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OFDMBASED ONTHE FRACTIONAL FOURIER

TRANSFORM

AHMED AMIN, JOHN SORAGHAN, AND STEPHAN WEISS


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A b s t r a c t

Nowadays there is a great demand on fast mobile communication systems, especially

multimedia services like video calls, audio/video entertainment and wireless internet connections.

When we investigate the physical layer for all these standards we will found Orthogonal frequency-

division multiplexing (OFDM) as the main communication scheme.

OFDM is widely used in order to combat the severe effects offrequency dispersive channels;

the distortion in eachindependent subchannel can be easily compensated by simplegain and phase

adjustments.However, when the channel istimefrequency-selective (that is, doubly selective), as it

usually happens in the rapidly fadingwireless channel due to fast mobility, this traditional

methodology fails.

Our research work investigates the use of the Fractional Fourier Transform based OFDM in an

attempt to provide enhance ability to combat ICI compared to conventional DFT based OFDM

systems. In the proposed FrFT-OFDM system, the traditional sinusoidal subcarrier signal bases are

replaced by chirp subcarrier signal bases using the inverse discrete FrFT (IDFrFT). The received

signals are equalized by the multiplicative-filter in the fractional Fourier domain (FrFd) using the

known optimal transform order which give better performance than the traditional OFDM. The work

also propose a new FrFT-OFDM equalizer schemeby using the classical minimum meansquared

error(MMSE) scheme in the fractional domain, we found that replacing DFT by IDFrFT improve the

OFDM performance specially in doubly selective multipath channel environment.


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1. IntroductionHigh data rate transmission is needed by many applications however reducing the symbol

duration to increase the bit rate will cause intersymbol interference (ISI). To reduce the ISI the symbol

duration must be much larger than the delay spread of wireless channels.Multicarrier techniques

transmit data with much larger symbol duration by dividingthe stream into several parallel bit streams.

Each of thesubchannels has a much lower bit rate and is modulated onto adifferent carrier.

Orthogonal frequency-division multiplexing(OFDM) is a special case of multicarrier

modulation withequally spaced subcarriers and overlapping spectra. TheOFDM waveforms are chosen

orthogonal to each other in the frequency domain. So the substreams are essentially free of

intersymbol interference (ISI).this give the OFDM systems enormous popularity[1]; However, when

the channel is doubly selective (that is, timefrequency-selective), as it usually happens in the rapidly

fading wireless channel, this traditional methodology fails. That interchannel interference may degrade

an OFDM system performance. It is important to notice that when the channel is doubly selective the

entire conceptual framework of the basic Fourier-domain channel partitioning scheme loses its

optimality[2]. Many efforts has been researched where orthogonality was somehow sacrificed for

timefrequency localization of the transmitted signal set andwhere robustness to doubly dispersive

channel distortions was the main goal. However, the problem was not attacked at the cause, because

exponential (Fourier-like) signal sets were still used, both at the transmitter and at the receiver. In this

work, we investigate a new methodology that employs a signal set specially considered for the

synthesis/analysis of nonstationary (time-varying) signals.

The optimal transmission/reception communication system over the doubly dispersive channel

should be able to diagonalize nonstationary signals, where the subchannelcarrier frequencies should

be time-varying and ideally decomposethe frequency distortion of the channel at anyinstant in time. In

other words the bases for the OFDM system should be frequency varying with variation that is

matched with the channel frequency variation to compensate the channel frequency distortion. This

optimal approach presents significantchallenges both in terms of conceptual and

computationalcomplexity.Such bases are associated to the fractional Fourier transform whosetime


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frequency properties are well known in the signalprocessing community[3]. In a similar fashion as the

Fourier harmonic analysisemploy sinusoidal function to decompose periodic signals, fractional Fourier

techniques employ chirp harmonics for thedecomposition of signals with time-varying periodicity.

The main point of the methodology we investigate relies on that the analysis/synthesis methods ofthe

fractional Fourier type are implemented with a complexitythat is equal to traditional fast Fourier

transform (FFT) computational procedures.

We investigate the use of a multicarrier system that uses the chirp type signals as the

orthogonal signal bases with the use ofthe multiplicative-filter in the FrFdusing the known optimal

transform order which give better performance than the traditional OFDM,then we will propose the

use of the MMSE symbol estimation schemeasan equalizer for the received symbols.

The remainder of the report is organized as follows. InSection 2, we introduce the fractional

Fourier transform is introduced. In Section3,we describe the system model. Section 4includes

simulation results that compare the investigated technologywith the traditional OFDM system also

contain the results from the proposed MMSE equalizer scheme. In Section 5 includes conclusions.

Section 6 includes Future work .References are provided in Section 7.


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2. THE FRACTIONAL FOURIERTRANSFORM

The fractional Fourier transform (FRFT) was introduced in [3] as a generalization of the

Fourier transform. The transform immediately appeared useful in many signal processing applications.

One of the FRFT definitions is that A Fractional Fourier transform is a rotation operation on the time

frequency distribution by angel . for =0, there will be no change after applying fractional Fourier

transform, and for =/2, fractional Fourier transform becomes a Fourier transform, which rotates the

time frequency distribution with /2. For other value of , fractional Fourier transform rotates the time

frequency distribution according to . Figure 2.1reports the results of the fractional Fourier transform

with different values of [4].

Figure 2.1The results of the fractional Fourier transform with different values of [4]


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The transformation kernel of the continuous FRFT is defined as:

ETETEE

csc2cot22

,tujutj

eAutK

! (2.1)

Where is the rotation angel for transformation process and

a_

E

EET

E

sin

2/4/][sin jsignjeA

!

(2.2)

The forward FRFT is defined as:

_ a g

g

!! dtutKtxuXutxf ),()()()( EEE (2.3)

g

g

! duutKuXtx ),()()( EE (2.4)

The domains of the signal for 0


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3. SYSTEM MODEL3.1 Chirp signals as OFDM bases

Consider the baseband representation of the multicarrier system is given by:

(3.1)

Where f,n(t)is given by :

v

! tTjn

Tntj

T

jtf n )/2(cot

2

)/2)(sin(exp

cos)sin()(

22

, TETEEE

E

,

(3.2)

The function f,n(t) is chosen to produce an impulse in the fractional Fourier domain that:

(3.3)

In Figure 3.1 and Figure 3.2 we show two basis from the f,n(t)set for =/2 *100000 , N =256

sampled at 10 kHz and T = 0.05 sec in Figure 3.3 we show the spectral energy distribution of the two

bases signals.

In Figure 3.4 we show the Wigner distribution in time and frequency domain for the 1st basis

signal.From the figure we can see the transformation in the time frequency domain to intermediate

domain which is the Fractional domain. In Figure 3.5 we show the Wigner distribution in time and

frequency domain for the 1st basis signal and the 20thbasis signal.In Figure 3.6we show a 3D

representation for the Wigner distribution in time and frequency domain for the 1stbasis signal and the

50thbasis signal.

The basis signals are chirp signals with chirp rate = -cot the frequencies of the basis are dependent on

time and equal to:

E

T

[E cot2

n, tn ! (3.4)


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Figure 3.1OFDM Fractional basis where = /2 *100000 , N = 256 and n = 0

Figure 3.2OFDM basis where = /2 *100000 , N = 256 and n = 20

0 50 100 150 200 250 300 350 400 450 500-5

-4

-3

-2

-1

0

1

2

3

4

5

Time *100 m icroseconds [S]

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0-5

-4

-3

-2

-1

0

1

2

3

4

5

Tim e *100 m icroseconds [S ]


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Figure 3.3Spectral Energy Distribution of the two bases signals

Figure 3.4the Wigner distribution in time and frequency domain for the 1stbasis signal

-5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 50000

50

10 0

15 0

Frequancy

M

ag

nitude

Base 1

Base 20

mp

2

1

0

-1

ime

1

microseconds

Frequency

50 100 150 200 250 300 350 400 450 500-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000


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Figure 3.5the Wigner distribution for the 1stbasis signal and the 20

thbasis signal

Figure 3.6the Wigner distribution for the 1stbasis signal and the 20thbasis signal

3.2 The FrFT-based OFDM systemThe FrFT based OFDM system is shown in figure 5 which is like the FFT based OFDM system but

with a selector for the optimum orderof FrFT added to the usual OFDM system

mp

1

0

-1

Time

microseconds

Freue

ncy

50 1 00 1 50 2 00 2 5 0 3 00 3 5 0 40 0 4 5 0 50 0-5000

-4000

-3000

-2000

-1000

0

1 0 0 0

2 0 0 0

3 0 0 0

4 0 0 0

5 0 0 0


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Figure 3.7The FrFT based OFDM system

The subcarriers for the OFDM system are modulated by the Inverse Discrete FrFT (IDFrFT) where the

transmitted data vector

.and the subcarriers vector which can be calculated from (2.9):

(3.5)

At the receiver the subcarriers signals are demodulated using the Discrete FrFT (DFrFT) where thesignal vector after demodulation:

EEEEndHy ~

~!

(3.6)

Where EEE ! FHFH~

is the equivalent channel matrix in the FrFT and EEE nFn !~ is the noise vector in

the Fractional domain

Time FrequencyDomain Channel

Distortion

IFrFTS/P CP P/SH

n+S/PFrFTCP

RemoveFilter

Data

Multiplicative filter UpdateP/S

EstimatedDat

a

Fractional FourierDomain

Inverse Fractional FourierTransformation

Fractional Fourier

Transformation

d x

yd


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3.2.1 Optimal filtering in FrFT:When the channel model is doubly selective channel with large Doppler frequency we can use this

equalizer.

The signal observationmodel is given by:

nxHy ! (3.7)

wherex is the transmitted OFDM symbol vector , y is the received vector after the dispersive channel

( H is the matrixcharacterizing the degradation process) ,and n is the additive noise,We assume that

inputand output processes and noise are finite length randomprocesses and that we know the

correlation matrix of the inputprocess and noise We will further assume that the noiseis independent of

the input process and is zero mean. We consider an estimate of the form[6]:

yFxg

E

0! (3.8)

whereE

F andE

F are discrete fractional Fourier transformmatrices of order and , respectively

andg

0 is a diagonalmatrix whose diagonal consists of the elements of the vectorg. This estimation

corresponds to amultiplicative filter in the th

fractional Fourier domain.as we mention before If =/2,

E

F corresponds to theDFT matrix, and the estimation corresponds to that obtainedby conventional

Fourier domain filtering.

The multiplicative filter design criterion is the mean square error (MSE),which is defined as:

? AxxxxE

H

e

12!W (3.9)

whereN is the size of the input vectorx The problem is thento find the vectorg, which minimizes 2e

W

In order to solve this discrete time problem, we first definethe cost function Jdto be equal to the MSEdefined in (3.9),which is also equal to the error in the th domain:

? A

? AEEEE

xxxxE

xxxxE

J

H

H

d

1

1

!

!

(3.10)

Where

xFxE

E ! and yFx gE

E0!

(3.11)

It is easily find the components of the optimalvector to be


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jjR

jjRg

yy

yx

opt,

,

EE

EE! j=1, 2, N (3.12)

The above correlation matrices can be obtained from the input vector and noise correlation matrices asEE

EE

! FHRFRH

xxyx

(3.13)

EEEE

! FRHRHFRnn

H

xxyy

(3.14)

Equation (3.12) provides the solution to our minimization problemin the discrete time setting.

3.2.2 The MMSE equalizer in the FrFT:First of all when we review the literatures we found that it is the first time to use this method with

OFDM based on FrFT which give better performance for the OFDM scheme.

A nondiagonal subcarrier coupling matrix introduces ICI, which is the case when the dispersive

channel is multipath doubly selective channel (Rayleigh dispersive channel with large Doppler

frequency) complicating the symbol estimation task.

Figure 3.8The FrFT based OFDM system with MMSE equalizer

Consider that theMMSEequalizer in FrFd is GMMSE so the equalizer signal can be written as[7]

Time FrequencyDomain Channel

Distortion

IFrFTS/P CP P/SH

n+S/PFrFTCP

RemoveSymbolestimationMMSE(Equalizer)

Data

P/SEstimatedD

ata

Fractional FourierDomain

Inverse Fractional FourierTransformation

Fractional Fourier

Transformation

d x

yd


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EEyGd MMSE.

! (3.15)

Where ? A .1..,.........1,0 TKdddd ! EEEE From the MMSE principle, the equalizer G should get the minimum for the error function:

!

2

EE ddEJk (3.16)

This can be done by using the orthogonality principle, equalizer G fulfilling the following equation:

? A_ a 0 * ! EEE yddE (3.17)which equal

_ a _ a** .. EEEE ydEydE ! (3.18)Submitting (3.15)into (3.18), we can obtain the filter operator as:

_ a_ a**

..

E

EE

yyEydEGMMSE ! (3.19)

When assuming the channel transmission matrix isknown by the channel estimation and the

transmitteddata are i.i.d, the filter operator can be written as:

IHH

HGMMSE 2*

*

~~

~

WEE

E

! (3.20)

In the above two equalizers, in which order FrFd thereceived signals are equalized is a fundamental

andimportant problem we met. We can investigate different orders then selectthe order that give the

smallest error as the optimalorder to modulate and demodulate the subcarriersignals. So the

equalization is implemented in the FrFdwith this order.


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4. SIMULATION RESULTSIn this section, several simulation examples arepresented to illustrate the performance of the

proposedFrFT-OFDM systems and equalizers. Throughout oursimulation experiments, we investigate

an FrFT- OFDMor FT-OFDM systems with N = 128 subcarriers, cyclic prefixof CP = 32, the

generated symbols are i.i.d and aremodulated to complex 4-QAM signals.

4.1 Optimal filtering in FrFT:The channel model is based on Rayleigh dispersive channel with normalized Doppler

spreadsshiftfdT = 0.00125.Figure 4.1shows the investigation to determine the optimal order which give

the minimum error. Figure 4.2 illustrates the BER performance of the Fractional Fourier scheme as

compares to the classical scheme based on IFFT/FFT processing at different Doppler spreads

(0.000625 and 0.00125 normalized Doppler spreads) for an uncoded biterror rate (BER) averaged over

10000 multicarrierblocks, from which it is obvious that there is a greatimprovement in the

performance of the FrFT-OFDMsystem compared to the FT-OFDM system.

Figure 4.1the investigation to determine the optimal order which give the minimum error.

-0.5 0 0.5 1 1.5 2

10- 1 .9

10- 1 .8

10- 1 .7

Order

BitErrorRate


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Figure 4.2the BER performance of the Fractional Fourier scheme as compares to the classical scheme

based on IFFT/FFT processing at different Doppler spreads

4.2 The MMSE equalizer in the FrFT:The channel model is based on Rayleigh dispersive multipath channel with normalized Doppler

spreads shiftfdT = 0.01(which is much more than that used with the optimal filter in the FrFT).Figure

4.3shows the investigation to determine the optimal order which give the minimum error. Figure

4.4illustrates the BER performance of the Fractional Fourier scheme as compares to the classical

scheme based on IFFT/FFT processing at normalized Doppler spread = 0.005 for an uncoded biterrorrate (BER) averaged over 10000 multicarrierblocks, from which it is obvious that there is a

greatimprovement in the performance of the FrFT-OFDMsystem compared to the FT-OFDM

system.From the simulation its clear to say that when the system channel is rapidly Rayleigh Fading

dispersive channel the using of FrFT bases is more convenient then using FFT bases specially when

choosing the optimum Fractional order which give the minimum error where the FrFT bases is

matched with the frequency variations in the channel.

4 6 8 10 12 14 16 1810

-3

10-2

10-1

100

SNR per bit in dB

BitErrorRate(log

scale)

FFT OFDM

FrFT OFDM

Doppler Spread = 1000

Doppler Spread = 500


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Figure 4.3the investigation to determine the optimal order which give the minimum error with the MMSE

equalizer.

Figure 4.4BER performance of FrFT-based OFDM scheme with MMSE equalizer compared to the FT-

based OFDM scheme.

0 0. 2 0. 4 0 .6 0 .

1 1 . 2 1 . 4 1 .6 1 .

210

-3

10-2

10-1

O r

r

Bit

rr

rR

t

30 0

50 0

80 0

1 0 0 0

5 10 1 5 2 0 2 5 3 010

-4

10-3

10 -2

10-1

100

S NR i n

B

Bit

rr

r

R

t

FrFT

FF T


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5. CONCLUSIONSIn this work, we have introduced the idea whichillustrate that the using of frequency-varying basis

functionsare more appropriate for multicarriertransmission than the using of the traditional OFDM

carriers in the presence of rapidly fading channels. The basic idea we have introduced is basedon using

Fractional Fourier transform to generate a chirp-like signalas bases for the OFDM system these chirp

bases matches the time-varying characteristicsof the RF propagationchannel. Using recently

introduced schemes forfractional Fourier signal analysis, we have shown that, at noextra

computational cost, it is feasible to obtain an improvement in performance in rapidly fading channels.

Theproposed methodology in channels characterized bylarge Doppler spread remarkably outbid the

classicalFFT-based scheme. Also we present the optimal filtering and the MMSE equalizer in the

fractional domain.


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6. Future workA- Investigate using of Low-Complexity Equalization of OFDM inDoubly Selective Channels

with FrFT bases.

B- MIMO FrFT_OFDM.C- Mathematical derivation for the optimum order.


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7. REFERENCES[1] H. Taewon, Y. Chenyang, W. Gang, L. Shaoqian, and G. Ye Li, "OFDM and Its Wireless

Applications: A Survey," Vehicular Technology, IEEETransactions on, vol. 58, pp. 1673-

1694, 2009.[2] M. Martone, "A multicarrier system based on the fractional Fourier transform for time-

frequency-selective channels," Communications, IEEETransactions on, vol. 49, pp. 1011-1020, 2001.

[3] L. B. Almeida, "The fractional Fourier transform and time-frequency representations," Signal

Processing, IEEETransactions on, vol. 42, pp. 3084-3091, 1994.[4] http://en.wikipedia.org/wiki/Fractional_Fourier_transform.[5] H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, "Digital computation of the fractional

Fourier transform," Signal Processing, IEEETransactions on, vol. 44, pp. 2141-2150, 1996.[6] A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, "Optimal filtering in fractional Fourier

domains," Signal Processing, IEEETransactions on, vol. 45, pp. 1129-1143, 1997.[7] P. Schniter, "Low-complexity equalization of OFDM in doubly selective channels," Signal

Processing, IEEETransactions on, vol. 52, pp. 1002-1011, 2004.

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6 Monthes Report Word 97-2003 Format