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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