A novel approach for reducing the computational cost of GA in synthesizing low sidelobe unequally spaced linear arrays

A Novel Approach for Reducing the ComputationalCost of GA in Synthesizing Low Sidelobe UnequallySpaced Linear Arrays

Shuai Zhang, Shu-Xi Gong, Ying Guan, Bao Lu

National Key Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an,Shaanxi Province 710071, People’s Republic of China

Received 27 April 2010; accepted 24 October 2010

ABSTRACT: This article presents two approaches to reduce the computational cost of

genetic algorithm (GA) in generating low sidelobe linear arrays by optimizing the element

positions subject to the multiple design constraints of the number of elements, the array

aperture dimension, and the minimum interelement spacing. Many experimental results

indicate that only considering a few edge elements on both the ends of the array will have

the same effect on the low sidelobe pattern synthesis with the case of taking all the elements

into account. Firstly, based on this concept, the equation for calculating the number of the

edge elements is found by using the statistical method. With this equation the number of

gene variables of the chromosome of GA in the evolutionary process is considerably

reduced. Undoubtedly, the computational cost of GA is effectively reduced. Then, at the

same time of presenting this equation, a new manner of individual description is introduced.

Using it and modifying the optimization process of GA, the size of the searching space of

GA is reduced and the infeasible solutions during the evolutionary process are avoided.

Thus the computational cost of GA is further reduced. Example arrays with low sidelobe

pattern are synthesized to assess the efficiency and robustness of the proposed GA.

The simulated results demonstrate that the computational cost of GA is considerably

reduced and preferable sidelobe levels are obtained compared with that of previous works.

VC 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE 21:221–227, 2011.

Keywords: antenna arrays; nonuniformly spaced arrays; side lobe suppression; genetic algorithms

(GA)

I. INTRODUCTION

One of the main issues in array synthesis problems is to

find appropriate element positions to generate low side-

lobe level (SLL) radiation pattern. The challenge of deter-

mining optimum parameter values simultaneously stems

from the nonlinear and nonconvex dependency of the

array factor to the element positions. As genetic algorithm

(GA) [1] possesses an intrinsic flexibility to nonlinear

problem, it has been employed in the pattern synthesis of

unequally spaced arrays for years. In [2], Haupt succeeded

in synthesizing low sidelobe unequally spaced arrays sub-

ject to the minimum interelement spacing constraint with

binary GA. In this case the elements are placed on the

array aperture discretized by uniform grids, so that the

minimum interelement spacing constraint can be satisfied

by controlling the spacing of the adjacent grids. However,

when the coding scheme of GA is real coded, the mini-

mum interelement spacing control will become an intrac-

table problem. Chen [3] solved this problem by indirectly

describing the individuals and modifying the optimization

process of GA. As a stochastic search technique, however,

the major disadvantages of GA are premature convergence

and slow convergence speed, especially when they are

employed to solve complicated problems with a large so-

lution space and a number of local optima. To overcome

the problems, a novel multisection based crossover and a

self-supervised mutation process were developed in [4],

with which the computational cost of GA was reduced

and preferable results were obtained compared with that

Correspondence to: S. Zhang; e-mail: [email protected]

VC 2011 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20508Published online 4 March 2011 in Wiley Online Library

(wileyonlinelibrary.com).

221

of [3]. In a word, to improve the performance of GA,

many previous works only focus on modifying the optimi-

zation process [4, 5] or combining GA with other optimi-

zation methods [6–8]. Actually, which is neglected by the

authors, the computational cost of GA can also be reduced

by simplifying the optimization model. Thus this article

will discuss how to improve the performance of GA in

synthesizing low sidelobe pattern by simplifying the opti-

mization model of unequally spaced linear arrays.

For the model with the multiple design constraints of

the number of elements, the array aperture dimension and

the minimum interelements spacing, the studies of [9–11]

show that the elements of the low SLL linear arrays are

partially tapered distributed and some interior elements

are uniformly distributed with the spacing of the value of

the minimum interelement spacing constraint. This phe-

nomenon demonstrates that only considering a few edge

elements on both the ends of the array has the same effect

on the low SLL pattern synthesis with the case of taking

all the elements into account. Consequently, based on this

concept, the equation for calculating the number of the

edge elements is found by using the statistical method.

With the equation the number of gene variables of GA in

the evolutionary process is considerably reduced.

Undoubtedly, the model is simplified and the computa-

tional cost of GA is effectively reduced. Then, at the

same time of presenting this equation, a new manner of

individual description is introduced to further simplify the

model. Using it and modifying the optimization process of

GA, the size of the searching space of GA is reduced and

the infeasible solutions during the evolutionary process

are avoided. Therefore, the model is further simplified and

the computational cost of GA is further reduced.

The other advantage of the proposed method is that the

concept of it can also be used to improve the performance

of other evolutionary methods, such as differential evolu-

tion algorithm [12], simulated annealing algorithm [13],

particle swarm optimization algorithm [14], etc. To illus-

trate the efficiency of the proposed GA, three low SLL

arrays synthesized by it are compared with that of literature

[2, 3, 15, 16]. The comparisons demonstrate that the pro-

posed GA is superior to the previous GAs and other meth-

ods in terms of convergence speed and solution quality.

II. FITNESS FUNCTION OF GA

Consider a linear array of isotropic point sources placed

along the x-axis with the first element at the origin of the

coordinate system (see Fig. 1). The far-field radiated by

the array in the x-z plane is given by

EðhÞ ¼ EeðhÞ �XNn¼1

In expðjkdn sin hÞ (1)

where y is the angle measured from the z-axis; k is the

wavenumber, and k ¼ 2p/k; k is the wavelength; dn is the

position of the nth element; N is the number of elements;

L is the array aperture dimension; Ee(y) ¼ 1 is the ele-

ment pattern; in this article, all the element currents are

supposed to be identical, viz., assume In ¼ 1, for all n.The goal of optimization is to find the appropriate ele-

ment positions to minimize the maximum SLL of the nor-

malized radiation pattern in the x-z plane, so the fitness

function of GA is defined by

fitnessðd1; d2; � � � ; dNÞ ¼ minfmaxEðhÞEmax

��g (2)

where the optimal variable D ¼ {d1, d2, ��, dN} acts as an

individual (the element positions vector of the array), and

the coding scheme of GA is real-coded; Emax is the peak

value of the main beam; �p/2 � y p/2, and the valid

region of y excludes the main beam; thus, max|E(y)/Emax|

is the maximum SLL of the normalized radiation pattern

in the x-z plane.

III. IMPROVED GA

A. Derivation and Validation of the Calculation Formulaof Aperture Release Probability (ARP)In this article, aperture release probability (ARP, which is

denoted by l) is defined as the ratio of the number of the

interior elements to the number of the total elements. The

aim of this section is to find the equation to calculate the

number of the interior and the edge elements. In deriving

the equation, the interelement spacing of a uniform array

duni and the design constrain of the minimum interelement

spacing d0 are both normalized by k. Therefore, duni ¼ L/(N � 1)k, where subscript ‘‘uni’’ stands for uniform; min

{di � dj} � d0, 1�j < i�N.Since it can not be derived based on the ordinary

antenna theory, the statistical method is employed here to

construct the equation of l. Thus, the element placement

form of many low SLL linear arrays is studied in the fol-

lowing paragraphs. The design constrain of d0 ¼ 0.5 is

Figure 1 Geometry of unequally spaced linear arrays.

222 Zhang et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 2, March 2011

considered in many previous works [2–4, 14–16], for

comparison duni taken in this article is larger than 0.5.

Firstly, we take the arrays whose number of elements

increases from 30 to 100 with the step of 10 when duni is0.60, 0.65, 0.70, 0.75, and 0.80, respectively as examples

to study the variation of l with the number of elements

N. As shown in Figure 2, l changes little with N when

duni is prescribed. Therefore, we can assume that the l for

a prescribed duni is the arithmetical average of ls of the

arrays with various N.Then the variation of the l with duni is studied. As

shown in Figure 3, the discrete points denote the arithmet-

ical average of l of various duni. We note that the track of

l varying with duni approaches to a line. The function of

the line, which is obtained by using Least-Squares

Method, shows that the l varies with duni as follows:

l ¼ �0:924 � duni þ 1:054 ¼ �0:924 � L

ðN � 1Þkþ 1:054

(3)

To study the effect of average l on SLL, as shown in

Table I, the SLL of the actual l is compared with that of

the l calculated by (3). d0 ¼ 0.5 and the unit of the SLLs

is dB. It can be seen that the SLLs obtained by this way

approach to that of the actual l, thereby validating the

accuracy of (3).

Then the number of the edge elements N0 is obtained

by

N0 ¼ ð1� lÞ � N (4)

Since the number of elements must be an integer, the

value of N0 must be rounded up and down. In this article,

the digits to the right of the decimal point are dropped.

With equation (4) the number of the gene variables of

chromosome of GA in the evolutionary process is reduced

to 1 � l of its original value, thereby saving much com-

putational cost.

B. New Manner of Individual DescriptionLet dl ¼ 0, dN ¼ L/k thus the array aperture dimension is

always L. According to the standard GA, individual can be

produced with a combination of N � 2 random numbers

between 0 and L/k, but testing their validity (whether satisfythe minimum interelement spacing constraint) is so ineffi-

cient that it may make the problem intractable. To meet the

minimum interelement spacing constraint, the interelement

spacing constraint vector P is introduced and expressed as

P ¼ f0; d0; 2d0; � � � ; ðN � 2Þ � d0; 0gT (5)

Figure 2 The variation of l with N.Figure 3 The variation of l with duni.

TABLE I Comparison of the SLL of the Actual l and the l Calculated by (3)

duni N 30 40 50 60 70 80 90 100

0.60 l of (3) 20.26 20.11 19.93 19.72 19.80 19.91 20.11 20.45

Actual l 20.40 20.37 20.14 19.94 19.93 20.13 20.29 20.54

0.65 l of (3) 22.37 22.94 23.05 23.32 23.16 23.25 23.34 23.49

Actual l 22.58 23.11 23.27 23.67 23.46 23.47 23.52 23.63

0.70 l of (3) 22.43 21.90 21.53 22.57 21.97 21.91 21.90 21.95

Actual l 22.87 22.24 21.72 22.65 22.09 22.06 21.95 22.13

0.75 l of (3) 20.60 20.31 20.49 20.74 20.83 20.67 20.75 20.86

Actual l 20.78 20.46 20.67 20.89 20.91 20.88 20.95 20.94

0.80 l of (3) 20.12 19.85 19.78 19.81 20.02 20.09 20.31 20.52

Actual l 20.27 20.08 20.16 20.05 20.17 20.21 20.41 20.74

A Novel Approach for Reducing the Computational Cost of GA 223

International Journal of RF and Microwave Computer-Aided Engineering DOI 10.1002/mmce

Upon subtracting P from D, we get

D0 ¼ D� P (6)

where D0 ¼ {d

01, d

02, ��, d

0N}, and the value of the elements

of D0 can be obtained by equation (10). Instead of D by

D0 being the chromosome whose elements are gene varia-

bles, the region of (N � 1) � d0 will exclude antenna ele-

ment, thereby the practical array aperture dimension for

placing elements is reduced to Y.

Y ¼ L� ðN � 1Þk � d0 (7)

Obviously, the size of the searching space of GA is down-

sized to e of its original size. Thus the computational cost

of GA is at least reduced to e of its original value.

e ¼ Y

L¼ L� ðN � 1Þk � d0

L¼ 1:054� l� 0:924 � d0

1:054� l(8)

According to equation (4), the number of elements required

to be considered in the evolutionary process is reduced to

N0. Firstly, get N0 � 2 random real numbers among the range

of [0, Y/k] and reset them in an increasing order. Then the

random real number sequence can be expressed as

s ¼ fs1; s2; � � � ; sN0�2gT (9)

Consequently, the value of the gene variables of D0is

obtained by

d0n ¼

0; n¼ 1

sn�1; n ¼ 2; 3; � � �N0=2sN0=2; n ¼ N0=2þ 1; � � � ;N � N0=2� 1

sn�NþN0�1; n ¼ N � N0=2;N � 1

L=k; n ¼ N

8>>>><>>>>:

(10)

where d0n is normalized by k; let d

01 ¼ 0, d

0N ¼ L/k thus

the array aperture dimension is maintained as L; let d0n ¼

sN0/2 (n ¼ N

0/2 þ 1, N

0/2 þ 2, ��, N � N

0/2 � 1) thus the

interelement spacing of the interior elements is always d0.

C. Improved Optimization Process of GAThe new manner of individual description is demonstrated

in the above section. Produced by this way, the individuals

will meet the multiple design constraints, and the interele-

ment spacing of the interior elements can be assured as d0.However, the standard GA may break the sort order of the

elements of individual and produce infeasible solution (the

solution which doesn’t satisfy the minimum interelement

spacing constraint) when the conventional crossover and

mutation strategies are applied. To overcome this problem,

the optimization process of GA is redefined as follows.

1. Selection: The tournament selection and elitism are

employed in the process of creating the new

generation;

2. Genetic preprocessing: Subtracted the interelement

spacing constraint vector P from each element posi-

tions vector D (representing the individual) of the pop-

ulation after ‘‘selection’’ (see Eq. (6));

3. Modified crossover and mutation: Since the interior

elements, the first and the last elements don’t need to

be considered in the evolutionary process, the gene var-

iables representing the positions of these elements are

ignored in the optimization process of crossover and

mutation. Thus, the optimal variables in the evolution-

ary process are not the individuals themselves, but the

ones produced by ‘‘genetic preprocessing.’’

4. Genetic post processing: Firstly, the gene variables of

each optimal variable produced by the ‘‘modified cross-

over and mutation’’ are reset in an increasing order.

Then, the individuals of the new generation are

obtained by adding P to each ordered optimal variable.

With the above redefination of the optimization pro-

cess, the individuals can be guaranteed to be a feasible so-

lution, and the size of the searching space of GA is

reduced to e of its original value. Furthermore, the number

of gene variables is reduced to e of its original value and

the interelement spacing of the interior elements is main-

tained as d0. Therefore, the computational cost of the

improved GA is at least reduced to r of that of standard

GA.

r ¼ ð1� lÞ � e � 1:054� l� 0:924 � d0¼ 0:924 � ðduni � d0Þ ð11Þ

D. Flow of Improved GAThe following describes the details of each step of the

improved GA.

Step 1. Initialize starting population

Step 2. Evaluate fitness

Step 3. Selection

Step 4. Genetic preprocessing

Step 5. Improved crossover and mutation

Step 6. Genetic postprocessing

Step 7. Iterate until termination criteria is met. Otherwise,

jump to step 2.

IV. SIMULATED RESULTS

To illustrate the robustness and efficiency of the proposed

GA, three simulated results are presented in this section to

compare with that of literature [2, 3, 15, 16]. Basic pa-

rameters of GA are set as follows: population includes 50

individuals; the number of generation is 100; single-point

crossover strategy is employed; the crossover and muta-

tional probabilities are 85% and 0.5%, respectively. A

FORTRAN program is written for a PC (3.4 GHz Core 2

Duo processor, 2GBytes). In the program, the radiation

pattern in the y-region (�p/2 � y � p/2) is sampled by

1001 points.

224 Zhang et al.


A. First ExampleA linear array of 17 elements with the minimum interele-

ment spacing constraint of d0 ¼ 0.5 is considered in liter-

ature [3]. GA is used and the goal of optimization is to

minimize the maximum SLL and obtain pencil beam pat-

tern with abrupt skirt. Its aperture is 9.744 k and the opti-

mal SLL is �19.79 dB. We can figure out its l is 41%

and its duni is 0.609. Using improved GA, the l calculated

by (3) is 49% which approaches to that of literature [3],

thereby validating the accuracy of (3). The computational

cost of improved GA calculated by (11) is reduced to

10% of that of standard GA.

Convergence characteristics are averaged from five in-

dependent runs. It takes only 50 seconds to complete a

single trail. Figure 4 shows the average and single pro-

gress of the improved GA optimization as the function of

the number of generations. The optimal SLL of five trials

averaged is �19.78 dB and the difference of the optimal

SLL between the best and the worst single trial is only

0.23 dB, so the improved GA is high in stability. More-

over, the starting value of the convergent curve of 5 trials

averaged is �18.67 dB, which approaches to the optimal

SLL of literature [3]. The element positions vector of the

optimal array is D ¼ {0, 0.89, 1.73, 2.42, 3.04, 3.54,

4.04, 4.54, 5.04, 5.54, 6.04, 6.54, 7.08, 7.59, 8.27, 9.09,

9.744}. As shown in Figure 5, the maximum SLL of the

optimal pattern is �19.95 dB which is better than that of

literature [3]. These simulated results demonstrate that the

improved GA has good convergence characteristic and

produces preferable SLL compared with that of [3].

B. Second ExampleIn [3], GA is used to optimize the element positions to

minimize the maximum SLL of a linear aray of 37 ele-

ments subject to the design constraint of d0 ¼ 0.5. Its

aperture is 21.996 k and the optimal SLL is �20.56 dB.

We can figured out its l is 41% and its duni is 0.611.

Using improved GA, the l is 48% which approaches to

that of literature [3], and the computational cost is

reduced to 10% of that of standard GA.




gress of the improved GA. The optimal SLL of five trials

averaged is �20.77 dB and the difference of the optimal


0.24 dB. Furthermore the starting value of the convergent

curve of five trials averaged is �19.94 dB, which

approaches to the optimal SLL of literature [3]. The ele-

ment positions vector of the optimal array is D ¼ {0,

0.52, 1.13, 2.59, 3.31, 4.05, 4.79, 5.48, 6.04, 6.54, 7.04,

7.54, 8.04, 8.54, 9.04, 9.54, 10.04, 10.54, 11.04, 11.54,

12.04, 12.54, 13.04, 13.54, 14.04, 14.54, 15.04, 15.59,

16.14, 16.68, 17.23, 17.78, 18.54, 19.52, 20.43, 21.49,

21.996}. The corresponding pattern is depicted in Figure

7, we note that the maximum SLL is �20.81 dB, which is

better than that of literature [3].

C. Third ExampleIn [2], GA is used to optimize the element placement to

minimize the maximum SLL of a 154-element thinned

linear array, in which the elements are distributed on a

aperture of 100 k discretized by a uniform grid of 0.5 k.Its optimal SLL is �22.09 dB. We can figured out its l is

Figure 4 Convergence characteristics of improved GA.

Figure 5 The optimal radiation pattern.

Figure 6 Convergence characteristics of improved GA.



44%, its duni is 0.654, and its d0 is 0.5. The same array is

also synthesized by PSO and iterative FFT technique in

the recent literature [15] and [16], respectively. Its l is

43% and 49% and its optimal SLL is �22.40 dB and

�22.92 dB, respectively. Using improved GA, the l is

45% which approaches to that of literature [2, 15, 16],

and the computational cost is reduced to 14% of that of

standard GA.




gress of the improved GA. The optimal SLL of five trials

averaged is �23.36 dB, which is better than the optimal

SLL of literature [2, 15, 16]. The difference of the optimal


0.51 dB. The element positions vector of the optimal array

is D ¼ {0, 0.50, 1.29, 1.81, 2.40, 3.79, 4.47, 5.01, 5.51,

6.26, 8.54, 9.12, 9.87, 11.49, 12.03, 13.00, 13.50, 14.62,

15.31, 16.53, 17.07, 17.62, 18.97, 19.55, 20.09, 20.59,

21.20, 21.71, 22.22, 23.11, 23.70, 24.41, 24.91, 25.42,

26.23, 26.91, 27.80, 28.31, 29.25, 29.89, 30.63, 31.13,

31.63, 32.13, 32.63, 33.13, 33.63, 34.13, 34.63, 35.13,

35.63, 36.13, 36.63, 37.13, 37.63, 38.13, 38.63, 39.13,

39.63, 40.13, 40.63, 41.13, 41.63, 42.13, 42.63, 43.13, 43.63,

44.13, 44.63, 45.13, 45.63, 46.13, 46.63, 47.13, 47.63, 48.13,

48.63, 49.13, 49.63, 50.13, 50.63, 51.13, 51.63, 52.13, 52.63,

53.13, 53.63, 54.13, 54.63, 55.13, 55.63, 56.13, 56.63, 57.13,

57.63, 58.13, 58.63, 59.13, 59.63, 60.13, 60.63, 61.13, 61.63,

62.13, 62.63, 63.13, 63.63, 64.13, 64.63, 65.13, 65.63, 66.13,

66.63, 67.29, 68.06, 68.57, 69.23, 69.86, 70.40, 71.05, 71.78,

72.70, 73.48, 74.06, 74.69, 75.24, 75.79, 76.53, 77.24, 78.00,

79.55, 80.15, 80.68, 81.54, 82.29, 82.97, 83.51, 84.24, 85.59,

86.31, 87.30, 88.91, 89.90, 90.44, 91.07, 91.57, 93.65, 95.72,

96.39, 97.12, 97.84, 98.62, 99.33, 100}. As shown in Figure

9, the maximum SLL of the optimal pattern is �23.73 dB.

Thus the improved GA has higher efficiency and produces

preferable SLL compared with that of [2, 15, 16].

The above numerical results demonstrate that the

improved GA runs well with great efficiency and consid-

erable stability. Compared with the synthesis techniques

described in literature [2, 3, 15, 16], the improved GA

costs fewer computing resources and produces better

results.

V. CONCLUSION

This article describes a novel method for reducing the

computational cost of GA in synthesizing low sidelobe

level (SLL) unequally spaced linear arrays with multiple

design constraints. As shown in the examples, aiming at

simplifying the optimization model, the improved GA

saves much running time and achieves preferable results

compared with that of previous works. Certainly, the best

way for improving the performance of GA is to improve

the evolution process and simplify the optimization model

simultaneously, which will be the task in the next step.

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers for their

detailed and constructive comments that helped to improve

the quality of this manuscript. This work is supported by the

Central Universities of China under Grant JY10000902009,

and the Natural Science Foundation of China under Grant

60801042.Figure 8 Convergence characteristics of improved GA.

Figure 9 The optimal radiation pattern.Figure 7 The optimal radiation pattern.

226 Zhang et al.


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BIOGRAPHIES

Shuai Zhang was born in Hubei

province, China. He received the

B.S. degree in electromagnetic field

and microwave technique from

Xidian University, Xi’an, in 2007.

He is currently working toward the

Ph.D. degree at Xidian University in

electromagnetic field and microwave

technique. His current research interests include antennas,

arrays, calculation and synthesis of the radiation and scat-

tering patterns of array antennas, and optimization meth-

ods in electromagnetics.

Shu Xi Gong was born in Hebei

province, China, in 1957. He was

currently the professor and tutor of

Doctor in Xidian University. His

research interests include electromag-

netic theory, computational electro-

magnetics, antennas, antenna arrays,

and radiate wave propagation and

scattering in various media.

Ying Guan was born in Shaanxi

province, China. He received the

B.S. degree in electromagnetic field

and microwave technique from

Xidian University, Xi’an, in 2007.

He is currently working toward the

Ph.D. degree at Xidian University in

electromagnetic field and microwave

technique. His research interests

focus on the numerical methods in solving electromag-

netic problems.

Bao Lu was born in Shaanxi prov-

ince, China. He received the B.S.

degree in electromagnetic field and

microwave technique from Xidian

University, Xi’an, in 2004. He is cur-

rently working toward the Ph.D.

degree at Xidian University in electro-

magnetic field and microwave tech-

nique. His research interests focus on

electromagnetic scattering, frequency selective surfaces,

electromagnetic bandgap structures, and RCS prediction

and measurement programs.



Documents

A novel approach for reducing the computational cost of GA in synthesizing low sidelobe unequally spaced linear arrays