Optimized Element Positions for Prescribed RadarCross Section Pattern of Linear Dipole 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 1 March 2011; accepted 13 May 2011

ABSTRACT: Approximate closed-form expression is derived for the scattering from dipole

arrays based on the equivalent circuit theory. Then, a method is proposed for synthesis of

dipole arrays to produce desired scattering pattern using genetic algorithm (GA). In the

synthesis method, the element positions in an array are considered as the optimization

parameter and the derived expression is used to evaluate the fitness function of GA. To assess

the validity and efficiency of the proposed method, several linear dipole arrays are designed to

obtain scattering pattern with low sidelobe level (SLL). A good agreement between the pat-

terns calculated using the expression and simulated by FEKO validates the accuracy of the

presented expression. In addition, the numerical results show that the maximum SLL of the

scattering pattern is considerably reduced by optimization. VC 2011 Wiley Periodicals, Inc. Int J

RF and Microwave CAE 21:622–628, 2011.

Keywords: antenna arrays; dipole antennas; nonuniformly spaced arrays; scattering; radar cross

section

I. INTRODUCTION

Although extensive work has been done with regard to

dipoles scattering analysis and calculation, it is rare to

find published reports describing synthesis of desired scat-

tering pattern. In most of applications, it is found that the

field scattered by the dipoles cause interference or produce

unwanted radar return. Hence, there is a need for a

method to synthesize prescribed scattering pattern of

dipole antenna arrays.

Coe and Ishimaru [1] presented a method for produc-

ing a prescribed zero in the bistatic scattering pattern of

linear dipole array by optimizing the parameters of a net-

work connected to the antenna terminals. However, this

method might not be used to automatically control the ra-

dar cross section (RCS) levels of the main lobe and the

sidelobes. To overcome this problem, in a way the similar

to the radiation pattern multiplication method [2], Lu

et al. [3] indicated that the RCS of an array can also be

expressed as the product of an array factor and an element

factor. Then, based on the expression in Ref. 3, a method

was presented in Ref. 4 for synthesizing desired scattering

pattern by adjusting the dipole positions. However, this

method does not fully consider the mutual coupling

effects. Thus, the main aim of this article is to present an

accurate and efficient method for synthesis of desired scat-

tering pattern of dipoles.

Since the electromagnetic scattering is essentially a

re-radiation problem, the method for synthesizing radia-

tion pattern may be extended to the scattering problem.

Consequently, the methods for synthesis of desired radia-

tion pattern are reviewed here.

To block the interference from the sidelobes, many

researchers have focused their aims on minimizing the

maximum side lobe level (SLL) of the radiation pattern of

an array with evolutionary methods [5–10]. Classically,

using the well-known array radiation pattern multiplica-

tion method [2] to evaluate the fitness function of the

optimization algorithm, the subject can be achieved by

optimizing the element positions [6, 7], the excitation

phases or amplitudes [8, 9]. Obtained in this case, how-

ever, the works did not fully consider the mutual coupling

effects. To solve this problem, some researchers [9–12]

proposed to combine the optimization algorithm with a

numerical method and apply the numerical method to

evaluate the fitness function. In Ref. 9, differential evolu-

tion algorithm was used to optimize the amplitude weights

of elements to drive down the sidelobes, and the method

Correspondence to: S. Zhang; e-mail: 41539766@qq.com

VC 2011 Wiley Periodicals, Inc.

DOI 10.1002/mmce.20556Published online 26 September 2011 in Wiley Online Library

(wileyonlinelibrary.com).

622

of moments (MoM) [2] was used to evaluate the fitness

function. Similarly, Haupt and Aten [10] presented a

method for synthesizing low SLL radiation pattern of

dipole arrays, in which genetic algorithm (GA) was used

to call a fitness function evaluated by MoM. Although can

offer good accuracy, they can only be applied to small

arrays due to their large computation memory space and

time requirements. For larger arrays, on the order of tens

or hundreds of elements, even the most efficient hybrid

method (those combined an optimization method with a

numerical method) may still be infeasible. In this case, it

might prove fruitful to invoke approximate methods to

evaluate the fitness function to reduce the analysis time.

To this end, based on the equivalent circuit theory [13],

the element-by-element analysis technique [14, 15] was

proposed to calculate the radiation field of dipole arrays.

In this method, the elements of the impedance matrix are

self and mutual impedance of the dipoles and calculated

using the induced electromotive force (EMF) method [2].

Since the fitness function in Ref. 16 was evaluated by the

element-by-element method, the dipole array with low

SLL radiation pattern was quickly and accurately designed

by adjusting the element positions. In this article, we

extend the method in Ref. 16 to the synthesis of desired

scattering pattern of dipole arrays.

First, the approximate closed-form expression is

derived for the scattering from dipole arrays. Then, a

method is proposed for synthesis of dipole arrays to

produce desired scattering pattern using GA [17]. In the

synthesis method, the element positions in an array are

considered as the optimization parameter and the derived

expression is used to evaluate the fitness function of GA.

To validate the accuracy of the derived expression, the

scattering from several linear uniform dipole arrays

calculated using the expression are compared with those

simulated by the commercial EM software FEKO [18].

Finally, the dipole arrays are redesigned with the proposed

synthesis method to produce scattering pattern with low

SLL. The calculated and simulated results show that the

maximum SLL of the scattering pattern is considerably

reduced by optimization.

II. THE NOVEL SYNTHESIS METHOD FOR PRESCRIBEDRCS PATTERN OF LINEAR DIPOLE ARRAYS

A. Derivation of the Closed-form Expression for LinearDipole Arrays ScatteringConsider a linear array of dipoles oriented parallel to the

z-axis (Fig. 1). The array is placed along the x-axis with

the first dipole at the origin of the coordinate system. The

dipoles have an equal length of 0.5 k (l ¼ 0.5k and k is

the wavelength) and wire radius of 0.005 k. Each dipole

is terminated with a load ZL at its port.

Assume that the array is illuminated by a unit magnitude

y-polarized incident plane wave E*i ¼ expð�jk

*i � r*Þh. k*iis

the incident wave vector, and k*i ¼ �kðx sin h cos/þ

y sin h sin/þ z cos hÞ. Utilizing the definition of [19], the

induced voltage of the mth dipole can be obtained by

Vm ¼ h* � E*i ¼ h

* � exp½jkðr1 þ dm sin h cos/Þ�h (1)

where k is the free space propagation constant, k ¼ 2 p/kand k is the free space wavelength corresponding to the

operating frequency of the antenna element; r1 is the

distance from the center of the first dipole to the observation

point; dm (m ¼ 1, 2, … , N) is the distance from the origin

of the coordinate system to the center of the mth dipole,

and the unit of them are specified in terms of wavelength,

so the distance vector d*

m ¼ dmx; h*

is the effective height

of dipoles, and h* ¼ 2l

p z ¼ kp z; h ¼ x cos h cos/þ

y cos h sin/� z sin h. Thus, eq. (1) can be rewritten as

Figure 1 Geometry of a linear array of N half-wave dipoles.

RCS Pattern Optimization of Dipole Arrays 623

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

Vm ¼ hz � h expð�jk*i � r*Þ

¼ hz � ðx cos h cos/þ y cos h sin/

� z sin hÞ exp½jkðr1 þ dm sin h cos/Þ�¼ �h sin h expðjkr1Þ expðjkdm sin h cos/Þ ð2Þ

According to the equivalent circuit theory [2, 13], the rela-

tion between the induced voltage and the current at each

dipole terminal can be expressed as

V1 ¼ ðZsm11 þ ZLÞI1 þ � � � þ Zsm

1mIm þ � � � þ Zsm1NIN

..

. ... ..

. ... ..

.

Vm ¼ Zsmm1I1 þ � � � þ ðZsm

mm þ ZLÞIm þ � � � þ ZsmmNIN

..

. ... ..

. ... ..

.

VN ¼ ZsmN1I1 þ � � � þ Zsm

NmIm þ � � � þ ðZsmNN þ ZLÞIN

8>>>>>>><>>>>>>>:

(3)

where superscript ‘‘sm’’ stands for self and mutual imped-

ance; Zsmmm is the self-impedance of the mth dipole element;

Zsmmn is the mutual impedance between the mth and the nthdipole element; Im is the current at the mth dipole terminal.

Therefore, the terminal current at each dipole I ¼ {I1, I2, ���,Im, ��� IN} can be solved from (3) by matrix inversion.

I ¼ Z�1V (4)

where V ¼ {V1, V2, ���, Vm, ��� VN} is the induced voltages;

the impedance matrix Z is obtained as follows:

Z ¼ Zsm þ ZL (5)

The elements of the mutual coupling impedance matrix Zsm

are computed using the induced EMF method [2, 20]. ZL is

the generator terminating impedance matrix, so it is a diago-

nal matrix with elements ZL.The terminal current at each dipole can be calculated

using (4). The scattering from the mth dipole in the far-

zone due to the re-radiation of the terminal current Im is

given by [2]

E*s

m ¼ jg02p

f ðhÞIm expð�jk* � r*mÞ=rm (6)

where g0 ¼ 120p ohms is the intrinsic wave impedance in

free space; r*m is the distance vector from the center of

the mth dipole in the observation direction, which satisfies

r*m ¼ r

*1 � d

*

m and j r*mj ¼ rm; f(y) is the element factor

of a half-wave dipole calculated by [16]

f ðhÞ ¼ cosp cos h

2

� �= sin h (7)

Then (6) becomes

E*s

m ¼ j60f ðhÞIm expðjkdm sin h cos/Þ expð�jk* � r*1Þ=rm

(8)

Since rm � r1 ¼ r in the far-field, the total scattered field

of the array obtained by summing over all the elements is

expressed as

E*s ¼ j60f ðhÞ expð�jk

* � r*1Þ=r �XNm¼1

Im expðjkdm sin h cos/Þ

(9)

Then the approximate closed-form expression for the

monostatic RCS (MRCS, r) [2] of an N dipoles linear

array is given by

rðh;uÞ ¼ 4pr2 E*s��� ���2= E

*i��� ���2

¼ 4p 60f ðhÞ �XNm¼1

Im expðjkdm sin h cos/Þ�����

�����2

(10)

B. Validation of the ExpressionThe approximate expression for the scattering from dipole

arrays is derived above. Here some examples are analyzed

to assess the validity of Eq. (10).

1. 15 Dipoles Uniform Linear Array with ZL ¼ 50 andd0 ¼ 0.65k. Consider a uniform linear array of 15 half-

wave dipoles, in which the dipoles are all terminated with

a uniform load of 50-X impedance. The array aperture

dimension is L ¼ 9.1 k (k ¼ 1 m), so the element spacing

is d0 ¼ 0.65k. The frequency of the illuminating wave is

assumed to be the same as the operating frequency of the

array. The f-region is sampled by 1001 points. As shown

in Figure 2a, the MRCS pattern of the uniform linear

dipole array calculated using (10) is compared with the

one simulated by FEKO. Observe that the patterns

obtained by these two methods agree well in the whole

region of f-angle. In addition, the computer time required

to calculate this pattern is less than a second. All these

demonstrate that the calculation method of (10) has the

advantages of great efficiency and high accuracy.

It is also note that the maximum MRCS of 21.24

dBsm arises at boresight and the maximum SLLs (called

grating lobes [21]), which are only 6.06 dBsm below the

main beam arise at f ¼ 39.83� and 140.17� [these angles

can also be predicted using (10)]. These ‘‘grating lobes’’

arise because of the two way transit of the radar signal

and they may be significant in terms of increased detect-

ability. Consequently, it is significant to minimize the

maximum SLL of the MRCS pattern.

2. 15 Dipoles Uniform Linear Array with ZL ¼ 70 and d0¼ 0.65k. To investigate the effect of the terminal imped-

ance on the accuracy of (10), as an example, the terminal

impedance of the aforementioned array is reset as 70 X.The MRCS pattern of the array is depicted in Figure 2b.

Observe that the calculated and simulated results agree

well. The maximum MRCS of 19.72 dBsm arises at bore-

sight and the maximum SLLs are 5.96 dBsm below the

main beam and arise at f ¼ 39.83� and 140.17�. In addi-

tion, through the comparison of Figures 2a and 2b, it is

found that different load impedances only make the array

have different amplitude of MRCS. However, the shape of

the MRCS pattern is unchanged. Hence, no matter what

624 Zhang et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

impedance is connected to the element, the proposed

method remains working. In the following examples, the

dipoles are all terminated with 50-X impedance.

3. 15 Dipoles Uniform Linear Array with ZL ¼ 50 andd0 ¼ 0.1k. In this section, the effect of the element spac-

ing between two adjacent elements on the accuracy of

(10) is investigated. For the same array of 15 dipoles, the

element spacing is chosen to be as small as 0.1 k to

achieve strong mutual coupling effects. To illustrate the

advantage of consideration of the mutual coupling effects

and the stability of the proposed method, the MRCS pat-

tern calculated using the proposed method is compared

with that computed using the pattern multiplication

method (without mutual coupling) in Refs. 3 and 4 and

using FEKO in Figure 3.

Note that the results computed using the pattern multi-

plication method (dotted line) are consistent with those

simulated by FEKO (solid line) in the range of main

beam. Away from the main lobe, the pattern multiplica-

tion method fails since the mutual coupling cause signifi-

cant variation in the element patterns across the array.

Fortunately, as expected, the pattern calculated using the

proposed method (dashed line) agrees well with the one

simulated by FEKO. The comparisons above demonstrate

that the proposed method does lead to a more accurate

representation of the scattered field than the pattern multi-

plication method. A comparison between Figures 2a and 3

indicates that the scattering patterns calculated with the

proposed method for various interelement spacing have a

similar accuracy, which verifies the reliability and general-

ity of the proposed expression of (10).

4. 40 Dipoles Uniform Linear Array with ZL ¼ 50 andd0 ¼ 0.4k. In this section, the effect of the number of

array elements on the accuracy of (10) is investigated.

Consider a uniform linear array of 40 half-wave dipoles,

in which the elements are spaced by distance of d0 ¼ 0.4k(k ¼ 0.1 m). The MRCS pattern of the array is shown in

Figure 4. The maximum MRCS of 27.45 dBsm arises at

boresight and the maximum SLL is 13.38 dBsm below

the main beam. Differently from the MRCS pattern in

Figure 2, there are no grating lobes in the MRCS pattern

because of the element spacing is less than 0.5 k.

Figure 2 MRCS pattern of the uniform array of 15-dipole in

the y ¼ p/2 plane, as calculated by (10) (dotted line) and simu-

lated by FEKO (solid line): (a). ZL ¼ 50, (b). ZL ¼ 70.

Figure 3 MRCS pattern of a uniform linear array of 15-dipole,

in which the elements are spaced by distance of 0.1k.

Figure 4 MRCS pattern of the uniform array of 40-dipole in

the y ¼ p/2 plane, as calculated by (10) (dotted line) and simu-

lated by FEKO (solid line).

RCS Pattern Optimization of Dipole Arrays 625

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

Meanwhile, as expected, the calculated pattern is in good

agreement with the one simulated by FEKO.

The above examples demonstrate that (10) can rigor-

ously consider the mutual coupling between dipoles, so that

it can become the basis of an iterative technique to accu-

rately synthesize prescribed RCS pattern of dipole arrays.

C. Synthesis MethodAs expressed in (10), the MRCS pattern of dipole array is

a function of the element positions. Thus, similar to the

synthesis method for desired radiation pattern, a method is

proposed for synthesis of desired scattering pattern of

dipole arrays by adjusting the element positions using GA

[17]. In the proposed synthesis method, eq. (10) is used to

evaluate the fitness function of GA. Since (10) can take

the mutual coupling effects into account, the proposed

synthesis method is a more accurate iterative technique

compared with the one described in our previous work

[4]. Here, we only consider the MRCS pattern in the y ¼p/2 plane, so the fitness function of GA is defined as

fitnessðd1; � � � ; dNÞ ¼ minfmax½10 � lgð rðp=2;/Þj j=rmaxj jÞ�g ð11Þ

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

individual (which means the element positions vector of

the array), and the coding scheme of GA is real-coded; in

the optimization, the array aperture dimension remains a

constant; since the first element is located at the origin of

coordinate system (d1 ¼ 0), the coordinate of the Nth ele-

ment is always L (dN ¼ L, L is the array aperture dimen-

sion); 0 � f � p and the valid region of f excludes the

main beam of it; |rmax| is the peak value of the main

beam; thus max[10� lg(| r(p/2,f)|/|rmax)] is the maximum

SLL of the normalized MRCS pattern.

III. NUMERICAL EXAMPLE

To assess the validity of the proposed synthesis method, a

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

Duo processor, 2GBytes). In the program, basic parame-

ters of GA are set as follows: population includes 50 indi-

viduals; number of generation is 50; the crossover and

mutational probabilities are 85 and 0.5%, respectively; the

f-region is sampled by 1001 points; the tournament selec-

tion and elitism are used in the process of creating the

new generation.

A. Optimized Linear Array of 15 DipolesFor comparison, the aforementioned uniform linear array

of 15 dipoles in section II is redesigned with the proposed

synthesis method. Ten runs are made to investigate the

stability of the algorithm. The difference between the best

and the worst run is less than 0.75 dBsm, so the algorithm

is high in stability. The element positions vector of the

best run is d ¼ {0, 0.588, 1.336, 1.853, 2.376, 2.893,

3.794, 4.317, 4.829, 5.338, 5.846, 6.357, 7.036, 7.539,

9.1}. Figure 5 shows the MRCS pattern of the optimized

array using the proposed synthesis method. As expected,

the grating lobes are suppressed effectively by optimiza-

tion, and the maximum SLL of the scattering pattern is

13.45 dBsm below the main beam with the peak value of

20.09 dBsm. Thus, this optimized array has a maximum

sidelobe level 7.39 dBsm lower than the uniform array

(see Fig. 2a). Moreover, almost all the sidelobes are at the

same level of 6.64 dBsm, so the optimized array is ap-

proximate to the optimal one. In addition, the computer

time required to synthesize this pattern is only 5 min.

B. Optimized Linear Array of 40 DipolesTo illustrate the efficiency, stability, and generality of the

proposed synthesis method, a larger linear uniform array

of 40 dipoles depicted in Section II is redesigned with the

proposed synthesis method to obtain low sidelobe MRCS

pattern. After optimization, the element positions vector

of the optimal array is d ¼ {0, 0.694, 1.070, 1.576, 3.134,

3.483, 3.837, 4.053, 4.194, 4.732, 5.127, 5.361, 5.662,

6.046, 6.470, 6.734, 6.836, 7.130, 7.385, 7.819, 8.132,

8.471, 8.927, 9.357, 9.591, 10.023, 10.542, 10.777,

11.160, 11.376, 1.624, 11.989, 12.128, 12.604, 13.032,

13.424, 13.561, 14.399, 14.680, 15.600}. Figure 6 shows

Figure 5 Optimized MRCS pattern of the 15-dipole linear array

in the y ¼ p/2 plane.

Figure 6 Optimized MRCS pattern of the 40-dipole linear array

in the y ¼ p/2 plane.

626 Zhang et al.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

the MRCS pattern of the optimal array using the proposed

synthesis method. The maximum MRCS at boresight is

26.09 dBsm and the maximum sidelobe level of the

MRCS pattern is 7.89 dBsm. Thus, this optimized array

has a maximum sidelobe level 6.19 dBsm lower than the

uniform array (see Fig. 4). Moreover, the computer time

required to synthesize this pattern is only 21 min.

In above discussions, two dipole arrays are redesigned

with the proposed synthesis method to obtain low SLL

scattering pattern. Compared with the synthesis technique

described in Ref. 4, the proposed synthesis method can

fully consider the mutual coupling between elements.

Meanwhile, it is more efficient than the methods pre-

sented in Refs. 9–12, which applied numerical methods

to evaluate the fitness function of evolutionary

algorithms.

IV. CONCLUSIONS

An accurate expression for calculating the scattering from

linear dipole arrays is successfully derived referring to the

calculation of the field radiated by dipoles. Based on the

expression, a novel method is presented for synthesizing

prescribed scattering pattern of dipole arrays by adjusting

the element positions. Meanwhile, GA is used in the syn-

thesis method to minimize the maximum SLL of the scat-

tering pattern. Numerical results show that the maximum

SLL of the optimal array is considerably reduced com-

pared with that of the uniform array through optimization.

In addition, the concept of our method can be easily

extended to the calculation and synthesis of the scattering

pattern of planar dipole arrays.

ACKNOWLEDGMENTS

This work was supported by the Fundamental Research

Funds for the Central Universities of China under Grant

JY10000902009, and the Natural Science Foundation of

China under Grant 60801042.

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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 Micro-

wave Technique. His current research interests include

antennas, arrays, calculation, and synthesis of the radiation

and scattering patterns of array antennas, and optimization

methods 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 electromagnetic problems.

Bao Lu was born in Shaanxi province,

China. He received the B.S. degree in

Electromagnetic Field and Microwave

Technique from Xidian University,

Xi’an, in 2004. He is currently working

toward the Ph.D. degree at Xidian

University in Electromagnetic Field and

Microwave Technique. His research

interests focus on electromagnetic scattering, frequency selec-

tive surfaces, electromagnetic bandgap structures, and RCS

prediction and measurement programs.

International Journal of RF and Microwave Computer-Aided Engineering/Vol. 21, No. 6, November 2011

628 Zhang et al.