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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.