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Title: Uniform Linear Array Beampattern Gain Optmization using LMS Technique
Authors:
Address:
Farrukh Nagi, Nassar Ikram
Universiti Tenaga Nasional
Dept. of Mechanical Eng.
Km 7, Jalan Kjang-Puchong
43009 Kjang, Selangor, Malaysia
Email:
farrukh@uniten.my.edu, drfnagi@yahoo.com
Tel:
60 -03- 89287202
Fax:
60 -03- 89263506
Contact Author: Farrukh Nagi – Dr.
Topic area:
Subject area:
Statistical Signal and Array Processing
Array Processing - Optimization
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Uniform Linear Array Beampattern Gain Optimization using LMS
Technique
Farrukh Nagi
Universiti Tenaga Nasional
Km 7, Jalan Kjang –Puchong
43009 Kjang, Selangor, Malaysia
farrukh@uniten.edu.my, drfnagi@yahoo.com
Radiation Pattern – Beamforming
Abstract
2.
The work describe here uses optimization
technique to increase the gain of uniform linear
array beampattern when array elements fails.
The optimization criteria evaluates the weights of
the remaining elements so as to restore the
reference beampattern. LMS is used to optimize
the error between the reference and the iterated
beampattern.
The error is evaluated after
subtraction of failed beampattern from
the
optimized pattern – reducing computational
overload on the optimization algorithm.
In
general the results are promising near the
boresight of the array and are presented for
different number and configuration of the failed
elements.
Beamforming is a process of forming an output from
the weighted combination of signal from N elements
of ULA [3]sensor array. The beamforming output
from such array is given as
1.
Introduction
The performance of the beamforming process is
degraded when array elements fails to work in
unison. Long usage of phased array radar [1], [2] and
sonar in planes, ships or submarines render array
elements or its associated circuitry defective. The
maintenance and repair is time consuming if not
costly. The efficiency of such array can be restored
by optimization techniques.
Optimization techniques could be used to re-asses the
weights of the remaining array elements in order to
minimize the distortion of the beampattern. The
effect of failed array elements on the beampattern is
the overall reduction in array gain factor. Also, the
sidelobes losses their symmetry due to aperiodic [3]
inter-element distance between some of its elements,
figure 1.
The availability of powerful DSPs and their efficient
computation algorithms can be exploited to
implement the optimization techniques in real time
sonar and radar phased array systems [2]. The
optimization techniques will be demonstrated in this
work for ULA but the procedure can be generalized
for planar and other array configurations.
N
i(2n N 1) (  )
G( )   w(n) e
n1
(1)
Where
 = -90 to 90



180
  d sin 

  d sin 



  , Scanning angle
2

 , Steering angle
2

w(n)=[1,1,…..,N], Uniform weight on each element

180
d, inter-element distance between array elements
Typically the quantity |G()|2 is known beam pattern,
which is simply the power of beamformer as a
function of .
3.
Failed Pattern
The array elements can be failed by zeroing the
weights w(n) in array element weighting/shading in
vector wf(m) as
wf(m) = [1,0,1,0,0,1, M=N-l]
index, m+ p = d, 2d, d, 3d, d, .
(2)
Where M=N-l, l is the numbers of the failed elements
and p is the index of the failed elements. The beam
pattern of the failed elements is evaluated from
equ.(2) as
M
G f ( )   w f (m)e i ( 2( m  p )  M 1) (  )
(3)
m 1
In equ.(3) index m+p caters for increased
inter-element distance in multiple of d, figure 1.
The corresponding weights vector wf is described by
equ. (2).
2
e(n)
w(n)
Effect of more than one element failure can be
investigated. Failing elements at extreme ends will
only reduce the aperture of array and will give effect
of only reducing size N of the array.
d
w=0
w=0
w=0
Gf()
wopt(m)
f(w)=Gopt()
Optimization function
LMS
Figure 1. Increased inter-element array distance
Figure 2. Optimization process expressed in equation (5)
Optimization Function and Pattern
The optimization process evaluates the optimized
weights wopt =[ wopt1 wop2. . . . M] for the failed
beampattern function The optimization function
consist of failed element beampattern f(wopt)=Gf().
The optimization algorithm then evaluates the weight
wopt (m) of only working elements of the array. The
optimized weights are so determined that f(wopt) is
optimized to Gopt() with wopt and approaches
closest to G().
M
f (wopt )  Gopt ( )   wopt (m) ei ( 2( m  p )  M 1) (  )
( 4)
m 1
5.
GA()
wf(m)
3d
2d
4.
G()
6. Performance measure
After completion of optimization process the
comparison is made between failed and optimized
beam pattern to evaluate the improvement of the
optimized pattern over the failed pattern. For this
purpose ratio of powers under -3db beamwidth of
main beam and the rest of patterns is defined as SNR
of the beam pattern , see figure 3, and



  (G f ( )  BW 3db) 
SNR f , opt  20 log 10 
 BW 3db
(7)
Optimization Function
The beam pattern optimization is to based upon
certain criterion. The criterion is chosen so as to
restore the model-reference beampattern in the best
possible way. LMS [4] is utilised to accomplish the
optimized procedure
-3db
The optimization function in equ. (4) is used for the
iterative-search of weights wopt in LMS algorithms as
e(n)  GA( )
2
 G f ( )
2
wopt (m)  wopt (m  1)  2G f ( , m)e(n) w f (m)
( 5)
Where  is the adaptation step size and wf are the
weights updated to wopt in such a way as to reduce the
error e(n). Figure 2 illustrates the LMS optimization
procedure expressed in equ. (5)
The failed pattern Gf() is subtracted from the
optimized function prior to least square error e(n).
GA( )  f ( w opt )  G f ( )
( 6)
Equation (6) reduces the computation burden of
optimization algorithm and helps in its faster
convergence to the solution
Figure 3. –3db beamwidth used in SNR equ. (7)
The rationale behind this SNR is that main beam is
pointing toward desired direction and sidelobes give
rise to undesirable noise entering in receiver from
direction other than desired.
3
7.
Results
Sixteen element ULA optimization results were
evaluated at different steering angles with different
combination of
failing
elements, figure 3.
Matlab’s optimization toolbox [5]
‘leastsq’
command was used for implementation of least
square algorithm shown in Figure 2. The reference
pattern is obtained from |G()| –equ. (1), failed
|Gf()| –equ. (3) and optimized beampattern |Gopt()| equ.(4).
Purposely-failed elements in equ. (2) are represented
by ‘0.000’ in the weight column at the left of the
figures 4.
Figures 5-7 shows S/N vs steering angle. The S/N of
the three radiation patterns in figure 4 are evaluated
using equ.(5). The S/N at different steering angles
were compiled and are shown in figures 5-7
Figure 6. SNR vs scanning angle of reference, optimized
and failed radiation patterns. Failed elements numbers 3, 7.
Figure 4. LMS optimization results - reference, failed and
optimized beampatterns. Failed elements 2, 9, 14.
Figure 5. SNR vs. scanning angle of reference, optimized
and failed radiation patterns. The SNR in figure 4 can be
read at –20 degree steering angle.
Figure 7. SNR vs scanning angle of reference, optimized
and failed radiation patterns. Failed element number 6.
4
8.
Conclusion
From figures 5-7 it can be observed that LMS
optimization is effective near the boresight scanning
angle, as the scanning angle moves away from the
boresight the optimization fails to deliver good S/N.
For single element in figure 7 the optimization is not
effective.
Also the distribution of the failing
elements does not effects the optimization process
figure 5-6.
In view of above results and comments it would be
wise to optimize the radiation pattern. In case of
failure of the elements the best results whether failed
or optimized pattern could be used for
receiving/transmitting the signal.
The LMS optimization algorithms is applied to
ULA, the same can be applied with out losing
generality on two-dimensional array [3]. Further
investigation can be carried out on determining the
effect of failing element distribution on the
beampattern, and using other optimization
techniques.
References
1)
Statistical and Adaptive signal processing
Processing by D.G. Manolakis, V.K. Ingle
and S.M. Kogon, McGraw Hill, Boston
USA, 2000.
2)
Phased Array Antennas by R.C. Hansen,
John Wiley & Sons, Inc. Newark, USA,
1998
3)
Antenna Theory Analysis and Design by C.
A. Balanis –2ed, Wiley & Sons, Inc.
NewYork, USA, 1997.
4)
Engineering Optimization by S. S. Rao,
Wiley & Sons, Inc. NewYork, USA, 1996
5)
Optimization toolbox for Use with
MATLAB – User Guide, MathWorks, Inc.
Natick, Mass. USA, 1996.
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