![ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010
Optimal Synthesis of Array Pattern for Concentric
Circular Antenna Array Using Hybrid
Evolutionary Programming
Durbadal Mandal 1, Sakti Prasad Ghoshal 2 and Anup Kumar Bhattacharjee 1
1
Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur,
West Bengal, India- 713209 Email: durbadal.bittu@gmail.com, akbece12@yahoo.com
2
Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India- 713209 Email:
spghoshalnitdgp@gmail.com
Abstract—In this paper one optimization heuristic search of the solution space increase, ii) frequent convergence to
technique, Hybrid Evolutionary Programming (HEP) is local optimum solution or divergence or revisiting the same
applied to the process of synthesizing three-ring Concentric suboptimal solution, iii) requirement of continuous and
Circular Antenna Array (CCAA) focused on maximum differentiable objective cost function, iv) requirement of
sidelobe-level reduction. This paper assumes non-uniform the piecewise linear cost approximation, and v) problem of
excitations and uniform spacing of excitation elements in each
convergence and algorithm complexity. So, in this work,
three-ring CCAA design. Experimental results reveal that the
design of non-uniformly excited CCAAs with optimal current for the optimization of complex, highly non-linear,
excitations using the method of HEP provides a considerable discontinuous, and non-differentiable array factors of
sidelobe level reduction with respect to the uniform current CCAA designs, one heuristic search technique such as a
excitation with d=λ/2 element-to-element spacing. Among the novel hybrid evolutionary programming technique (HEP)
various CCAA designs, the design containing central element [9] is adopted. Each optimal CCAA design should have an
and 4, 6 and 8 elements in three successive concentric rings optimized set of non-uniform current excitation weights,
proves to be such global optimal design with global minimum which, when incorporated, results in a radiation pattern
SLL (-40.22 dB) as determined by HEP. with significant sidelobe level reduction.
Index Terms—Concentric Circular Antenna Array, Non-
uniform Excitation, Sidelobe Level, Hybrid Evolutionary II. PROBLEM FORMULATION
Programming Geometrical configuration is a key factor in the design
process of an antenna array. For CCAAs, the elements are
I. INTRODUCTION arranged in such a way that all antenna elements are placed
In array pattern synthesis, the main objective is to find in multiple concentric circular rings, which differ in radii
the physical layout of the array that produces the radiation and in number of elements. Fig. 1 shows the general
pattern closest to the desired pattern. Over the past few configuration of CCAA with M concentric circular rings,
decades [1-8] many synthesis methods are concerned with where the mth (m = 1, 2,…, M) ring has a radius rm and the
suppressing the SLL while preserving the gain of the main corresponding number of elements is Nm. If all the
beam. Low sidelobes in the array factor are usually elements in all the rings are assumed to be isotopic sources,
obtained through amplitude excitation weighting the signal the radiation pattern of this array can be written in terms of
at each element. The antenna arrays have been widely used its array factor only.
in phase array radar, satellite communications and other Referring to Fig.1, the array factor, AF (φ , I ) for the
domains [6]. In the satellite communications in order to CCAA in x-y plane may be written as (1) [7]:
improve the ability of antenna array to resist interference
M Nm
AF(φ, I ) = ∑∑Imi exp j(kr cosφ −φmi ) +αmi )]
[ m (
and noise the pattern of the antenna array should have low
sidelobes, controllable beamwidth and the pattern synthesis
in azimuth angles. The traditional optimization methods m=1 i=1
cannot bear the demand of such complex optimization (1)
problem.
Classical optimization methods have several
disadvantages such as: i) highly sensitive to starting points
when the number of solution variables and hence the size
Corresponding author: Durbadal Mandal
Email: durbadal.bittu@gmail.com
17
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![ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010
III. THE PROPOSED ALGORITHMS
A. Evolutionary Programming (EP)
In general, evolutionary computation programming is a
very rich class of multi-agent stochastic search (MASS)
algorithms based on the Neo-Darwinian paradigm of
natural evolution, which can perform exhaustive searches
in complex solution space [9]. These techniques start first
with searching a population of feasible solutions generated
stochastically; then, stochastic variations are incorporated
to the parameters of the population in order to evolve the
solution to a global optimum. Thus, these methods provide
a rigorous stochastic search in the entire domain taking into
account maximum possible interactions among them.
Fig. 1. Concentric circular antenna array (CCAA). B. Basic Evolutionary Programming (BEP)
where I mi denotes current excitation of the ith element of The efficiency of any optimization algorithm for finding
a global optimum for a given function depends on how
the mth ring. k = 2π λ , λ being the signal wave-length. φ effectively the balance has been maintained between two
symbolize the azimuth angle from the positive x axis to the contradictory requirements, exploitation of already
orthogonal projection of the observation point respectively. discovered regions, and exploration of new and unknown
The angle φ mi is the element to element angular separation regions in the search space. An evolutionary programming
measured from the positive x-axis. As the elements in each method, which models evolution at the level of computing
ring are assumed to be uniformly distributed, we have species for the same resources, uses mutation as the sole
operator for the advancement of generation, and the
φmi = 2π ((i −1) N ) ;
m
m = 1,⋅ ⋅ ⋅, M ; i = 1,⋅ ⋅ ⋅, N m (2) amount of exploitation and exploration is decided only
through the mutation operator. Usually, in the BEP [9], the
α mi is the phase difference between the individual mutation operator produces one off-spring from each
elements in the array which is a function of angular parent by adding a Gaussian random variable with zero
separation φmi and ring radii rm. mean and a variance proportional to the individual fitness
score. The value of standard deviation, which is the square
α mi = − Krm cos(φ0 − φmi ) ; m = 1,⋅ ⋅ ⋅, M ; i = 1,⋅ ⋅ ⋅, N m (3) root of the variance, decides the characteristics of off-
spring produced with respect to its parent. A standard
where φ0 is the value of φ where peak of the main lobe is deviation close to zero will produce off-spring that has
obtained. more probability of resembling its parent, and a much less
After defining the array factor, the next step in the design probability of being largely or altogether different from it.
process is to formulate the objective function which is to As the value of the standard deviation departs from zero,
be minimized. The objective function “Cost Function” the probability of resemblance of off-spring with its parent
(CF ) may be written as (4): decreases, and the probability of producing altogether
different off-spring increases. With this feature, the
AF (φmsl 1 , I mi ) + AF (φmsl 2 , I mi ) standard deviation essentially maintains the trade-off
CF = WF 1 × between exploration and exploitation in a population
AF (φ0 , I mi ) (4) during the search.
+ WF 2 × BWFN computed − BWFN (I mi = 1) The main steps of the algorithm are as follows:
Step 1: Initialization:
where φ0 is the angle where global maximum is attained Initialize a population pool. Let Ii= [I1,i I2,i I3,i …. Ij,i ....
In,i], where j=1 to n; n is the number of current excitation
in φ ∈ [− π , π ] . φmsl 1 is the angle where the maximum
weights.
sidelobe is attained in the lower band and φmsl 2 is the angle Step 2: Creation of BEP based off springs
where the maximum sidelobe is attained in the upper band. An ith off-spring vector I′i is created (mutation process)
BWFN is angular width between the first nulls on either from each parent vector Ii by adding to each component j
side of the main beam. In (4) the two beamwidths, of Ii a Gaussian random variable N (0, σj(i)) with zero
BWFN computed and BWFN (I mi = 1) are the first null mean and a standard deviation σj (i) proportional to scaled
CF values of the parent trial solution as given by the
beamwidths for the non-uniform excitation case and for the
following equations:-
uniform excitation respectively. WF 1 and WF 2 are the I′i= [I′1,i I′2,i I′3,i …………. I′j,i…….. I′n,i] (5)
weighting factors. where I′j,i = Ij,i + N (0, σj (i))
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![ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010
An FBM operated off-spring vector Ij,i″ is created from
σ j (i ) = α ×
CF (i )
min_ CF
( )
× I j , i max − I j , i min
each parent Ij,i by mutation, which is essentially
algebraically adding (depending on the sign as computed)
where “ min_ CF ” is minimum CF value in the
to each component (j) of Ii a Gaussian random variable
population np and α is the strategy parameter with zero mean and the standard deviation σFBM j,i.
experimentally chosen within prefixed limits and remains Ij,i″ =Ij,i + N (0, σFBM j,i) × Dirj,i (9)
fixed for the maximum number of BEP run cycles. Then, evaluation, competition and selection are similarly
Population of off-spring vectors is also np. So, a total performed among the BEP generated off-springs and FBM
population pool of 2×np vectors is formed and all will generated off-springs (instead of parents as in BEP), the
undergo mutual competition and selection through random total population pool being the same as 2×np. Effective
evaluation of a quantity called ‘score’ (Step 3). strategy parameters, α , β are fixed by experimentation.
Step 3: Evaluation, Competition and Selection
The np parent trial vectors Ii and np corresponding off- V. NUMERICAL RESULTS
spring vectors I′i contend for surviving with each other
within the competition pool of size 2×np. The score for ith This section gives the experimental results for various
trial vector in 2×np pool is evaluated by CCAA designs obtained by HEP technique. For this
np optimization technique ten three-ring (M=3) CCAA
wi = ∑ w j (6) designs for each of two cases as a) without central element
j =1
feeding and b) with central element feeding in three-ring
w j =1, if ( CF (i ) < CF (t ( j ))) else w j = 0, where j varies CCAA are assumed. Each CCAA maintains a fixed
from 1 to np , i varies from 1 to 2×np and t(j)= [2×np×u spacing between the elements in each ring (inter-element
+1], u is random number ranging over [0, 1] and [y] spacing being 0.55λ, 0.61λ and 0.75λ for first ring, second
denotes the greatest integer less than or equal to y. ring and third ring respectively). These spacings are the
After competing, the 2×np trial solutions including the means of the values determined for the ten designs for non-
parents and the off-springs are ranked in descending order uniform spacing and non-uniform excitations in each ring
of the score obtained in (6). The first np survive and are using 25 trial generalized optimization runs for each
transcribed along with their objective functions CF into design. For all sets of experiments, the number of elements
the survivor set as the basis of the next generation. A for the inner most ring is N1, for the outermost ring is N3,
maximum number of BEP run cycles ‘Nm’ is given. The whereas the middle ring consists of N2 number of elements.
search process is stopped as the final count of run cycles For all the cases, φ0 = 00 is considered so that the centre of
reaches ‘Nm’. Then, the optimal cycle is determined for the main lobe in radiation patterns of CCAA starts from the
which the CF is the grand lowest and the solution is origin. After several experimentations, the best proven
optimal. parameters are: i) WF 1 and WF 2 are fixed as 18 and 1
respectively. ii) Initial population = 120, iii) Maximum
IV. HYBRID EVOLUTIONARY PROGRAMMING (HEP) number of iteration cycles = 400, and iv) α = 0.4, β = 0.6 .
Steps 1 and 2 are the same. Step 3 involves creation of Sets of three-ring CCAA (N1, N2, N3) designs
another set of off-springs using FBM operator (FBM based considered for both without and with central element
off-springs), which will be actually competing with BEP feeding are (2,4,6), (3,5,7), (4,6,8), (5,7,9), (6,8,10),
based off-springs instead of parent solutions as in BEP (7,9,11), (8,10,12), (9,11,13), (10,12,13), (11,13,15). HEP
algorithm. generates a set of normalized, optimized non-uniform
In the HEP [9] algorithm, the primary factor is the current excitation weights for each CCAA design. Some of
introduction of a new mutation mechanism, which is called the optimal results as determined by HEP are shown in
a fitness-blind mutation (FBM) operator. The FBM Tables II-III. Table I depicts SLL values and BWFN values
operator uses a standard deviation (σFBMj,i) for the Gaussian for all corresponding uniformly excited ( I mi =1) CCAA
distribution, which is made proportional to the absolute designs.
value of its genotype distance (Diffj,i) from the fittest
parent in that generation by another strategy multiplying A. Analysis of Radiation Pattern of Optimal CCAA
factor β as given below: Figs. 2-3 depict the substantial reductions in SLL with
Diffj, i =absolute value of (Ijopt - Ij, i) (7) non-uniform optimal current excitations as compared to the
where j is the number of parameters to be optimized and case of uniform non-optimal current excitations. All
Ijopt is the parameter corresponding to minimum CF CCAA design sets having central element feeding (Case
among np trial vectors for the current BEP run cycle and Ij,i (b)) yield much more reductions in SLL as compared to the
are the parameters of ith trial vector of the same cycle. same not having central element feeding (Case (a)). As
σFBM j, i = β × Diffj,i (8) shown in Tables II-III, SLL reduces to -32.4 dB for Case
(a) and -40.22 dB (grand lowest SLL) for Case (b), with
The ‘+’or ‘-‘sign of (Ijopt - Ij,i) is also computed as Dirj,i. the CCAA having N1=4, N2=6, N3=8 elements (Set No.
The signs are not necessarily the same for different III). BWFN become narrower for non-uniform optimal
parameters denoted by j.
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![ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010
Table III much more reductions in SLL as compared to the same not
Current Excitation Weights, SLL and BWFN For Non-Uniformly Excited
having central element feeding, (ii) The CCAA design
CCAA Design Sets (Case (b)) Using HEP
having N1=4, N2=6, N3=8 elements along with central
Set Current excitation weights for the CF SLL BW element feeding gives the grand maximum SLL reduction
No. array elements ( I11 , I12 ,…., I mi ) (dB) FN
(deg)
(-40.22 dB) as compared to all other designs, which one is
I 0.0008 0.3528 0.3569 0.3323 1.81 -29.54 128.8 thus the grand optimal design among all the three-ring
0.0724 0.3329 0.0758 0.1377 designs. Thus, the proposed HEP technique proves to be a
0.1405 0.0411 0.1319 0.1291 promising evolutionary optimization technique for the
0.0495 global optimization of antenna array problem.
III 0.3439 0.5294 0.4568 0.5139 0.39 -40.22 93.2
0.4515 0.3531 0.3480 0.0396
0.3428 0.3411 0.0300 0.1236 REFERENCES
0.2012 0.1318 0.0177 0.1428
0.2002 0.1390 0.0189 [1] C. Stearns and A. Stewart, An investigation of concentric
V 0.1939 0.2532 0.2000 0.1735 1.39 -28.8 58.8 ring antennas with low sidelobes, IEEE Trans. Antennas
0.2011 0.2319 0.3748 0.3846 Propag. 13(6) (Nov 1965), 856–863.
0.0016 0.2338 0.4292 0.2103 [2] R. Das, Concentric ring array, IEEE Trans. Antennas
0.0127 0.3924 0.4618 0.1867 Propag. 14(3) (May 1966), 398–400.
0.4303 0.3675 0.1651 0.2062
0.1539 0.3414 0.4194 0.1625 [3] N. Goto and D. K. Cheng, On the synthesis of concentric-
0.1976 ring arrays, IEEE Proc. 58(5) (May 1970), 839–840.
VII 0.4078 0.1970 0.4477 0.1998 1.02 -31.52 58.1 [4] L. Biller and G. Friedman, Optimization of radiation patterns
0.5073 0.2234 0.4604 0.1883 for an array of concentric ring sources, IEEE Trans. Audio
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Electroacoust. 21(1) (Feb. 1973), 57–61.
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0.0108 0.2729 0.0710 0.1091 [5] M D. A. Huebner, Design and optimization of small
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0.0961 0.1405 0.1487 0.2681 455–458.
0.1353 0.0947 0.0398
[6] C. A. Balanis, Antenna Theory Analysis and Design, John
Wiley & Sons, New York, 1997.
VI. CONCLUSION [7] R.L.Haupt, “Optimized element spacing for low sidelobe
concentric ring arrays,” IEEE Trans. Antennas Propag., vol.
In this paper, the optimal design of a non-uniformly 56(1), pp. 266–268, Jan. 2008.
excited CCAAs with uniform inter-element spacing and [8] M. Dessouky, H. Sharshar, and Y. Albagory, “Efficient
with / without central element feeding has been described sidelobe reduction technique for small-sized concentric
using the hybrid evolutionary optimization technique, circular arrays,” Progress In Electromagnetics Research,
vol. PIER 65, pp. 187–200, 2006.
HEP. Experimental results reveal that the design of non-
[9] A. K. Swain and A. S. Morris, “A Novel Hybrid
uniformly excited CCAA offer a considerable SLL Evolutionary Programming Method for Function
reduction along with the reduction of BWFN as well as Optimization,” Evolutionary Computation, 2000.
compared to the case of corresponding uniformly excited Proceedings of the 2000 Congress on, vol. 1, pp. 699-705,
CCAA. The main contribution of the paper is twofold: (i) 2000.
All CCAA designs having central element feeding yield
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Optimal Synthesis of Array Pattern for Concentric Circular Antenna Array Using Hybrid Evolutionary Programming
- 1. ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010 Optimal Synthesis of Array Pattern for Concentric Circular Antenna Array Using Hybrid Evolutionary Programming Durbadal Mandal 1, Sakti Prasad Ghoshal 2 and Anup Kumar Bhattacharjee 1 1 Department of Electronics and Communication Engineering, National Institute of Technology, Durgapur, West Bengal, India- 713209 Email: [email protected], [email protected] 2 Department of Electrical Engineering, National Institute of Technology, Durgapur, West Bengal, India- 713209 Email: [email protected] Abstract—In this paper one optimization heuristic search of the solution space increase, ii) frequent convergence to technique, Hybrid Evolutionary Programming (HEP) is local optimum solution or divergence or revisiting the same applied to the process of synthesizing three-ring Concentric suboptimal solution, iii) requirement of continuous and Circular Antenna Array (CCAA) focused on maximum differentiable objective cost function, iv) requirement of sidelobe-level reduction. This paper assumes non-uniform the piecewise linear cost approximation, and v) problem of excitations and uniform spacing of excitation elements in each convergence and algorithm complexity. So, in this work, three-ring CCAA design. Experimental results reveal that the design of non-uniformly excited CCAAs with optimal current for the optimization of complex, highly non-linear, excitations using the method of HEP provides a considerable discontinuous, and non-differentiable array factors of sidelobe level reduction with respect to the uniform current CCAA designs, one heuristic search technique such as a excitation with d=λ/2 element-to-element spacing. Among the novel hybrid evolutionary programming technique (HEP) various CCAA designs, the design containing central element [9] is adopted. Each optimal CCAA design should have an and 4, 6 and 8 elements in three successive concentric rings optimized set of non-uniform current excitation weights, proves to be such global optimal design with global minimum which, when incorporated, results in a radiation pattern SLL (-40.22 dB) as determined by HEP. with significant sidelobe level reduction. Index Terms—Concentric Circular Antenna Array, Non- uniform Excitation, Sidelobe Level, Hybrid Evolutionary II. PROBLEM FORMULATION Programming Geometrical configuration is a key factor in the design process of an antenna array. For CCAAs, the elements are I. INTRODUCTION arranged in such a way that all antenna elements are placed In array pattern synthesis, the main objective is to find in multiple concentric circular rings, which differ in radii the physical layout of the array that produces the radiation and in number of elements. Fig. 1 shows the general pattern closest to the desired pattern. Over the past few configuration of CCAA with M concentric circular rings, decades [1-8] many synthesis methods are concerned with where the mth (m = 1, 2,…, M) ring has a radius rm and the suppressing the SLL while preserving the gain of the main corresponding number of elements is Nm. If all the beam. Low sidelobes in the array factor are usually elements in all the rings are assumed to be isotopic sources, obtained through amplitude excitation weighting the signal the radiation pattern of this array can be written in terms of at each element. The antenna arrays have been widely used its array factor only. in phase array radar, satellite communications and other Referring to Fig.1, the array factor, AF (φ , I ) for the domains [6]. In the satellite communications in order to CCAA in x-y plane may be written as (1) [7]: improve the ability of antenna array to resist interference M Nm AF(φ, I ) = ∑∑Imi exp j(kr cosφ −φmi ) +αmi )] [ m ( and noise the pattern of the antenna array should have low sidelobes, controllable beamwidth and the pattern synthesis in azimuth angles. The traditional optimization methods m=1 i=1 cannot bear the demand of such complex optimization (1) problem. Classical optimization methods have several disadvantages such as: i) highly sensitive to starting points when the number of solution variables and hence the size Corresponding author: Durbadal Mandal Email: [email protected] 17 © 2010 ACEEE DOI: 01.IJEPE.01.03.79
- 2. ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010 III. THE PROPOSED ALGORITHMS A. Evolutionary Programming (EP) In general, evolutionary computation programming is a very rich class of multi-agent stochastic search (MASS) algorithms based on the Neo-Darwinian paradigm of natural evolution, which can perform exhaustive searches in complex solution space [9]. These techniques start first with searching a population of feasible solutions generated stochastically; then, stochastic variations are incorporated to the parameters of the population in order to evolve the solution to a global optimum. Thus, these methods provide a rigorous stochastic search in the entire domain taking into account maximum possible interactions among them. Fig. 1. Concentric circular antenna array (CCAA). B. Basic Evolutionary Programming (BEP) where I mi denotes current excitation of the ith element of The efficiency of any optimization algorithm for finding a global optimum for a given function depends on how the mth ring. k = 2π λ , λ being the signal wave-length. φ effectively the balance has been maintained between two symbolize the azimuth angle from the positive x axis to the contradictory requirements, exploitation of already orthogonal projection of the observation point respectively. discovered regions, and exploration of new and unknown The angle φ mi is the element to element angular separation regions in the search space. An evolutionary programming measured from the positive x-axis. As the elements in each method, which models evolution at the level of computing ring are assumed to be uniformly distributed, we have species for the same resources, uses mutation as the sole operator for the advancement of generation, and the φmi = 2π ((i −1) N ) ; m m = 1,⋅ ⋅ ⋅, M ; i = 1,⋅ ⋅ ⋅, N m (2) amount of exploitation and exploration is decided only through the mutation operator. Usually, in the BEP [9], the α mi is the phase difference between the individual mutation operator produces one off-spring from each elements in the array which is a function of angular parent by adding a Gaussian random variable with zero separation φmi and ring radii rm. mean and a variance proportional to the individual fitness score. The value of standard deviation, which is the square α mi = − Krm cos(φ0 − φmi ) ; m = 1,⋅ ⋅ ⋅, M ; i = 1,⋅ ⋅ ⋅, N m (3) root of the variance, decides the characteristics of off- spring produced with respect to its parent. A standard where φ0 is the value of φ where peak of the main lobe is deviation close to zero will produce off-spring that has obtained. more probability of resembling its parent, and a much less After defining the array factor, the next step in the design probability of being largely or altogether different from it. process is to formulate the objective function which is to As the value of the standard deviation departs from zero, be minimized. The objective function “Cost Function” the probability of resemblance of off-spring with its parent (CF ) may be written as (4): decreases, and the probability of producing altogether different off-spring increases. With this feature, the AF (φmsl 1 , I mi ) + AF (φmsl 2 , I mi ) standard deviation essentially maintains the trade-off CF = WF 1 × between exploration and exploitation in a population AF (φ0 , I mi ) (4) during the search. + WF 2 × BWFN computed − BWFN (I mi = 1) The main steps of the algorithm are as follows: Step 1: Initialization: where φ0 is the angle where global maximum is attained Initialize a population pool. Let Ii= [I1,i I2,i I3,i …. Ij,i .... In,i], where j=1 to n; n is the number of current excitation in φ ∈ [− π , π ] . φmsl 1 is the angle where the maximum weights. sidelobe is attained in the lower band and φmsl 2 is the angle Step 2: Creation of BEP based off springs where the maximum sidelobe is attained in the upper band. An ith off-spring vector I′i is created (mutation process) BWFN is angular width between the first nulls on either from each parent vector Ii by adding to each component j side of the main beam. In (4) the two beamwidths, of Ii a Gaussian random variable N (0, σj(i)) with zero BWFN computed and BWFN (I mi = 1) are the first null mean and a standard deviation σj (i) proportional to scaled CF values of the parent trial solution as given by the beamwidths for the non-uniform excitation case and for the following equations:- uniform excitation respectively. WF 1 and WF 2 are the I′i= [I′1,i I′2,i I′3,i …………. I′j,i…….. I′n,i] (5) weighting factors. where I′j,i = Ij,i + N (0, σj (i)) 18 © 2010 ACEEE DOI: 01.IJEPE.01.03.79
- 3. ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010 An FBM operated off-spring vector Ij,i″ is created from σ j (i ) = α × CF (i ) min_ CF ( ) × I j , i max − I j , i min each parent Ij,i by mutation, which is essentially algebraically adding (depending on the sign as computed) where “ min_ CF ” is minimum CF value in the to each component (j) of Ii a Gaussian random variable population np and α is the strategy parameter with zero mean and the standard deviation σFBM j,i. experimentally chosen within prefixed limits and remains Ij,i″ =Ij,i + N (0, σFBM j,i) × Dirj,i (9) fixed for the maximum number of BEP run cycles. Then, evaluation, competition and selection are similarly Population of off-spring vectors is also np. So, a total performed among the BEP generated off-springs and FBM population pool of 2×np vectors is formed and all will generated off-springs (instead of parents as in BEP), the undergo mutual competition and selection through random total population pool being the same as 2×np. Effective evaluation of a quantity called ‘score’ (Step 3). strategy parameters, α , β are fixed by experimentation. Step 3: Evaluation, Competition and Selection The np parent trial vectors Ii and np corresponding off- V. NUMERICAL RESULTS spring vectors I′i contend for surviving with each other within the competition pool of size 2×np. The score for ith This section gives the experimental results for various trial vector in 2×np pool is evaluated by CCAA designs obtained by HEP technique. For this np optimization technique ten three-ring (M=3) CCAA wi = ∑ w j (6) designs for each of two cases as a) without central element j =1 feeding and b) with central element feeding in three-ring w j =1, if ( CF (i ) < CF (t ( j ))) else w j = 0, where j varies CCAA are assumed. Each CCAA maintains a fixed from 1 to np , i varies from 1 to 2×np and t(j)= [2×np×u spacing between the elements in each ring (inter-element +1], u is random number ranging over [0, 1] and [y] spacing being 0.55λ, 0.61λ and 0.75λ for first ring, second denotes the greatest integer less than or equal to y. ring and third ring respectively). These spacings are the After competing, the 2×np trial solutions including the means of the values determined for the ten designs for non- parents and the off-springs are ranked in descending order uniform spacing and non-uniform excitations in each ring of the score obtained in (6). The first np survive and are using 25 trial generalized optimization runs for each transcribed along with their objective functions CF into design. For all sets of experiments, the number of elements the survivor set as the basis of the next generation. A for the inner most ring is N1, for the outermost ring is N3, maximum number of BEP run cycles ‘Nm’ is given. The whereas the middle ring consists of N2 number of elements. search process is stopped as the final count of run cycles For all the cases, φ0 = 00 is considered so that the centre of reaches ‘Nm’. Then, the optimal cycle is determined for the main lobe in radiation patterns of CCAA starts from the which the CF is the grand lowest and the solution is origin. After several experimentations, the best proven optimal. parameters are: i) WF 1 and WF 2 are fixed as 18 and 1 respectively. ii) Initial population = 120, iii) Maximum IV. HYBRID EVOLUTIONARY PROGRAMMING (HEP) number of iteration cycles = 400, and iv) α = 0.4, β = 0.6 . Steps 1 and 2 are the same. Step 3 involves creation of Sets of three-ring CCAA (N1, N2, N3) designs another set of off-springs using FBM operator (FBM based considered for both without and with central element off-springs), which will be actually competing with BEP feeding are (2,4,6), (3,5,7), (4,6,8), (5,7,9), (6,8,10), based off-springs instead of parent solutions as in BEP (7,9,11), (8,10,12), (9,11,13), (10,12,13), (11,13,15). HEP algorithm. generates a set of normalized, optimized non-uniform In the HEP [9] algorithm, the primary factor is the current excitation weights for each CCAA design. Some of introduction of a new mutation mechanism, which is called the optimal results as determined by HEP are shown in a fitness-blind mutation (FBM) operator. The FBM Tables II-III. Table I depicts SLL values and BWFN values operator uses a standard deviation (σFBMj,i) for the Gaussian for all corresponding uniformly excited ( I mi =1) CCAA distribution, which is made proportional to the absolute designs. value of its genotype distance (Diffj,i) from the fittest parent in that generation by another strategy multiplying A. Analysis of Radiation Pattern of Optimal CCAA factor β as given below: Figs. 2-3 depict the substantial reductions in SLL with Diffj, i =absolute value of (Ijopt - Ij, i) (7) non-uniform optimal current excitations as compared to the where j is the number of parameters to be optimized and case of uniform non-optimal current excitations. All Ijopt is the parameter corresponding to minimum CF CCAA design sets having central element feeding (Case among np trial vectors for the current BEP run cycle and Ij,i (b)) yield much more reductions in SLL as compared to the are the parameters of ith trial vector of the same cycle. same not having central element feeding (Case (a)). As σFBM j, i = β × Diffj,i (8) shown in Tables II-III, SLL reduces to -32.4 dB for Case (a) and -40.22 dB (grand lowest SLL) for Case (b), with The ‘+’or ‘-‘sign of (Ijopt - Ij,i) is also computed as Dirj,i. the CCAA having N1=4, N2=6, N3=8 elements (Set No. The signs are not necessarily the same for different III). BWFN become narrower for non-uniform optimal parameters denoted by j. 19 © 2010 ACEEE DOI: 01.IJEPE.01.03.79
- 4. ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010 current excitations as compared to the case of uniform non- 5 optimal current excitations for all the design sets in both the test cases. For the same optimal CCAA set, the BWFN 4 values are 77.10 for Case (a), and 93.20 for Case (b) against 90.30 (Case (a)), 95.40 (Case (b)) respectively for the corresponding uniformly excited CCAA having the same 3 number of elements. So, this technique yields maximum C F reductions of BWFN also for this optimal CCAA. 2 B. Convergence profile for HEP 1 The minimum CF values are recorded against the number of iteration cycles to get the convergence profile. 0 Fig. 4 portrays the convergence profile of minimum 0 50 100 150 200 250 Iteration Cycle 300 350 400 CF for N1=4, N2=6, N3=8 with central element feeding. Fig. 4. Convergence profile for HEP in case of non-uniformly excited The programming is written in MATLAB 7.5 version on CCAA (N1=4, N2=6, N3=8 with central element feeding) core (TM) 2 duo processor, 3.00 GHz with 2 GB RAM. Table I 0 SLL And BWFN FOR Uniformly Excited ( I mi =1) CCAA Sets -10 N o rm a liz e d a rr a y fa c to r (d B ) Set No. of Without central With central -20 No. elements in element (Case (a)) element (Case (b)) each rings -30 (N1,N2,N3) SLL BWFN SLL BWFN -40 (dB) (deg) (dB) (deg) I 2, 4, 6 -12.56 128.4 -17.0 140.0 -50 II 3, 5, 7 -13.8 107.2 -15.0 116 -60 Uniform Excitation (without central element feeding) III 4, 6, 8 -11.23 90.3 -12.32 95.4 Uniform Excitation (with central element feeding) IV 5, 7, 9 -11.2 78.2 -13.24 81.6 -70 HEP (without central element feeding) V 6, 8, 10 -10.34 68.4 -12.0 71.1 HEP (with central element feeding) VI 7, 9, 11 -10.0 61.0 -11.32 63.0 -80 -150 -100 -50 0 50 100 150 VII 8, 10, 12 -9.6 54.8 -10.76 56.4 Angle of Arival (Degrees) VIII 9, 11, 13 -9.28 50.0 -10.34 51.3 IX 10, 12, 14 -9.06 46.0 -10.0 47.0 Fig. 2. Radiation pattern for a uniformly excited CCAA and corresponding X 11, 13, 15 -8.90 42.0 -9.8 43.2 HEP based non-uniformly excited CCAA (N1=4, N2=6, N3=8). Table II 0 Current Excitation Weights, SLL and BWFN For Non-Uniformly Excited CCAA Design Sets (Case (a)) Using HEP -10 N o r m a liz e d a r r a y fa c to r (d B ) Set Current excitation weights for the CF SLL BW -20 No. array elements ( I11 , I12 ,…., I mi ) (dB) FN (deg) -30 I 0.6436 0.6539 0.6168 0.0902 1.83 -29.0 126.7 0.6119 0.1027 0.2256 0.2286 -40 0.0816 0.2298 0.2301 0.0980 III 0.0192 0.4230 0.0233 0.4009 0.88 -32.4 78.3 -50 0.2530 0.2507 0.6606 0.2746 -60 0.2473 0.6098 0.2956 0.4095 Uniform Excitation (without central element feeding) Uniform Excitation (with central element feeding) 0.3052 0.1664 0.3213 0.4082 -70 HEP (without central element feeding) 0.3124 0.1514 HEP (with central element feeding) V 0.2539 0.1643 0.2535 0.1615 1.78 -25.96 59.3 -80 0.1837 0.4012 0.4195 0 -150 -100 -50 0 50 100 150 0.3601 0.5418 0.3092 0 Angle of arival (degrees) 0.4333 0.5739 0.1976 0.4085 Fig. 3. Radiation pattern for a uniformly excited CCAA and corresponding 0.4174 0.2424 0.3586 0.2639 HEP based non-uniformly excited CCAA (N1=8, N2=10, N3=12). 0.3406 0.4188 0.2275 0.2457 VII 0.3356 0.1697 0.2629 0.2719 1.53 -27.30 51.3 0.3039 0.1342 0.3562 0.4342 0.3223 0.0657 0.0444 0.1575 0.1418 0.1738 0.0445 0.0721 0.3286 0.2282 0.1509 0.1533 0.3812 0.0685 0.1776 0.1759 0.1427 0.1064 0.3736 0.1897 0.1265 0.1526 20 © 2010 ACEEE DOI: 01.IJEPE.01.03.79
- 5. ACEEE Int. J. on Electrical and Power Engineering, Vol. 01, No. 03, Dec 2010 Table III much more reductions in SLL as compared to the same not Current Excitation Weights, SLL and BWFN For Non-Uniformly Excited having central element feeding, (ii) The CCAA design CCAA Design Sets (Case (b)) Using HEP having N1=4, N2=6, N3=8 elements along with central Set Current excitation weights for the CF SLL BW element feeding gives the grand maximum SLL reduction No. array elements ( I11 , I12 ,…., I mi ) (dB) FN (deg) (-40.22 dB) as compared to all other designs, which one is I 0.0008 0.3528 0.3569 0.3323 1.81 -29.54 128.8 thus the grand optimal design among all the three-ring 0.0724 0.3329 0.0758 0.1377 designs. Thus, the proposed HEP technique proves to be a 0.1405 0.0411 0.1319 0.1291 promising evolutionary optimization technique for the 0.0495 global optimization of antenna array problem. III 0.3439 0.5294 0.4568 0.5139 0.39 -40.22 93.2 0.4515 0.3531 0.3480 0.0396 0.3428 0.3411 0.0300 0.1236 REFERENCES 0.2012 0.1318 0.0177 0.1428 0.2002 0.1390 0.0189 [1] C. Stearns and A. Stewart, An investigation of concentric V 0.1939 0.2532 0.2000 0.1735 1.39 -28.8 58.8 ring antennas with low sidelobes, IEEE Trans. Antennas 0.2011 0.2319 0.3748 0.3846 Propag. 13(6) (Nov 1965), 856–863. 0.0016 0.2338 0.4292 0.2103 [2] R. Das, Concentric ring array, IEEE Trans. Antennas 0.0127 0.3924 0.4618 0.1867 Propag. 14(3) (May 1966), 398–400. 0.4303 0.3675 0.1651 0.2062 0.1539 0.3414 0.4194 0.1625 [3] N. Goto and D. K. Cheng, On the synthesis of concentric- 0.1976 ring arrays, IEEE Proc. 58(5) (May 1970), 839–840. VII 0.4078 0.1970 0.4477 0.1998 1.02 -31.52 58.1 [4] L. Biller and G. Friedman, Optimization of radiation patterns 0.5073 0.2234 0.4604 0.1883 for an array of concentric ring sources, IEEE Trans. Audio 0.3547 0.2062 0.0260 0.0298 Electroacoust. 21(1) (Feb. 1973), 57–61. 0.3548 0.1008 0.3343 0 0.0108 0.2729 0.0710 0.1091 [5] M D. A. Huebner, Design and optimization of small 0.0752 0.2589 0.1412 0.1478 concentric ring arrays, in Proc. IEEE AP-S Symp. (1978), 0.0961 0.1405 0.1487 0.2681 455–458. 0.1353 0.0947 0.0398 [6] C. A. Balanis, Antenna Theory Analysis and Design, John Wiley & Sons, New York, 1997. VI. CONCLUSION [7] R.L.Haupt, “Optimized element spacing for low sidelobe concentric ring arrays,” IEEE Trans. Antennas Propag., vol. In this paper, the optimal design of a non-uniformly 56(1), pp. 266–268, Jan. 2008. excited CCAAs with uniform inter-element spacing and [8] M. Dessouky, H. Sharshar, and Y. Albagory, “Efficient with / without central element feeding has been described sidelobe reduction technique for small-sized concentric using the hybrid evolutionary optimization technique, circular arrays,” Progress In Electromagnetics Research, vol. PIER 65, pp. 187–200, 2006. HEP. Experimental results reveal that the design of non- [9] A. K. Swain and A. S. Morris, “A Novel Hybrid uniformly excited CCAA offer a considerable SLL Evolutionary Programming Method for Function reduction along with the reduction of BWFN as well as Optimization,” Evolutionary Computation, 2000. compared to the case of corresponding uniformly excited Proceedings of the 2000 Congress on, vol. 1, pp. 699-705, CCAA. The main contribution of the paper is twofold: (i) 2000. All CCAA designs having central element feeding yield 21 © 2010 ACEEE DOI: 01.IJEPE.01.03.79