Synthesis of Cosecant Array Factor Pattern Using Taguchi Method for
Smart Antenna System
Amara Prakasa Rao1 and N.V.S.N. Sarma2
1,2 Dept of Electronics and Communication Engineering, National Institute of Technology, Warangal, India
Abstract—This paper describes design of smart antenna array
beamforming using Taguchi algorithm. A 24-element equally
spaced linear array is synthesized. Based on the concept of the
orthogonal array, this method effectively reduces the number of
tests required in an optimization process. It effectively obtains
the optimum weights for efficient beamforming performance.
Numerical results are presented to verify the convergence, and
computational efficiency of the proposed method.
I.
I NTRODUCTION
15
where k is the wave number, an and ϕn are the excitation
amplitude and phase of the nth element, respectively. dn is
the distance between the position of the nth element and the
centre, θ is the scanning angle from array axis. Fig.2 shows
the functional block diagram of smart antenna system. A smart
array is commonly defined as a multiple antenna elements,
whose signals are processed adaptively in order to optimize
its radiation pattern automatically in response to the radio
channel.
AT
M
S
IN
D
IA
The demand for wireless mobile communication services is
increasing at a rapid pace throughout the globe. As the communication services are becoming more demanding, requirements
for the antenna radiation performance increase greatly. The
synthesis of antenna array systems with the desired radiation
pattern is a nonlinear optimization problem. To find a solution
many analytical techniques such as Woodward Lawson, Taylor method, Fourier transform method, and DolphChebyshev
methods are used[1]. But some beamforming patterns such as
cosecant beam patterns[2], pencil and sector beam patterns[3]
with nulls in the undesired directions and low side lobe levels
are complex to realize. Traditional methods are not efficient
to obtain the optimal solution. So in order to get the best
results the optimization methods are used to synthesize the
complicated beam patterns. In modern technology, shapedbeams are widely used in satellite and radar based applications.
Cosecant-squared pattern(CSP) is one such pattern which is
generally employed for long-range systems requiring higher
gain near the horizon with low gain at higher elevation angles.
Fig. 1. Geometry of the 2N-element symmetric linear array placed along the
x-axis.
20
Keywords—Cosecant Array factor, Beamforming, Optimization
method, Orthogonal array, Taguchi method, Smart antenna system.
In this work, Taguchi method is used to optimize the
magnitude of the array elements to obtain the desired radiation
pattern. The paper is organized as follows. Section II describes
the mathematical model for the problem and also presents
the implementation of the Taguchi method. Simulated results
are discussed in Section III and conclusions are mentioned in
Section IV.
II.
M ATHEMATICAL M ODEL
A linear array of 2N isotropic equispaced elements are
placed symmetrically along the x axis. The array geometry
is shown in Fig.1. Due to the symmetry, the array factor in the
azimuth plane can be written as:
AF (θ) = 2
N
X
n=1
an ej(kdn cos(θ)+ϕn )
(1)
Fig. 2.
Functional Block diagram of Smart Antenna System.
The optimized ranges of all magnitudes and phases are
set to 0 to 1 and 0 to π, respectively. The desired cosecant
pattern is characterized by equations given below[2].
f1 (θ) = 1 , f or 900 ≤ θ < 970
= 1.22csc(cosθ)csc(cos990 ), f or970 ≤ θ < 1200
(2)
=1.22csc(cos1350 ) − csc(cos990 ), f or 1200 ≤ θ ≤1270
−25
20
, elsewhere
csc(cosθ) − csc(cos990 )
f2 (θ) =
, f or95.80 ≤ θ ≤ 1200
1.122
(3)
Taguchi method is developed based on the concept of the orthogonal array(OA), which can effectively reduce the number
of test iterations required in an optimization process[4]-[6].
The Taguchi method process flow chart is shown in Fig.3.
the array is 0.5λ; the excitation magnitudes of each symmetric
element are the same but phases are reversed. So 12 excitation
values are computed based on cosecant beam array factor. The
parameters of the Taguchi method are taken as follows: an
OA(27, 12, 3, 2); number of rows/experiments: 27; number of
columns: 12; Levels: 3; Strength: 2; Reduction rate: 0.9. Using
Taguchi method, the optimization of a linear array quickly
converges to the design parameters, as shown in Fig.4. The
fitness drops significantly during the first 10 iterations. The
optimized cosecant array factor pattern is depicted in Fig.5.
All side lobe levels are below −22.8dB. The desired cosecant
shaped array factor pattern is successfully achieved by using
orthogonal array based Taguchi method. Fig.6 illustrates the
convergence of the magnitude values of the elements in the
array verses the number of runs of 27 experiments. After 50
runs, the final optimum excitation values of the elements in
the array are given in Table I.
Fig. 3.
D
IA
20
15
=10
The Taguchi method process flow chart.
Fig. 4.
Convergence curve of fitness function.
AT
M
S
IN
The pseudo structure of Taguchi process is as follows[8]:
1. Problem Initialization: Select suitable Orthogonal Array(OA) which mainly depends on the number of parameters
to be optimized.
2. Design of Input parameters: Fix the three levels of each parameter, then match OA with level values of Input parameters.
3. Conduct Experiments: Calculate S/N ratio based on fitness
function for each experiment in the OA.
4. Identify Optimal level values and conduct confirmation
experiment: Based on value of S/N, the optimal level for each
parameter is identified. The process will stop, if the result
converges.
5. Reduce the Optimization Range: If it does not converge, the
optimization range is modified for the next iteration.
To obtain the desired pattern in antenna array system,
a suitable fitness function is defined based on control parameters such as size, weight, amplitude, phase, directivity,
gain, side lobe level and null position. To design the adaptive beamforming linear array with minimum side lobe level
cosecant pattern the following fitness function is used[2].
0
θ=180
P
Fitness=W1
(AF (θ) − f1 (θ))[ 1+sgn(AF2(θ)−f1 (θ)) ]4θ
θ=00
+W2
0
θ=120
P
θ=95.80
(θ))
(f2 (θ) − AF (θ))[ 1+sgn(f2 (θ)−AF
]4θ(4)
2
where 4θ is the angular interval, which is set to 0.10 . Both
W 1 and W 2 are set to 0.1 in this study.
III.
S IMULATION R ESULTS
Consider a 24-element symmetrically, equally spaced linear
array with the following specifications:inter element spacing in
Fig. 5. Normalized radiation pattern for the 24-element linear array with
cosecant array factor using Taguchi method.
[6]
Wei-Chung Weng, Fan Yang, Veysel Demir, and Atef Elsherbeni, “Optimization using Taguchi method for electromagnetic applications, 2006
Antennas and Propagation, EuCAP, pp.1-6, 2006.
[7] A.Akdagli and K.Guney, Shaped-beam pattern synthesis of equally and
unequally spaced linear antenna arrays using a modified tabu search
algorithm, Microwave and Opt.Tech.Lett., vol.36, no.1, pp.16-20, January
2003.
[8] Wei-Chung Weng, Fan Yang, V Demir, and A Elsherbeni, Electromagnetic Optimization Using Taguchi Method:A Case Study of Linear Antenna
Array Design, IEEE-1-4244-0123-2/06,pp.2063-2065, 2006.
TABLE I.
15
Fig. 6. Convergence of the magnitude values of the elements in the array
verses the number of iterations.
O PTIMIZED E XCITATION MAGNITUDES AND PHASES OF THE
IN
C ONCLUSION
S
IV.
Optimized Phases(degree)
25
57.37
86.6
91.5 25
86.4
122.4
135.7
143.8
148
180
158
180
D
IA
Optimized magnitudes
0.9
0.8
0.5
0.35
0.34
0.3
0.25
0.15
0.15
0.2
0.1
0.15
AT
M
In this paper, the performance analysis of adaptive beamforming of linear array using Taguchi method is demonstrated.
The cosecant beam pattern is considered in the fitness function,
to optimize the excitation values of the array elements to
exhibit the desired radiation pattern. It is found that Taguchi
method is easy to implement and it converges to the optimal
values quickly with few numbers of runs/experiments. This
method is a good candidate for optimizing electromagnetic
problems. Results show the effectiveness of the proposed
method in comparison with published results[7].
R EFERENCES
[1]
[2]
[3]
[4]
[5]
20
ELEMENTS
Element numbers
1
2
3
4
5
6
7
8
9
10
11
12
C.A.Balanis, Antenna Theory: Analysis and Design, 3rd ed. New Jersey:
John Wiley & Sons Inc., 2005.
Min-Chi Chang and Wei-Chung Weng, Synthesis of Cosecant Array
Factor Pattern using Particle Swarm Optimization, National Chi Nan
University, Puli 54561, Taiwan.
D.Gies and Y.Rahmat-Samii, Particle swarm optimization for reconfigurable phase-differential array design, Microwave and Opt. Tech. Lett.,
vol.38,no.3, pp.168-175,August 2003.
Genichi Taguchi, Introduction to quality engineering, White Plain,
NY:Uni Pub,1986.
Wei-Chung Weng, Fan Yang, Veysel Demir, and Atef Elsherbeni, “Electromagnetic optimization using Taguchi method: A Case study of linear
antenna array design, Antennas and Propagation Society International
Symposium, IEEE, pp.2063-2065, 2006.