Linear Antenna Array synthesis with Decreasing Sidelobe and Narrow Beamwidth

ACEEE Int. J. on Communications, Vol. 03, No. 01, March 2012

Linear Antenna Array synthesis with Decreasing
Sidelobe and Narrow Beamwidth
Sudipta Das1, Mangolika Bhattacharya2, Atanu Sen1 and Durbadal Mandal3
1
Department of Electronics and Communication Engineering, Dumkal Institute of Engineering and Technology,
Murshidabad, West Bengal, India
Email: sudipta.sit59@gmail.com, atanusen007@gmail.com
2
Department of Electrical and Electronics Engineering, Aryabhatta Institute of Engineering and Management,
Durgapur, West Bengal, India
Email: mangolika@gmail.com
3
Department of Electronics and Communication Engineering, National Institute of Technology,
Durgapur, West Bengal, India
Email: durbadal.bittu@gmail.com

Abstract—Synthesizing arrays with low sidelobe and pencil uniformity in both the current distribution and the inter-
beam radiation profile is under investigation for decades. A element distance profile is proposed. The establishment of
variety of array structures are available, but the simplest and the paper follows Design Equation in section II, A brief
useful structure is that of a linear array. Here, two basic
discussion on Differential Evolution Algorithm in section III,
symmetric Linear Antenna Array structures are assumed. The
required array structure is assumed to provide low sidelobe
Discourse on results in section IV and ends up summing and
and pencil beam profile. Departure from a uniformity in concluding on the results in section V.
current and location profile has shown quiet appreciable
improvement in the radiation pattern. The simulations are II. DESIGN EQUATIONS
carried out using Differential Evolution Algorithm employing
Figures 1 and 2 denote two Symmetric Linear Antenna Array
Best of Random mutation strategy (DEBoR).
structures placed along z-axis, without and with a centre
element respectively. Symmetric linear broadside array
Index Terms—Non-uniform Current-Location profile, Low
Sidelobe-Narrow Beam synthesis, Low Current Tapering, structure without any centre element, have a generalized Array
Differential Evolution Algorithm,Best of Random Mutation. Factor as given by [11]
N
I. INTRODUCTION AFw ( I , x, )  2 I n cosKx n cos   (1)
n 1
From past few decades since the concept of using arrays
instead of a single element has been evolved, researchers The Array Factor corresponding to a symmetric structure as
have took the challenge to provide various array designs to considered in the text with a centre element is given by,
provide a radiation characteristics according to the N
requirements [1-10]. Synthesizing an array depends on several AFc ( I , x, )  I 0  2 I n cosKx n cos  (2)
matters, like requirements on the radiation pattern, directivity n 1
pattern etc. Radiation pattern depends on the number and Where, N is the number of elements on one side of the array
the type of elements being used, the physical and electrical
axis,
structure of the array etc. Numerous variations over the
antenna structures as well as the type of elements are I n is the excitation amplitude of the nth element from the
available, but for simplicity only one kind of elements are
used in the whole array structure [11]. Aiming towards a centre, and I  I 1 , I 2 ,  , I n , , I N  is the current
radiation pattern with low sidelobe and narrow beam, this vector defining the current distribution over the aperture.
paper deals with linear array geometries with omnidirectional
Current vector is normalized to its maximum value. I 0 is the
antenna elements without any inter-element phasing. For
simplicity in formulation of pattern characteristics two current amplitude for the centre element.
symmetric structures of linear arrays are assumed, one without x n is the distance of the nth element from the centre, and
and the other with a centre element and the whole array is
assumed to be broadside. The design goal in this paper is to x  x1 , x 2 ,  , x n , , x N  is the location vector..
find-out a probable profile of locating elements and the current
distribution over the linear array aperture, that could provide Location vector is expressed in terms of the wavelength  .
low and decreasing sidelobes, narrow main beam, but with
2
the least sacrifice in the directional characteristics. In search K is the wave number, and  is the azimuth angle.
of a good radiation and directional characteristics, non- 
© 2012 ACEEE 10
DOI: 01.IJCOM.3.1.46


L  2 x N is the length of the array. The inter-element dis- The Cost Function CF is designed carefully so as to shape
the optimization as a minimization problem. It is given as
tance is assumed never to exceed the limit  / 2,   .
SLLU 2
CF    AF  k   TR
SLLC k
(4)
 BWFN C  BWFN U
Where,
SLLC is the peak sidelobe level (SLL) of the design evolved
with current generation in dB,
SLLU is the SLL value in dB of the uniform array with the
same length as found in the current iteration.
2
The term  AF  
k
k is used to impose null [11] in every

direction outside the main beam.
Figure 1. Schematic architecture of a symmetric Linear Array
Antenna structure of 2N elements placed along z-axis
BWFN C denotes the first-null beamwidth (BWFN) of the
radiation pattern with the parameters generated from current
iteration,
BWFN U denotes the BWFN of the radiation pattern of the
uniform array of same length as that of the non-uniform array
generated in the current iteration.

III. THE DIFFERENTIAL EVOLUTION ALGORITHM (DEA)
No Optimization technique has ever been superior on all
problems. However, according to the survey by . Roscca, G.
Oliveri and A. Massa [9], since Storn and Price had introduced
a special type of Evolutionary Algorithm namely, Differential
Evolution [6], a huge number of researches employing DEA
have been carried out in the past decade, of which a
Figure 2. Schematic architecture of a symmetric Linear Array significantly large portion were in the field of electromagnetics
Antenna structure of 2N+1 elements placed along z-axis and antenna synthesis. In [8] a new improved variety of DEA
The directivity of the non-uniform array is given by, is introduced, employing Best of Random selection strategy
in mutation portion. This is called as Differential Evolution
2
 N TOT  with Best of Random algorithm (DEBoR).
  Ik 
  The main idea of DEA is to use vector differences in the
D NU  N TOT NTOT  k 1  creation of new probable solutions. DEA aims at evolving a
sin K  z l  z  (3) population of S trial solutions to achieve a optimal solution
 1 I l I m K z  z m
l 1 m  l m in K max iterations. Each of S individual is identified by one
chromosome that codes a set of unknown descriptive of the
Where, N TOT is the total number of the element on the array problems solution. DEA proceeds in the following steps,

aperture, I j and z j are the current amplitude and location A. Initialization

of j th element on the aperture looking the array from one S chromosomes each with N genes are generated. To
end. generate a chromosome, z n , n  1,2,  N within its lower
N TOT  2 N , for a symmetric array without any centre min max
z n , n  1, 2,  N and upper z n , n  1, 2, . N
element, and
bounds a uniform random string rn , s  0,1 is used in the
(k )
N TOT  2 N  1 , for a symmetric linear array with a centre
following manner,
element.
© 2012 ACEEE 11


The steps namely Mutation, Crossover and Selection are
min max

min
z n, s  z n, s  rn, s z n, s  z n, s  continued until the maximum iteration is reached.
Where r varies with n, n  1,2, N ,
IV. RESULTS AND DISCUSSIONS
s, s  1, 2, S and k , k  1, 2,  K max . Cost Function This segment establishes the modelled results for different
CF  z  is designed carefully for multi-objective optimization broadside symmetric linear antenna array designs optimized
by DEBoR profiency. Eight groups of linear array structures
process. Maximum number of iterations K max , Scaling Factor are studied. Sets for harmonious structures without core
elements include 14, 18, 22 and 26 elements, while for centred
F and Crossover probability CR is carefully initialized. elemented symmetric structures are taken as 15, 19, 23 and
F and CR are random variables in 0.4  F  1 and 27. It is found that the best conclusions are received for an
starting population of 120 chromosomes. Utmost number of
CR  0.3, 0.9 .
generations is limited to 400. Scaling Factor F for mutation
B. Mutation is selected as 0.49. Binary Crossover is chosen with Crossover
3 out of S chromosomes are randomly selected and Probability CR as 0.35.
recombined according to the Best of Random (BoR) strategy A. Numerical Data and Resultant Patterns
[8] to create a mutant vector v as below,
,
Parameters related to the physical, electrical and directional
(k ) (k )
v s  z  F  z y  z y  (k ) (k )
 properties are Tabulated in Table I, II and III. Inter-element
distance for a uniform array and Location of elements for a
y
non-uniform array is considered as physical parameter, while
y is the number of the differential variations.  ,  y and excitation amplitude of each element pair is considered as the
electrical parameter. Current amplitude for the uniform array
 y are the indices of randomly chosen secondary parent is considered to be 1. Radiaton parameters are considered as
the SLL and BWFN. Directivity is commemorated for the non-
(k ) (k )
z  , and two randomly chosen donor vectors ( z  y and uniform array as radiation parameter. Table I records the SLL
and BWFN and Peak Directivity of the array sets. The initial
(k )
z  y ), of which, 
inter-element spacing is considered as . For symmetric
CF z     min CF z , CF z  z
(k ) (k )
y
(k )
y
k
s is called the
2
linear array sets without and with centre element Table II and
primary parent. III record the location of element pairs and the respective
C. Crossover current amplitudes and compares the SLL and BWFN with
that of the representing consistent array with same length.
Binary Crossover is adopted for generating new offspring
Peak Directivity as proposed by the optimal non-uniform set
(k )
solution vector u n , s to compete with the corresponding is marked in each of the Tables .
Figures 3 and 4 equates the radiation patterns of non-
k
primary parent z s . The process is done as uniform array groups to their representing equally long
uniform array sets for 26 and 27 elements respectively.
v nks) if r1(nk, s)  CR
(
Radiation patterns are diagrammed in linear dB scale in  -
u (k )
  (,k )
otherwise , plane, since linear arrays as regarded in these papers execute
n,s
 z n,s
omni-directional properties in  - plane.
(k )
Where, r1 n , s is a string of uniformly distributed random T ABLE I. SLL, BWFN, AND DIRECTIVITY FOR UNIFORM LINEAR ARRAY SETS

numbers over the range 0, 1 . is kept under the same bounds WITH INTER-ELEMENT SPACING AS /2
as specified for .
D. Selection
Based on the fitness is operated on and in order to keep
better solutions for the next iteration and discard the relatively
poor one.

zs
k 1 u ( k )
  sk )
 (k )
if CF u s  CF z s   (k )

(
z x  
if CF z (sk )  CF u (sk )  
© 2012 ACEEE 12


TABLE I. LOCATION OF ELEMENTS , C URRENT APLITUDES, SLL, BWFN, AND Figure 4 and the Tables I and III show that for 27 element
DIRECTIVITY FOR NON-UNIFORM SYMMETRIC LINEAR ARRAY SETS WITHOUT array the SLL value of the non-uniform array is reduced to -
CENTRE ELEMENT
20.23 dB against -13.42 dB of the uniform arrays. Directivity
of the non-uniform array is 27.07 while that for the initial
array was 27. BWFN of the optimized set is 7.46o, while those
for the initial and equally long uniform arrays are 8.50o and
6.29o respectively.

Figure 3. Θ-Plane radiation Patterns of 26 element broadside
symmetric linear antenna arrays
TABLE I. LOCATION OF ELEMENTS , CURRENT APLITUDES , SLL, BWFN, AND
DIRECTIVITY FOR NON-UNIFORM SYMMETRIC LINEAR ARRAY SETS WITH A C ENTRE
ELEMENT

Figure 4. Θ-Plane radiation Patterns of 27 element broadside
symmetric linear antenna arrays

Figures 5 and 6 portray that an adept selection of position of
elements and current distribution over the array aperture can
remarkably mend the resultant radiation pattern, even if
compared to equally long uniform array. The results are
tabulated in Tables I, II and III.
From the Tables I and II and Figure 3 it is clearly viewed
that, for a 26 element linear array, radiation pattern of both Figure 5. Optimal Current distribution over the array aperture of
the uniform arrays have -13.42 dB whereas, the non-uniform 26 element broadside linear antenna array
array inhibits the sidelobe to -22.02 dB. The Directivity of the
non-uniform array is 26.97 and that of the initial array is 26.
The BWFN of the non-uniform array is 8.49o, and those for
initial and equally long uniform arrays are 8.82o and 6.90o.

© 2012 ACEEE 13


CONCLUSIONS
The modelled resultants have demostrated that
considerable meliorations can be detected with the current-
location non-uniformity. Less dwindling of currents have
rendered overall sidelobes in reducing manner and the main-
beamwidth in between the initial and equally long uniform
arrays. Thus, it defeats the problems due to unceasing
sidelobes in a large and the peak directivity is restrained to
be leastwise to that of the initial array. Hence, Differential
Evolution with Best of Random mutation scheme has
executed successfully to work on the linear array synthesis
problem.
Figure 6. Optimal Current distribution over the array aperture of
27 element broadside linear antenna array REFERENCES
B. Convergence Profile of DEBoR [1] P. K. Murthy, and A. Kumar, “Synthesis of Linear Antenna
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Figure 8.Convergence Profile of Differential Evolution Algorithm [9] P. Roscca, G. Oliveri and A. Massa, “Differential Evolution as
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Figures 7 and 8 registers the convergence profiles of Magazine, Vol 53, No. 1, February, 2011, pp. 38-49.
Differential Evolution Algorithm as the respective optimization [10] Mangolika Bhattacharya, Sudipta Das, Durbadal Mandal, Anup
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Synthesis Based on Angular Positions”, 2011 International
value of the Cost Function CF in each iteration is recorded
Conference on Network Communication and Computer (ICNCC
for the respective optimization process. These curve display 2011), New Delhi, India, held since 19th to 20 th March, 2011, in
that the best up to the current iteration in each process is press.
saved and thus results are prohibited from deterioration. The [11] C. A. Balanis, Antenna Theory Analysis and Design, 2nd
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found converging after 337 and 167 iterations. The
programming has been written in MATLAB language using
MATLAB 7.5 on core (TM) 2 duo processor, 1.83 GHz with 2
GB RAM.
© 2012 ACEEE 14

Linear Antenna Array synthesis with Decreasing Sidelobe and Narrow Beamwidth

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Linear Antenna Array synthesis with Decreasing Sidelobe and Narrow Beamwidth