Investigation on the Pattern Synthesis of Subarray Weights for Low EMI Applications

IOSR Journal of Electronics and Communication Engineering (IOSR-JECE)
e-ISSN: 2278-2834,p- ISSN: 2278-8735.Volume 12, Issue 1, Ver. III (Jan.-Feb. 2017), PP 01-10
www.iosrjournals.org
DOI: 10.9790/2834-1201030110 www.iosrjournals.org 1 | Page
Investigation on the Pattern Synthesis of Subarray Weights for
Low EMI Applications
T Vidhyavathi1
, G S N Raju2
, G V K Varma3
1
Department of ECE, Gayatri Vidya parishad College of Engineering (Autonomous), Visakhapatnam,
Andhrapradesh, India.
2
Department of ECE, Andhra University college of Engineering (Autonomous), Visakhapatnam,
Andhrapradesh, India.
3
Sr. Engineer, Qual.comm, Sandiego, California, USA
Abstract: In modern radar applications, it is frequently required to produce sum and difference patterns
sequentially. The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method
where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose
optimal group weights will be introduced to the different array elements to obtain any type of beam depending
on the application. Optimization of excitation to the array elements is the main objective so in this process a
subarray configuration is adopted. However, Differential Evolution Algorithm is applied for optimization
method. The proposed method is reliable and accurate. It is superior to other methods in terms of convergence
speed and robustness. Numerical and simulation results are presented.
Key Words: Antenna Arrays, Sum and Difference, Pattern Synthesis, Subarray weighting configuration, Ultra
Sidelobe level suppression, Differential Evolution Method.
I. Introduction
The concepts of half-power beamwidth and peak directivity of a linear antenna array pattern are
introduced and it is applied to the case of pattern synthesis. For this purpose a well known technique called
Dolph-Chebyshev synthesis method [1] is widely used. Namara [2] proposed the excitation matching method
based on an expansion in terms of Zolotarev polynomials where each possible grouping, the corresponding
subarray coefficients are iteratively computed.
The design of monopulse radar systems [3]-[4] requires the synthesis of both the sum and the
difference pattern, which satisfy some specifications such as narrow beamwidth, low sidelobe level (SLL) and
high directivity. In order to properly solve the optimal compromise problem in monopulse radar tracking array
antennas, several techniques [5]-[6] based on sub arraying have been proposed to reduce the design complexity
of the feed network. Shindman et al. [7] developed the techniques for designing the minimum power sidelobes
for a main lobe array factor or difference pattern array factor. Although several methods [8]-[10] for
implementing monopulse antennas have been proposed, interest has been shown to the methods that use proper
feed networks for the design of a single feed for both sum and difference patterns instead to two independent
feeds.
Differential Evolution Algorithm (DEA) is most powerful global optimizer and has been successfully
applied to array pattern synthesis and also in other areas of the applied electromagnetic problems. In addition to
the above, proper amplitude excitation weights of the elements are used to control the beam width and sidelobe
level [11]-[12].
The sum pattern amplitude coefficients are obtained by using Dolph-Chebyshev synthesis method
where as the difference pattern excitation coefficients will be optimized in this present work. For this purpose
optimal group weights will be introduced to the different array elements to obtain any type of beam depending
on the application. Optimization of excitation to the array elements is the main objective so in this process a
subarray configuration is used to reduce the design complexity of feeding network.
The excitation coefficients for the sum pattern are calculated from the Dolph-Chebyshev method with
fixed SLL = -40dB. This problem has already studied and final difference pattern was optimized by Differential
Evolution Strategy (DES).
II. Differential Evolution Algorithm
2.1. Initialization:
Differential Evolution is very simple to understand and easy to implement, which begins with a population of D
dimensional vectors and denotes by Ns shown in the following equation (1)
Xi,G, i=1, 2, …, Ns (1)

Investigation on the Pattern Synthesis of Subarray weights for Low EMI Applications
Where the index „i‟ denotes the population and „G‟ denotes the generation to which the population belongs. DE
strategy depends on three main operators that are mutation, crossover and selection. Schematic representation of
DE strategy is shown in the figure 1.
Figure 1: Schematic Representation of DE Strategy.
2.2. Mutation:
The mutation process at each generation begins by randomly selecting three individual variables that are
Xr1
, Xr2
, Xr3
in the population set of Ns elements.
For each target vector Xi,G , a mutant vector is generated according to the following equation
ViG+1 = Xi,G + F. Xr2,G − Xr3,G (2)
Where the indexes r1, r2, r3 Є {1, 2, 3 … Ns} are randomly selected such that r1 ≠ r2 ≠ r3 ≠ i, F is a real and
constant factor Є [0, 2] which controls the strengthening of the discrepancy variation (Xr2,G – Xr3,G).
2.3. Crossover:
Once mutation part is completed, in order to increase the range of the perturbed parameter vectors crossover is
introduced. In this stage the parent vector is mixed with the mutated vector to produce a trail vector Uji,G+1. The
trail vector is described as follows.
Ui,G+1 = U1i,G+1, U2i,G+1, … … … , UDi,G+1 (3)
Uji,G+1 =
Vji,G+1 if randm b(j) ≤ CR or j = k
Xji,G if randm b j > CR or j ≠ k
(4)
Where j Є {1, 2… D}, k is a random parameter index chosen once for each „i‟.
CR is the crossover constant Є [0, 1].
randm b(j) is the jth
evaluation of a uniform random number generator with outcome Є [0,1]. Figure 2 gives an
example of crossover mechanism for 7-dimensional vectors.
Figure 2: Illustration of the crossover process for D=7 parameters.
2.4. Selection:
Selection takes place with a contest held between the one with best fitness function and the target
vector, which are allowed to enter the next generation. The trail vector Ui,G+1 is compared to the target vector
Xi,G using the greedy criterion to decide whether it should become a member of generation G+1 or not.
Population for the next generation is selected according to the following rule.
Xi,G+1 =
Ui,G+1 if f Ui,G+1 ≤ f Xi,G
Xi,G otherwise
(5)
Where j= 1, 2, 3… Ns.
For this problem, the ith
element of the population at the kth
iteration is indicated by Vk(i), (i=1, . . . ,Ns), which
has the following hybrid structure with integer and real variables Vk(i)= ( g1... gp , c1... cN). The key Parameters
of the DE algorithm CR and F should be accurately chosen in order to avoid a premature convergence to local
minima or to a slow convergence rate.

III. Mathematical Formulation
3.1 Dolph-Chebyshev Array Design Procedure:
Consider an array of isotropic elements positioned symmetrically along the X-axis. Suppose the distance
between any two adjacent elements is „d‟, and the array is operated at λ/2, a symmetric linear array is shown in
figure 3.
Figure 3: Geometry for N element linear array
Its excitation coefficients are related to Tchebyshev polynomials. The recursion formula for rth
Tchebyshev
polynomial Tr (z) is given below
Tr (z) = 2z T r-1 (z) – T r-2 (z) (6)
Here every polynomial can also be computed using
T r (z) = cos (r cos-1
(z)) -1 ≤ z ≤ +1 (7)
T r (z) = cos (r cosh-1
(z)) z < -1, z > +1
With this symmetric amplitude excitation the array factor of an array with even or odd number of elements is
more than a sum of cosine terms.
To show the approach, consider a linear array of N equally-spaced elements whose array factor AF (θ).
AF N = ancos 2 n − 1 u N = 2M (even)
M
n=1
(8)
Where, u =
πd
λ
cosθ
„an‟ are the amplitude excitation coefficients
„λ‟ is wave length in integer times of fundamental frequency,
„θ‟ defines the angle at which AF(θ) is calculated with respect to the broadside direction.
„d‟ is the inter-element distance,
„M‟ the number of elements.
3.2 Array synthesis methodology:
The objective of the synthesis is to construct a reduced subarray configuration able to synthesize as better as
possible this pattern. To avoid the implementation of several designing network arrays, a subarray configuration
is adopted. The sum Pattern of array factor AFs (θ) is obtained starting by a set of excitation coefficients, an
s
,
(n= -N. . .-1, 1, . . ., N) which are assumed to be symmetric i.e., a-n
s
=an
s
, (n=1,. . ., N) and are fixed.
In this case, the array space factor is given by
AFs θ = an
s
cos⁡[
1
2
2n − 1 k d cosθ]
N
n=1
(9)
The „N‟ number of elements for the array is grouped into „P‟ subarrays in order to construct the
difference pattern. Each subarray has a weighting coefficient gcn
, p = 1,..,P, and in order to create a difference
pattern, the group of the antennas must be optimized.
In particular, if cn = p, then the nth
element is to be connected to the pth
subarray. If cn = 0, then the element is
not considered in the process of synthesis.
The excitation coefficients of the difference pattern can be obtained by multiplying each coefficient of the sum
pattern to the coefficient of the corresponding subarray group weight.
Formally,
an
d
= gcn
. an
s
, cn =1, . . . , p , . . . P.
Here gcn
denotes the Kronecker function, i.e.,
gcn
= 1 if cn = p
gcn
= 0 elsewhere
(10)

Where gp is the group weight of the pth
subarray.
A typical configuration for a subarrayed linear array structure with N isotropic elements for sum pattern
excitation levels of an
s
and difference pattern excitation levels of an
d
are shown in figure 4.
Figure 4: Geometry for N element linear array
Due to the problem of symmetry, only one half of the array is considered in the synthesis problem. In
particular, the elements of the array are grouped in to „P‟ subarrays. Since the excitations of the difference
pattern must be antisymmetric, i.e. a-n
d
=-an
d
, (n=1,..., N) to ensure a deep null at the broad side.
In this case, the array space factor is given by
AFd θ = an
d
sin
1
2
2n − 1 kdcosθ (11)
N
n=1
Where „k‟ is a wave number of the propagation medium equals to 2π/λ
an
s
Amplitude sum pattern excitation coefficients,
an
d
Amplitude difference pattern excitation coefficients, and
„θ‟ defines the angle at which „F‟ is calculated with respect to a direction orthogonal to the array.
The sum patterns are obtained for SLL= -40dB and the difference patterns are achieved using proposed method.
This problem can now be deal with an optimization, where the clustering into subarrays „cn‟ and the subarray
group weights „gp‟ are the optimization variables in order to optimize the difference pattern with maximum
directivity. Specifications such as Sidelobe levels and beamwidth for the difference pattern can be optimized
with a proper selection of the element grouping and subarray weights. Only half of the array is needed to be
considered, since both the patterns are symmetric. The optimized radiation patterns are obtained with fixed
SLL‟s for various numbers of subarrays.
3.3 Objective function:
To this end, the optimized objective function used for calculating the fitness function associated with the
symmetrical linear array can be formulated as follows
Fitness = PSLL0 − SLLd (12)
Where PSLL0 = Max∀θЄS 20log10
E θ
Emax θ
(13)
SLLd = Desired sidelobe level
Emax θ = Peak value of the Main beam
„θ‟ = Steering angle from the broad side of the array, −90 ≤ θ ≤ 90
„S‟ = Space spanned by the angle excluding the mainlobe.
IV. Numerical Simulation Results And Discussions
In order to validate the effectiveness of Differential Evolution Method we first examine a linear array
with 8 subarrays of 40 elements and 60 elements that are spaced λ/2 distance apart. The excitations for the sum
pattern an
s
are calculated from the dolph-chebyshev method using equation (8) with SLL=-40dB.
In order to construct the difference pattern, the N number of elements for the array is grouped into P subarrays.
The excitations for the difference pattern an
d
are determined by using differential evolution with the equation
(9). Figure (5) reports the behavior of the cost function versus the number of iterations.
First investigate a linear array of 40 and 60 elements that are spaced λ/2 distance apart with 8 sub arrays for
sum SLL=-40dB. The optimized radiation patterns of difference array space factor for 8 subarrays are obtained
by using the equation (11) and the results are shown in figure (6) and figure (10). As can be seen from these

figures, the resulting first SLL‟s for difference pattern is reduced to FSLL= -54.53dB and FSLL= -54.91dB for
40 and 60 elements.
The radiation patterns of 40, 60 elements for 10, 6, 4, 2 subarrays are computed that are reported in figures (7),
(8) and figures (11), (12) and also the corresponding subbarray configurations and its group weights are listed in
tables (1), (2), (3) and (4).
Finally, for completeness, table (5) presented that 10 subarrays of difference pattern of reduced FSLL=-
54.97dB and table shows that the number of elements increases for increased subarrays, the difference patterns
of FSLL‟s may be reduced. So finally DES has a more robust exploration ability to reach the optimal point in
the search space.
Figure: 5 Behavior of fitness function of 40 elements with SLL = -40dB.
Figure: 6 Sum pattern for a 40 elements Dolph-chebyshev array space factor with SLL=-40dB and the
corresponding optimized difference pattern for P=8 subarrays
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
4
-50
-45
-40
-35
-30
-25
-20
-15
number of iterations
Fitnessvalue
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
Azimuth Angle  (in deg)
PatternIEIindB
Sum pattern
Difference pattern

Figure: 7 optimized difference patterns for various values of P obtained by DE when the sum pattern is
predetermined from the Dolph-chebyshev synthesis.
Table: 1: Subarray configurations of 40 element with SLL = -40dB
Subarray number Optimal clustering in to subarrays and here showing only half of the array
P=10 [10,2,1,1,7,5,8,1,1,5,1,1,1,1,1,1,1,1,2,10]
P=8 [3,5,8,8,8,1,1,2,1,8,1,8,1,2,1,1,8,1,5,3]
P=6 [4,3,2,1,6,6,6,1,5,6,1,6,6,2,1,2,1,1,3,4]
P=4 [2,3,1,4,4,4,4,4,1,1,1,1,4,4,1,1,4,1,3,2]
P=2 [2,2,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,2,2]
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
PatternIEIindB
P=10 subarrays
P=6 subarrays
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
PatternIEIindB
P=4 subarrays
P=2 subarrays

Table: 2 Subarray weights for the arrays described in table 1
P=10 P=8 P=6 P=4 P=2
G1 1.0000 0 1.0000 0.9957 0.8359
G2 0.8056 0.9955 1.0000 0.3213 0.3281
G3 1.0000 0.3120 0.7958 0.8105
G4 0 1.0000 0.3084 1.0000
G5 1.0000 0.8028 0
G6 1.0000 1.0000 1.0000
G7 0.9885 0
G8 0 0.9969
G9 0
G10 0.3164
Figure: 9 Behavior of fitness function of 60 elements with SLL = -40dB.
Figure: 10 Sum pattern for a 60 elements Dolph-chebyshev array space factor with SLL=-40dB and the
corresponding optimized difference pattern for P=8 subarrays
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-50
-45
-40
-35
-30
-25
-20
-15
Number of iterations
Fitnessfunction
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
PatternIEIindB
Sum pattern
Difference pattern

predetermined from the Dolph-chebyshev synthesis
Table: 3 Subarray configurations of 60 element with SLL = -40dB
Subarray number Optimal clustering in to subarrays and here showing only half of the array
P=10 [7,2,10,1,1,10,1,1,1,10,1,1,10,1,10,1,1,1,10,10,1,10,10,1,1,1,4,9,6,8]
P=8 [8,7,5,1,3,3,1,1,1,2,1,1,1,1,1,2,1,1,3,6,2,4,2,3,1,4,5,7,7,8]
P=6 [1,2,3,5,6,4,6,6,4,6,6,5,5,6,6,6,6,6,4,5,6,6,6,6,5,5,6,2,2,1]
P=4 [3,2,1,2,2,4,1,2,2,1,1,1,1,1,1,4,1,4,1,1,1,4,4,2,4,2,1,4,3,3]
P=2 [2,2,2,1,1,1,1,1,1,1,1,1,2,1,2,1,2,1,1,1,1,1,1,1,1,1,1,2,2,2]
Table: 4 Subarray weights for the arrays described in table 3
P=10 P=8 P=6 P=4 P=2
G1 0 0.9857 0.2182 0.9992 0.3818
G2 0.7642 0.9789 0.6445 0.9487 0.2244
G3 0 1.0000 0.8784 0.3980
G4 0.8928 0.9702 0 0.9984
G5 1.0000 0.8533 0
G6 0.5874 0 0
G7 0.2797 0.6053
G8 0.1618 0.2088
G9 0.8666
G10 0
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
PatternIEIindB
P=10 subarrays
P=6 subarrays
-80 -60 -40 -20 0 20 40 60 80
-60
-50
-40
-30
-20
-10
0
PatternIEIindB
P=4 subarrays
P=2 subarrays

Table: 5 Optimized difference pattern FSLL‟s obtained by DE with various values of P sub arrays for SLL = -
40dB
Indices P Number of elements 40 Number of elements 60
P=10 - 54.79 dB - 54.97 dB
P=8 - 54.53 dB - 54.91 dB
P=6 -54.43dB - 51.04 dB
P=4 -41.45dB - 54.67 dB
P=2 -30.46dB -35.16 dB
V. Conclusion
In the present context of growing need in the field of electromagnetic problems a relatively simple and
straight forward optimization method Differential Evolution Algorithm has been demonstrated. By employing
the above, jamming and EMI problems can be reduces effectively. The design of monopulse antennas for which
a newly proposed process based on subarray configuration has been reported and discussed with the help of
latest results. The method has been checked with several array configurations and successfully synthesizes the
sum and difference pattern in a linear array antenna with sufficiently ultra sidelobe level. Hence, the optimal
solutions found by DE are equivalently as good as those obtained in 60 element array. Numerical simulation
results of several array synthesis problems shows that the DES performs much better than PSO, Ant Colony
Optimization and most of the other methods.
References
[1]. R. S. Elliot, “Antenna Theory and Design,” Prentice-Hall, New York, 1981.
[2]. D. A. MC Namara, “synthesis of Sub-Arrayed Monopulse Linear Arrays through Matching of Independently Optimum Sum and
Differences Excitations,” Proc. IEE Microwave Antennas Propagation., Vol. 135, No. 5, PP. 371-374, October 1988.
[3]. C.A. Balanis, “Antenna Theory Analysis and Design,” John Wiley and Sons, INC., United States of America, 1982.
[4]. G.S.N. Raju, Antennas and Wave Propagation, Pearson Education, 2005.
[5]. P. Lopez, F. Ares, J.A. Rodriguez and E. Moreno, “Subarray weighting for the difference patterns of monopulse antennas: joint
optimization of subarray configurations and weights,” IEEE Trans. Antennas and Propagation, Vol.49, No. 11, PP. 1606-1608,
Nov. 2001.
[6]. D. G. Kurup, M. Himdi and A. Rydberg, “synthesis of uniform amplitude unequally spaced antenna arrays using the differential
evolution algorithm,” IEEE Transactions on Antennas and Propagation, Vol. 51, No. 9, PP. 2210-2217, September 2003.
[7]. D A. Shindman, “An optimization technique for sum and difference array factors,” IEEE Transactions on Aerospace and
Electronic Systems, Vol. 40, No. 1, PP. 360-370, January 2004.
[8]. O. M. Bucci, L. Caccavale and T. Isernia, “Optimal far-field focusing of uniformly spaced array subject to arbitrary upper bounds
in non target directions,” IEEE Transactions on Antennas and Propagation, Vol. 50, No. 11, PP. 1539-1554, November 2002.
[9]. O. M. Bucci, M. D‟urso and T. Isernia, “Optimal synthesis of difference patterns subject to arbitrary sidelobe bounds by using
arbitrary array antennas,” Proc. IEE Microwave Antennas and Propagation, Vol. 152, No. 3, PP. 129-137, June 2005.
[10]. M. Alvarez-Folgueiras, F. Arespena and J. Rodriguez-Gonzales, “Optimal compromise among sum and difference patterns in
monopulse antennas: Use of subarrays and distributions with common aperture tail,” Journal of electromagnetic waves and
applications, Vol. 23, Nos. 17-18, PP. 2301-2311, 2009.
[11]. P. Rocca, R. Azaro, L. Manica and A. Massa, “A hybrid approach to the synthesis of subarrayed monopulse linear arrays,” IEEE
Trans. Antennas and Propagation, Vol. 57, PP. 280–283, Jan. 2009.
[12]. Y. Chen, S. Yang and Z. Nie, “The application of a modified differential evolution with strategy adaptation,” IEEE Antennas and
Wireless Propagation Letters, Vol.10, 2011.
Dr. T Vidhyavathi received her B.Tech degree from JNTU Hyderabad in the department of Electronics and
Communication Engineering during the year 2006 and the Master of Technology in Radar
and Microwave Engineering from Andhra University College of Engineering (A) in the
year 2008. Her Ph.D degree awarded in the year 2015 from the department of Electronics
and Communication Engineering, Andhra University College of Engineering (A).
Currently, she is working as an Assistant Professor in the department of ECE, Gayatri
Vidya Parishad College of Engineering (A), Visakhapatnam, Andhra Pradesh, India. Her
Research interests include Antenna Array Processing, Electromagnetic Theory and Wave
Propagation, EMI/EMC, Soft Computing and Engineering applications of convex
optimization. She is a member of IEEE and also a life member of SEMCE (India).
Dr. G S N Raju received his B.E, M.E with distinction also first rank from Andhra University and Ph.D from
IIT- Kharagpur. At present, he is the Honorary Distinguished Professor in department of
Electronics and Communication Engineering, AU College of Engineering (A). He was the
former Vice – Chancellor of Andhra University. He is in teaching and research for the last
35 years in Andhra University. He guided 49 Ph.D.s in the fields of Array Antennas,
Electromagnetics and Wavepropagation, EMI/EMC, Microwave and Radar
Communications, Electronic circuits. Published about 394 technical papers in National/
International Journals/ Conference Journals and transactions. He is the recipient of The
State Best Teacher Award‟ from the Government of Andhra Pradesh in 1999, „The Best

Researcher Award‟ in 1994, „Prof. Aiya Memorial National IETE Award‟ for his best Research guidance in
2008 and Dr. Sarvepalli Radhakrishnan Award for the Best Academician of the year 2007, He was a visiting
Professor in the University of Paderborn and also in the University Karlsruhe, Germany in 1994. He held the
positions of Principal, Andhra University College of Engineering (A), Visakhapatnam, Chief Editor of National
Journal of Electromagnetic Compatibility. Prof. Raju has published 11 textbooks on Antennas and Wave
Propagation, Electromagnetic Field Theory and Transmission Lines, Electronics Devices and Circuits,
Microwave Engineering, Radar Engineering and Navigational Aids. Prof. Raju has been the best faculty
performer in Andhra University with the performance index of 99.37%..
Dr. G V K Varma received the B.Tech. degree from the Indian Institute of Technology Delhi, New Delhi,
India, the Master‟s degree in electrical engineering from Texas A&M University, College
Station, Texas and the Ph.D. degree in electrical engineering from the University of Texas
at Dallas, Richardson, Texas. He is currently working in Qualcomm, San Diego as a
Senior Engineer. His research interests are in the broad areas of wireless communications,
software engineering, algorithms and networking.

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