

)1(2

1

, N

mUj

U ec2

, 0, ……, 0]

m = 0, 1, 2,….., N-1…(13)

Where

)(1

)(1

0

2

,

ipdtetpc m

T

tij

mim

is the exponential Fourier series coefficient with

1

2

,

imc And )( fPm is the Fourier transform

of )(tpm .

If we replace )(tpm by its expression in (10), the

equivalent low pass of the OFDM signal becomes in a

matrix form

KKUNKLKLk

kk

SsssS

sPS

,~,......,~,~~)14.......(

~

,1,1,

is as defined previously, and

)15.....(...........

.

1

1

2

2

1

0

PD

PD

DP

P

P

is an )( UL shaping matrix, and m

i PD is

the ith cyclic shift of the vector mp .

The definition of the matrix P indicates that each sub-

carrier pulse of the OFDM scheme has a different shape

and all these pulse shapes are derived by cyclic shifts of

613


the same pulse. This will also reduce the PAPR of the

OFDM transmitted signal since the peak amplitude of the

different pulse shapes will never occur at the same time

instant unless p (t) is a rectangular pulse [32].

As shown in Figure-6, the OFDM transmitter

system with pulse shaping can be equivalently represented

by a discrete shaping matrix P followed by a regular

OFDM scheme. The OFDM transmitted signal with pulse

shaping is generated as follows. The information is first

passed through a shaping processor, which consists of

multiplying the input sequence, S by the transpose of

the shaping matrix P. The output is then modulated using

a regular OFDM scheme with L+N+U sub-carriers giving

the signal x (t) as defined in (10). The entities of the

shaping matrix are directly obtained from the selected set

of time waveforms that reduce the PAPR of OFDM

signals [3, 34].

Fig.6: Discrete representation of OFDM modulation

schemes with time-limited waveforms.

The ISI and ICI within an OFDM block are

avoided by adding a cyclic prefix (CP). The cyclic prefix

is simply a copy of the M last symbols of the N samples

and is added to the front of the OFDM block, making the

signal to appear periodic in the receiver as shown in

Figure-7.

Fig.

7: Adding the cyclic prefix, CP.

There is a trade off between bit rate and low ISI

and the optimum length of the cyclic prefix depends on

the delay spread of the channel. The length of cyclic

prefix is chosen larger than expected delay spread of the

channel.

The purpose of this paper is to look at designing

an optimum set of waveforms for the OFDM scheme by

varying the related parameter of p(t). The OFDM system

with pulse shaping was introduced and the system

performance was evaluated assuming that each sub-carrier

uses the same pulse shape. In our model the system

performance based on choosing the set of pulses that give

performance of OFDM system in terms of PSD and MI.

The modulation index (M.I.) of pulse shaped

OFDM signal is calculated by expression [28] as

)16..(..........100..minmax

minmax

VV

VVIM

Where minmax &VV are the maximum and minimum

amplitude of pulse shaped OFDM transmitted signal. We

evaluated various minmax &VV for above said pulse

shapes and presented in Table-1. The pulse shaped OFDM

waveforms are plotted in Figure-8 to Figure-12.

2 4 6 8 10 12 14

x 10-7

-150

-100

-50

0

50

100

150

Time(sec)

Am

plit

ude

Fig-9-Time response of signal s(t)

data 1

Fig.8: Time Response of OFDM signal for Rectangular pulse

2 4 6 8 10 12 14

x 10-7

-50

-40

-30

-20

-10

0

10

20

30

40

50

Time(sec)

Am

plit

ude


data 1

Fig.9: Time Response of OFDM signal for Blackman pulse

2 4 6 8 10 12 14

x 10-7

-60

-40

-20

0

20

40

60

Time(sec)

Am

plit

ude


data 1

Fig.10: Time Response of OFDM signal for Gaussian pulse

2 4 6 8 10 12 14

x 10-7

-60

-40

-20

0

20

40

60

Time(sec)

Am

plit

ude


data 1

Fig.11: Time Response of OFDM signal for Hamming pulse

614


2 4 6 8 10 12 14

x 10-7

-80

-60

-40

-20

0

20

40

60

80

Time(sec)

Am

plit

ude


data 1

Fig.12: Time Response of OFDM signal for Hanning pulse

The pulse shaped OFDM waveforms is then analyzed

for frequency domain response and the PSD in each case

as mentioned above are plotted in Figure-13 to Figure-17.

The PSD of each carrier at frequency defined by the

expression as:

17.....2 SincfPk

where

)20....(;2/'

)19........(

);18.......(

maxminminmax

'

kkkkkkk

andk

ff

ff

u

ck

sk

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

5

10

15

20

25

Frequency(Hz)

Magnitude

Fig-10-s(t) FFT

0 20 40 60 80 100 120 140 160 180-120

-100

-80

-60

-40

-20

Frequency (MHz)

Pow

er/

frequency (

dB

/Hz)

Welch Power Spectral Density Estimate

Fig.13: Frequency Response & PSD of signal for Rectangular pulse.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

2

4

6

8

10

Frequency(Hz)

Magnitude

Fig-10-s(t) FFT

0 20 40 60 80 100 120 140 160 180-140

-120

-100

-80

-60

-40

Frequency (MHz)

Pow

er/

frequency (

dB

/Hz)

Fig-10-s(t) FFT

data 1

data 2

Fig.14: Frequency Response & PSD of signal for Blackman pulse.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

2

4

6

8

10

Frequency(Hz)

Magnitude

Fig-10-s(t) FFT

0 20 40 60 80 100 120 140 160 180-140

-120

-100

-80

-60

-40

Frequency (MHz)

Pow

er/

frequency (

dB

/Hz)


data 1

data 2

Fig.15: Frequency Response & PSD of signal for Gaussian pulse.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

2

4

6

8

10

12

Frequency(Hz)

Magnitude

Fig-10-s(t) FFT

0 20 40 60 80 100 120 140 160 180-140

-120

-100

-80

-60

-40

Frequency (MHz)

Pow

er/

frequency (

dB

/Hz)


data 1

data 2

Fig.16: Frequency Response & PSD of signal for Hamming pulse.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 108

0

2

4

6

8

10

12

Frequency(Hz)

Magnitude

Fig-10-s(t) FFT

0 20 40 60 80 100 120 140 160 180-140

-120

-100

-80

-60

-40

Frequency (MHz)

Pow

er/

frequency (

dB

/Hz)


data 1

data 3

Fig.17: Frequency Response & PSD of signal for Hanning pulse.

V. RESULTS AND DISCUSSION

The modulation index after getting the OFDM for

applied five different pulses shape i.e. rectangular,

Blackman, Gaussian, Hamming and Hanning pulse has

been investigated and reported in Table-1.

The modulation index of OFDM for Rectangular

pulse is lower but the modulation index of OFDM signal

for Gaussian pulse is higher.

Various parameters of Magnitude Spectrum and

Power Spectral Density of OFDM transmitted signal for

different pulse shapes are tabulated in Table-2 and Table-

3 respectively.

Table-1: Simulation result of: Max and Min. Values &

modulation index of OFDM signal.

Sr.

No.

Types of

Pulse maxV minV M. I.

(%)

1 Rectangular

Pulse

113.1370 8.483 86.05

2 Blackman

Pulse

45.8639 3.263 86.72

3 Gaussian

Pulse

54.2013 3.076 89.26

4 Hamming

Pulse

59.2022 4.249 86.61

5 Hanning

Pulse

60.0577 4.353 86.48

VI. CONCLUSION

A study on pulse shaped OFDM signal is made and

the power spectral density (PSD) and MI of different

pulse shaped OFDM signal is presented in this paper. The

OFDM waveform is then analyzed for MI & frequency

domain response and the PSD in each case.

It is also possible to design a set of time domain

waveforms that will reduce the PAPR of the OFDM

615


transmitted signal and improve its power spectrum

simultaneously. The effect of some of these sets of time

waveform on the OFDM system performance in terms of

MI & power spectral density (PSD) is investigated and the

data is tabulated to analyze and establish the superiority of

a specific pulse shape over the other depending on the

application or requirement.

In this paper, envelope of pulse shaped OFDM signal

are shown and the variations in modulation index with

respect to pulse shape or window function used are

tabulated. The results are enabling to analyze and

establish the superiority at a glance of a specific window

function applied.

This study definitely looks forward and it reveals that

this will act as the stepping stone especially in the designs

of 4G Mobile Communication System.

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Table-2: The Parameters of Magnitude Spectrum of OFDM Transmitted Signal With Different Pulse Shapes

Para. Time (Sec.) Rect. Blackman Hanning Gaussian Hamming

Min: 8.9286310 8.3295

510 3.1005510 8.2062

510 1.1100410 9.1606

510

Max: 3.6571810 20.7719 8.3305 10.9302 9.8474 10.7639

Mean 1.8286810 0.3850 0.1563 0.2049 0.1844 0.2017

Median 1.8286810 0.0025 0.0010 0.0013 0.0012 0.0013

Mode: 8.9286810 8.3295

510 3.1005510 8.2062

510 1.1100410 9.1606

510

Std 1.0557810 1.9815 0.8034 1.0534 0.9483 1.0370

Range 3.6571810 20.7719 8.3305 10.9301 9.8473 10.7638

Table-3: The Parameters of Power Spectral Density of OFDM Transmitted Signal With Different Pulse Shapes

Para. Time(sec.) Rect. Blackman Hanning Gaussian Hamming

Min: 0 -116.5818 -124.3722 -122.0185 -122.9310 -122.1541

Max: 182.8571 -35.1984 -43.1080 -40.7408 -41.6560 -40.8792

Mean 91.4286 -105.1980 -112.9823 -110.6529 -111.5612 -110.7903

Median 91.4286 -110.1268 -117.9741 -115.6195 -116.5291 -115.7503

Mode: 0 -116.5818 -124.3722 -122.0185 -122.9310 -122.1541

Std 52.7911 15.7749 15.7621 15.7525 15.7516 15.7482

Range 182.8571 81.3834 81.2642 81.2777 81.2750 81.2749

617

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