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OFDM BASED ON THE 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 O rthogonal frequency-

division multiplexing (OFDM) as the main communication scheme.

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

the distortion in each independent subchannel can be easily compensated by simple gain and phase

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

usually happens in the rapidly fading wireless 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 dividing the stream into several parallel bit streams.

Each of the subchannels has a much lower bit rate and is modulated onto a different carrier.

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

modulation with equally spaced subcarriers and overlapping spectra. The OFDM 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 and where 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 subchannel carrier frequencies should

be time-varying and ideally decompose the frequency distortion of the channel at any instant 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 significant challenges both in terms of conceptual and computational

complexity. Such bases are associated to the fractional Fourier transform whose timefrequency


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properties are well known in the signal processing community[3]. In a similar fashion as the Fourier

harmonic analysis employ sinusoidal function to decompose periodic signals, fractional Fourier

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

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

fractional Fourier type are implemented with a complexity that is equal to traditional fast Fouriertransform (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 of the multiplicative-filter in the FrFd using the known optimal

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

use of the MMSE symbol estimation scheme as an equalizer for the received symbols.

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

Fourier transform is introduced. In Section 3, we describe the system model. Section 4 includes

simulation results that compare the investigated technology with 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 2.1 reports the results of the fractional Fourier transform

with different values of [4].

Figure 2.1 The 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:

csc2cot22

,tujutj

eAutK (2.1)

Where is the rotation angel for transformation process and

sin

2/4/][sin jsignjeA

(2.2)

The forward FRFT is defined as:

dtutKtxuXutxf ),()()()(

(2.3)

duutKuXtx ),()()( (2.4)

The domains of the signal for 0 < || < are defined fractional Fourier domains and = /2 in(2.3)

and (2.4) gives the well-known Fourier transform.

There are a lot of discrete FrFT (DFrFT) algorithms with different properties and accuracies in our

work we chose the DFrFT proposed in [5] because of, its transformation kernel and its inverse

transform is orthogonal and reversible.

We assume that the input function f(t) and the output function is F (u) of the FrFT have the chirp

period of order p with the period Tp=N t and Fp=M u and sampled signals are with the interval t

and u as;

x (n)= f (n. t) , X(m)= F(m. u) (2.5)

Where n = 0, 1 N-1 and m = 0, 1 M-1. When D. ,(2.1) can be converted as:

)(.... ....csc1

0

..cot2

..cot2

2222

nxeeetAmXtumnj

N

n

tnj

umj

(2.6)

When M = N in order to be reversible the following equation must be satisfied:

Mtu /sin.2. (2.7)

Eq.(2.6) can also be written as matrix vector multiplication,X = F . x (2.8)

Where = (), (),. , ( ) , = (), (), . , ( ) and F is a

matrix in the similar manner, the IDFrFT can be written as:


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x = F. X (2.9)

Where

F- = FH (2.10)


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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 :

tTjn

Tntj

T

jtf n )/2(cot

2

)/2)(sin(exp

cos)sin()(

22

,

,

(3.2)

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

FRFT[(un(sn ))]=const f,n(t) , n=,.....,,,,, (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 20th basis signal. In Figure 3.6 we show a 3D representation for

the Wigner distribution in time and frequency domain for the 1st basis signal and the 50th basis signal.

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

time and equal to:

cot2

n, tT

n (3.4)


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

Figure 3.2 OFDM 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 microseconds [S]

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

-4

-3

-2

-1

0

1

2

3

4

5

Time *100 microseconds [S]


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Figure 3.3 Spectral Energy Distribution of the two bases signals

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

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

50

100

150

Frequancy

Mag

nitude

Base 1

Base 20

Amp

2

1

0

-1

Time *100 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 3.5 the Wigner distribution for the 1st

basis signal and the 20th

basis signal

Figure 3.6 the Wigner distribution for the 1st basis signal and the 20th basis 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 order of FrFT added to the usual OFDM system

Amp

1

0

-1

Time *100 microseconds

Freque

ncy

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 3.7 The 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):

x = F. d (3.5)

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

y= Fr = FH x + Fn

ndHy ~

~

(3.6)

Where FHFH

~is the equivalent channel matrix in the FrFT and

nFn ~ is the noise vector

in the Fractional domain

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

equalizer.

TimeFrequency

Domain Channel

Distortion

IFrFTS/P CP P/SH

n+S/PFrFTCP

RemoveFilter

Data

Multiplicative filter UpdateP/S

Estimated

Data

Fractional Fourier

Domain

Inverse Fractional Fourier

Transformation

Fractional Fourier

Transformation

d x

y


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The signal observation model is given by:

nxHy (3.7)

where x is the transmitted OFDM symbol vector , y is the received vector after the dispersive

channel ( H is the matrix characterizing the degradation process) ,and n is the additive noise, We

assume that input and output processes and noise are finite length random processes and that we know

the correlation matrix of the input process and noise We will further assume that the noise is

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

yFxg

(3.8)

where

F and

F are discrete fractional Fourier transform matrices of order and , respectively

andg

is a diagonal matrix whose diagonal consists of the elements of the vector g . This estimation

corresponds to a multiplicative filter in the th fractional Fourier domain. as we mention before If

=/2,

F corresponds to the DFT matrix, and the estimation corresponds to that obtained by

conventional Fourier domain filtering.

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

xxxxEN

H

e 12 (3.9)

where N is the size of the input vector x The problem is then to find the vector g , which minimizes

2

e In order to solve this discrete time problem, we first define the cost function J d to be equal to theMSE defined in (3.9),which is also equal to the error in the

th domain:

xxxxE

N

xxxxEN

J

H

H

d

1

1

(3.10)

Where

xFx

and yFx

g

(3.11)

It is easily find the components of the optimal vector to be

jjR

jjRg

yy

yx

opt,

,

j=1, 2, N (3.12)

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


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FHRFR

H

xxyx

(3.13)

FRHRHFRnn

H

xxyy

(3.14)

Equation (3.12) provides the solution to our minimization problem in 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 3.8 The FrFT based OFDM system with MMSE equalizer

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

yGd MMSE.

(3.15)

Where .1..,.........1,0 TKdddd

From the MMSE principle, the equalizer G should get the minimum for the error function:

TimeFrequency

Domain Channel

Distortion

IFrFTS/P CP P/SH

n+S/PFrFTCP

RemoveSymbolestimationMMSE(Equalizer)

Data

P/SEstimated

Data

Fractional FourierDomain Inverse Fractional FourierTransformation

Fractional Fourier

Transformation

d x

y


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2

ddEJk (3.16)

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

0* yddE (3.17)

which equal

** ..ydEydE (3.18)

Submitting (3.15) into (3.18), we can obtain the filter operator as:

*

*

.

.

yyE

ydEGMMSE (3.19)

When assuming the channel transmission matrix is known by the channel estimation and the

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

IHH

HGMMSE 2*

*

~~~

(3.20)

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

and important problem we met. We can investigate different orders then select the order that give the

smallest error as the optimal order to modulate and demodulate the subcarrier signals. So the

equalization is implemented in the FrFd with this order.


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

proposed FrFT-OFDM systems and equalizers. Throughout our simulation experiments, we

investigate an FrFT- OFDM or FT-OFDM systems with N = 128 subcarriers, cyclic prefix of CP = 32,

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

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

shift fdT = 0.00125. Figure 4.1 shows 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 bit error rate (BER) averaged

over 10000 multicarrier blocks, from which it is obvious that there is a great improvement in the

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

Figure 4.1 the 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 4.2 the 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 shift fdT = 0.01(which is much more than that used with the optimal filter in the FrFT). Figure

4.3 shows the investigation to determine the optimal order which give the minimum error. Figure 4.4

illustrates 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 bit error rate(BER) averaged over 10000 multicarrier blocks, from which it is obvious that there is a great

improvement in the performance of the FrFT-OFDM system 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(l

ogscale)

FFT OFDM

FrFT OFDM

Doppler Spread = 1000

Doppler Spread = 500


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

equalizer.

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

based OFDM scheme.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

-3

10-2

10-1

Order

BitErrorRa

te

300

500

800

1000

5 10 15 20 25 3010

-4

10-3

10-2

10-1

100

SNR in dB

BitError

Rate

FrFT

FFT


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

functions are more appropriate for multicarrier transmission than the using of the traditional OFDM

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

using Fractional Fourier transform to generate a chirp-like signal as bases for the OFDM system these

chirp bases matches the time-varying characteristics of the RF propagation channel. Using recently

introduced schemes for fractional Fourier signal analysis, we have shown that, at no extra

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

The proposed methodology in channels characterized by large Doppler spread remarkably outbid the

classical FFT-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 in Doubly 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, IEEE Transactions 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, IEEE Transactions on, vol. 49, pp. 1011-

1020, 2001.[3] L. B. Almeida, "The fractional Fourier transform and time-frequency representations," Signal

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

Processing, IEEE Transactions on, vol. 52, pp. 1002-1011, 2004.
http://en.wikipedia.org/wiki/Fractional_Fourier_transformhttp://en.wikipedia.org/wiki/Fractional_Fourier_transformhttp://en.wikipedia.org/wiki/Fractional_Fourier_transform

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