Research Journal of Applied Sciences, Engineering and Technology 4(18): 3510-3515, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: May 08, 2012
Accepted: June 08, 2012
Published: September 15, 2012
Beamforming in Short Time Fractional Fourier Domain (FRFD)
Mostafa Rahmani and Hassan Ghassemian
Department of Electrical and Computer Engineering, Tarbiat Modares University, Tehran, Iran
Abstract: This study aims to generate a model for the problem of beamforming in fractional Fourier
domain, through which a general solution for obtaining weighted beamforming in fractional Fourier
domain for different criteria might be delivered. In order to generate this model, a description of fractional
delay concept is presented, through which the string vector in fractional Fourier domain is calculated and a
model for beamforming in fractional Fourier domain is generated which is similar to that used for obtaining
weighted beamforming in time domain for narrow-band signals. To complement our solution, a method for
obtaining the optimal fractional domain as well as direction of arrival in optimal fractional Fourier domain.
Keywords: Array processing, beamforming, direction of arrival, fractional fourier domain, fractional
fourier transform, steering vector
ordinary Fourier Transform (FT). Therefore, replacing
the FT with the FrFT should improve the performance
considerably. In this study, a novel Beamforming
algorithm of wideband LFM signals based on the FrFT
is proposed which can be applied to linear array.
Implementations of the proposed algorithm are also
developed, which are based on the spectral peak
searching. At the end of the study, numerical examples
and simulations are presented followed by a discussion
of the results.
INTRODUCTION
Beamforming is array signal processing technique
used in sensor arrays for directional signal transmission
or reception. This is achieved by combining elements in
the array in such a way that signals at particular
angle experience constructive interference and while
others experience destructive interference. Various
beamforming algorithms have been proposed
(Monzingo and Miller, 1980). The conventional
beamformer data back to the second world war and is a
mere application of Fourier-based spectral analysis to
spatio-temporally sampled data. Later, adaptive
beamformers (Gabriel, 1972) and classical time delay
estimation techniques were applied to enhance one's
ability to resolve closely spaced signal sources. The
extension of the time delay estimation methods to more
than one signal and the limited resolution of
beamforming together with an increasing number of
novel application, renewed interest of researchers in
statistical signal processing. However, for parametric
techniques, the intuitive notion of resolution is nontrivial to define in precise terms. This in turn, resulted in
the emergence of the parameter estimation approach as
an active research area. For the first time in 2003 Yetik
introduced Beamforming in FRFD (Yetik and Nehorai,
2003), than (Zhou and Tao, 2008) complicated this
algorithm (Tao, 2004). In this methods Beamforming
has been model as spatial filtering, in the other words
their work was an extension Osaka's work (Kutay et al.,
1997) on array signal processing. Our study tends to
solve the beamforming problem through parametric
method in FRFD. The motivation behind the proposed
method is the ability of the Fractional Fourier Transform
(FrFT) to process the chirp signals better than the
METHODOLOGY
Fractioanl fourier transform: Fractional Fourier
transform represents a powerful method in signal
processing which has been used over past two decades.
This transform, being a generalization of ordinary
Fourier transform, is indeed equal to signal rotation in
time-frequency domain in a given angle (Namias,
1980). Letting x(t ) be an arbitrary signal, its pth-order
FrFT is defined as:

X (u)  Fp[ x(t)]   x(t) K (t, u) dt

(1)
where, p(/2); F p [.] is operator signals of FRFT,
K (t , u) is kernel function:
 1 j cot

2


 t 2  u2

cot  tu csc ,
K (t,u)   exp j
2



 (t  u),

 (t  u),
  n
  2n
  (2n 1)
Corresponding Author: Mostafa Rahmani, Department of Electrical and Computer Engineering, Tarbiat Modares
University, Tehran, Iran
3510
Res. J. Appl. Sci. Eng. Technol., 4(18): 3510-3515, 2012
The FRFT is a linear transform, the ( , u ) is called
the FRFT domain. From the above relationship it is
obvious that the time and frequency are the 2n th and
2n  / 2th FRFT domains, respectively, meaning that
the FRFT is unified time-frequency transform and the
FRFT can demonstrate the characteristics for the signal
changing from the time domain to the frequency domain
having the order in the range from 0 increasing to 2 .
In practice the Discrete FRFT (DFRFT) is used.
The commonly used fast algorithm of DFRFT is
Osaka's algorithm (Ozaktas et al., 1996). In the fast
algorithm of the FRFT digital computation, the original
signal s(n) must be normalized. It assume that, the
original continuous signal is compact along the axes of
time and frequency, the time-domain representation of
the signal is confined to the interval [t / 2, t / 2] and
its frequency-domain representation is confined to the
interval [   f / 2,  f / 2 ] . So the signal's timebandwidth product is N  t  f , which is always
greater than unity because of the uncertainty relation.
Because time-domain and frequency domain have
different dimension. In order to compute the FRFT,
time-domain and frequency-domain must be
transformed to the domain whose dimension is “1”. It
introduces the scaling parameter s with the dimension
of time and introduces scaled coordinates:
x  t / s, v  fs
(2)
(x,v) are clearly
The new coordinates
dimensionless. The signal is confined to the interval
and
[t /( 2s ), t /( 2s )]
[   f s / 2,  f s / 2 ] ,
respectively under the new coordinates. In order to have
the two intervals equal to each other, it chooses
s  t / f , so that the length of both intervals are
equal to the dimensionless quantity x  tf . Both
intervals are normalized as [x / 2, x / 2]. In the newly
defined coordinates, according to the sampling theorem,
the signal can be represented in both domains with
N  x2 samples spaced 1/ x  1/ N apart.
According to the algorithm, the transform S ( , m)
of the digital signal
defined as below:
s(n) is DFRFT at p order is
A N jt m 2  2mnn 2  /(2x) 2  n 
s

e
2x  N
 2x 
  cot ,   csc
S ( , m) 
The FrFT has found many applications in digital
signal processing such as filtering (Aldirmaz and Durak,
2010; Erden et al., 1999), signal restoration (Erden
et al., 1999), system synthesis (Erden and Ozaktas,
1998), mutual intensity synthesis (Erden et al., 1996),
system decomposition (Kutay et al., 1999; Yetik et al.,
2000), optimum Wiener filtering (Kutay and Ozaktas,
1998), image restoration (Kutay and Ozaktas, 1998) and
perspective projections (Yetik et al., 2000). More
complete treatment of the FrFT and many of its optics
and digital signal processing applications can be found
in Ozaktas et al. (2000). The FrFT suggests a potential
improvement in any application where the ordinary FT
is used since it provides an extra degree of freedom
corresponding to the choice of the order s(n) . We can
attempt to improve the solution to any problem that
utilizes the FT by carrying the extra parameter
throughout the solution and then optimizing over this
parameter. The FrFT is most likely to improve the
solutions to problems where chirp signals are involved.
This is because a chirp signal forms a line in the timefrequency plane and therefore, there exists an order for
which such a signal is compact.
Array model: Let a uniform linear array as shown in
Fig. 1. This array has M sensors with a reference sensor
on the coordinate origin point. The i-th sensor
coordinate location is (i-1) d, (i = 1, 2, …, M). Assume
that wideband signal from far-field impinge on the
array. Thus, the observed signal at the i-th sensor can be
expressed as:
 
f s ri .u

x
n

s
n

(
)
(
)  n (t )  s ( n  qi )  n (t )
i

c

 q  f s ( i  1) d sin 
 i
c
where, n (t) is the additive white Gauss noise, Which is
assumed to be statistically independent with signal
sources, qi is the path delay, c is a wave velocity and 
is source direction.
Beamforming in fractional fourier domain: In order
for the model to be expressed parametrically, first the
concept of fractional delay should be defined and the
relations' must be extracted for the discrete state, by use
of which the beamforming of wideband signal in the
FRFD is modeled as beamforming of narrowband
signal in the Fourier domain.
(3)
3511 (4)
Res. J. Appl. Sci. Eng. Technol., 4(18): 3510-3515, 2012
2 f s q

cot n 
j
2
F p x(n  f s q)e N s



 X m.e
C() 

1
 j / N C( ) f s q2 B( ) f s qm

2
,
cot
csc
, B() 
s
s2
(8)
Modelling of beamforming in FRFD: According to
the definition fractional delay Eq. (8), we obtain xˆi (n)
fractional delay for each sensor as a below:
j
2 qi
cot n
N s2
Fig. 1: Uniform linear array
xˆi (n)  s(n  qi )e
Fractional delay: DFRFT for the delay signal and
Multiplication by an exponential signal can be
expressed as a below:
xˆi (n)  xi (n)Qi (n) , Qi (n)  e
X  m  ( f s q / s) cos e
(5)
Xˆ i (m)  S m.e
2 k
j
n

F  x ( n )e N  


p
X  m  sk sin  e
C ( ) 
 s2k 2

j 2 / N  
sin  cos   sk cos  m 


2


j
j
2 k
n
N
2 k
n
N
call is fractional delay:
.e
and if k  
fsq
cot  then:
s2
s2
Xˆ i (m)  Xˆ i , opt
(9)
, B( ) 
,
csc opt
s
(10)
,
s(n),
p
(m)  F xˆi (n)
popt
opt
(11)
If Xˆ  [ Xˆ 1 (m), Xˆ 2 (m),..., Xˆ M (m)] the output of
array can be compactly represented as a matrix form:
 s2k 2

j / N 
sin cos sk cosm
 2



(12)
where,
1

2
 j / N  1 C( )q12  B( )q1m 
j / N  C ( ) qM
 B( ) qM m  


A  e  2
,...,e  2



(13)
(7)
is the steering vector in FrFD. The output of
beamformer is:
3512 cot opt
Xˆ  AS ( m)  N ,
 ( f q)2

fq
j / N s 2 sin cos  s sin mkfs q sin2  
 2s

s


2 qi
cot n
N s2
1

j / N  C ( ) qi2  B ( ) qi m 
2

S (m)  Sopt (m)  F
2k
j
n

F p x(n  f s q)e N  


X m  sk sin  f s q / s cos.e
2 qi
cot n
N s2
where,
(6)
Now obtain DFRFT x ( n  f s q )e
, then we
gain k in term of q so that DRFRT it obtain in term of
X  (m ) , then x ( n  f s q ) e
j
j
Then according to the definition DFRFT Eq. (3), obtain
DFRFT Eq. (9) and extract the model for beamforming
in DFRFT:
F p x(n  f s q)
 ( f q)2

f q
j 2 / N  s
sin  cos   s sin m 
 2S 2

S


 xi (n)e
Res. J. Appl. Sci. Eng. Technol., 4(18): 3510-3515, 2012
Fig. 2: Output of conventional beamformer in FRFD an ordinary domain for signal along noise
M
M
i 1
i 1
y ( n)   x i ( n  q i )   a i xˆ i ( n  q i ) e
and the DFrFT of
Yopt (m)  F
popt
j
2 qi cot 
n
N
s2
SIMULATION
(14)
We demonstrate that the proposed method yields
improved results, that is smaller MSE. Assume a ULA
of M = 12 and the observed signal contain wideband
LFM component as a below:
y(n) can be obtain:
2 qi cot
M
n
j
2
ai xˆi (n  qi )e N s 

 i1

N/2

e jt m 2mn/ sn
2
T  0 . 01 , f s  4  10 4 .
2 q cot
M
  a xˆ (n  q )e  j N s n 
ii i
i
2
/ s2 / N
N nN / 2
2
 i1
 (15)
when Eq. (15) is a simplified:
M
Y opt ( m)   ai Xˆ i (m)e
i 1
1

 j / N  C ( ) qi2  B ( ) qi m 
2

(16)
In the FrFD, Original array model can be
degenerated into:
Xˆ  AS ( m )  N ,
Y  W Xˆ
(17)
Conventional beamformer: The received signal often
include some noise. The noise can be modeled as
complex, Gaussian distributed random number. We
assume that 3 dB SNR at each sensor. Figure 2 show
the output of conventional beamformer in Time domain
and conventional beamformer in optimum fractional
Fourier domain. The value of the SNR varies from -20
dB to 5dB with an interval of 1dB. At each level of the
SNR, we run 800 Monte Carlo simulations and the
RMSE obtained is shown in Fig. 3 and Compare of the
MSE for the Conventional FrFT Beamformer and
Conventional Beamformer
Adaptive Beamformer (MVDR): In the presence of
strong interference, the target signal may be masked by
3513 
f 0  5  10 3 ,   6  10 5.
Yopt (m) 
A

s ( t )  exp j 2 f 0 t  0 . 5  t 2
Res. J. Appl. Sci. Eng. Technol., 4(18): 3510-3515, 2012
MSE vs SNR
Output of MVDR Beamformer in Fractional Fourier Domain
1.5
Proposed
Orignal
16
1
12
0.5
10
Magnitud
Mean square error
14
8
6
4
0
-0.5
2
0
-20
-15
-10
-5
SNR (dB)
-1
5
0
-1.5
Fig. 3: Comparison of the MSE for the conventional FrFT
beamformer and conventional beamformer
50
100
150
200
Time (s)
250
300
350
400
Fig. 5: Output of MVDR beamformer in FrFD for signal
along noise and interference
Output of Conventional Beamformer in Fractional Fourier Domain
1
0.5
Magnitud
0
Conventional in
MVDER in Fractional
FourierDomain
MVDR in Fourier Domain
25
0
FourierDomain
Conventional in Fractional
FourierDomain
20
-0.5
15
-1
0
50
100
150
200
Time (s)
250
300
350
400
10
Output of Conventional Beamformer
1.5
5
Magnitud
1
0.5
0
-20
0
-15
-10
-5
0
5
10
15
-0.5
-1
0
50
100
150
200
Time (s)
250
300
350
Fig. 6: Comparison of the MSE for the conventional FrFT
beamformer, conventional beamformer, MVDR and
MVDR in FRFD
400
Fig. 4: Output of conventional beamformer in FrFD and
ordinary domain for signal along noise and
interference
the interference signal. Such problems are very challenging for a conventional beamformer and
therefore, adaptive beamformer are introduced to
address this problem. To overcome the interference
problem we can use the MVDR beamformer, a popular
adaptive beamformer. The MVDR Beamformer
preserve the signal arriving along a desired direction,
while trying to suppress signal coming from other
direction.
Figure 4 show Output of Conventional beamformer
in FRFD and Ordinary Domain for signal along noise.
Figure 5 show output of MVDR bean former in FrFD
for signal along noise and interference. The value of the
SNR varies from -20 to 5 dB with an interval of
1dB, at each level of the SNR, we run 800 Monte
Carlo simulations and the RMSE for the Conventional
FrFT Beamformer, Conventional Beamformer, MVDR
and MVDR in FrFD obtained is shown in Fig. 6. In this
figure can be clearly seen that, the beamforming for
wideband LFM signals with MVDR algorithm based on
FrFT has better performance, compared to MVDR
algorithm based on Frequency domain. Especially, its
effectiveness is more obvious in low SNR cases.
CONCLUSION
We have proposed a new method of beamforming
using the FrFT. The method was shown to be especially
useful (yielding smaller errors) for chirp signals. This
method can be useful also in active radar and sonar
3514 Res. J. Appl. Sci. Eng. Technol., 4(18): 3510-3515, 2012
when we chirp signals are transmitted to the target. The
simulations verified that the new method was better
than the conventional method in frequency domain.
REFERENCES
Aldirmaz, L. and S., Durak 2010. Adaptive fractional
fourier domain filtering. J. Sci. Direct Signal
Process., 90: 1188-119.
Erden, M.F., H.M. Ozaktas and D. Mendlovic, 1996.
Synthesis of mutual intensity distributions using
the fractional fourier transform. Opt. Commun.,
125: 288-301.
Erden, M.F. and H.M. Ozaktas, 1998. Synthesis of
general linear systems with repeated filtering in
consecutive fractional fourier domains. J. Opt. Soc.
Amer. A., 15: 1647-1657.
Erden, M.F., M.A. Kutay and H.M. Ozaktas, 1999.
Repeated filtering in consecutive fractional fourier
domains and its application to signal restoration.
IEEE Trans. Signal Process., 47: 1458-1462.
Gabriel, W.F., 1972. Spectral analysis and adaptive
array superresolution techniques. Proc. IEEE, 60:
926-935.
Kutay, M.A., H.M. Ozaktas, O. Arikan and L. Onural,
1997. Optimal Filtering in Fractional Fourier
domain. IEEE Transaction on Signal Process.,
45(5): 1129-11423.
Kutay, M.A. and H.M. Ozaktas, 1998. Optimal image
restoration with the fractional fourier transform. J.
Opt. Soc. Amer. A., 15: 825-833.
Kutay, M.A., H.M. Ozaktas and O. Arikan, 1999. The
fractional fourier domain decomposition. Signal
Process., 77(1): 105-109.
Monzingo, R.A. and T.W. Miller, 1980. Introduction to
Adaptive Arrays. Wiley-Interscience, New York,
pp: 555.
Namias, V., 1980. The fractional order fourier
transform and its application to quantum
mechanics. IMA J. Appl. Math., 25(3): 241-265.
Ozaktas, H.M., O. Arikan and M.A. Kutay and
B. Gözde, 1996. Digital Computation of the
Fractional Fourier Transform. IEEE Trans. Signal
Process., 44: 2141-2150.
Ozaktas, H.M., Z. Zalevsky and M.A. Kutay, 2000. The
Fractional Fourier Transform with Applications in
Optics and Signal Processing. Wiley, New York,
pp: 532, ISBN: 978-0-471-96346-2.
Tao, H.W., 2004. LCMV Beamforming Algorithm
Based on the Fractional Fourier Transform.
ICCCAS, pp: 826-830.
Yetik, I.S., H.M. Ozaktas, B. Barshan and L. Onural,
2000a. Perspective Projections in the SpaceFrequency Plane and Fractional Fourier transforms.
J. Opt. Soc. Amer. A., 17: 2382-2390.
Yetik, I.S., M.A. Kutay, H. Ozaktas and H.M. Ozaktas,
2000b. Continuous and discrete fractional fourier
domain decomposition. IEEE International
Conference on Acoustics, Speech and Signal
Processing, 2000, (ICASSP '00), Dept. of Electr.
Eng., Bilkent Univ., Ankara, 1: 93-96.
Yetik, I.S. and A. Nehorai, 2003. Beamforming using
fractional fourier transform. IEEE Trans. Signal
Process., 51(6): 1663-1668
Zhou, H. and R. Tao, 2008. Broadband Beamforming
of LFM Signal Based on the Fractional Fourier
Transform. International Conference on Signal
Processing, 2008. 26-29 Oct., Dept. of Electron.
Eng., Beijing Inst. of Technol., Beijing, pp: 296298.
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