design of steerable linear and planar array geometry with ... · design of steerable linear and...

IEICE TRANS. COMMUN., VOL.E88–B, NO.1 JANUARY 2005345

PAPER

Design of Steerable Linear and Planar Array Geometry withNon-uniform Spacing for Side-Lobe Reduction

Ji-Hoon BAE†a), Kyung-Tae KIM††, and Cheol-Sig PYO†, Nonmembers

SUMMARY In this paper, we present a noble pattern synthesis methodof linear and planar array antennas, with non-uniform spacing, for simulta-neous reduction of their side-lobe level and pattern distortion during beamsteering. In the case of linear array, the Gauss-Newton method is appliedto adjust the positions of elements, providing an optimal linear array in thesense of side-lobe level and pattern distortion. In the case of planar array,the concept of thinned array combined with non-uniformly spaced array isapplied to obtain an optimal two dimensional (2-D) planar array structureunder some constraints. The optimized non-uniformly spaced linear arrayis extended to the 2-D planar array structure, and it is used as an initialplanar array geometry. Next, we further modify the initial 2-D planar arraygeometry with the aid of thinned array theory in order to reduce the max-imum side-lobe level. This is implemented by a genetic algorithm undersome constraints, minimizing the maximum side-lobe level of the 2-D pla-nar array. It is shown that the proposed method can significantly reduce thepattern distortion as well as the side-lobe level, although the beam directionis scanned.key words: antenna array pattern synthesis, non-uniform spacing, opti-mization technique, side-lobe reduction

1. Introduction

In the field of antenna array pattern synthesis, a large num-ber of pattern synthesis techniques have been studied anddeveloped over more than 60 years. In general, these tech-niques can be classified into two categories: one method op-timizes the excitation of each element of the uniform arrayand the other adjusts the elements’ positions with uniformexcitation, resulting in a non-uniform array geometry. Theexcitation includes the amplitudes and phases of array ele-ments.

Traditional synthesis methods, such as Fourier trans-form, Talyor, Woodward-Lawson and Dolph-Chebyshev[1], belong to the first category. The advantage of thesemethods is that they can provide standard design rules foruniformly spaced array (USA) pattern synthesis. However,these traditional methods are restricted to only a pattern syn-thesis of arrays with uniformly spaced and isotropic antennaelements. Orchard et al. [2] presented a power pattern syn-thesis procedure of linear arrays with uniform element spac-ing. This method is based on Schelkunoff’s polynomial rep-resentation of the array pattern and it enables us to synthe-size an arbitrary beam pattern in the main lobe region as

Manuscript received December 12, 2003.Manuscript revised July 12, 2004.†The authors are with ETRI, 161 Gajeong-dong, Yuseong-gu,

Daejeon, 305-350, Korea.††The author is with the Dept. of EECS, Yeungnam University,

214-1 Daedogng, Kyongsan, Kyungbuk, 712-749, Korea.a) E-mail: [email protected]

well as in the side-lobe region by optimally placing rootsof the polynomial expression. The Orchard-Elliott synthesismethod, which yields shaped beams with a non-symmetricalcomplex distribution, has been extended and improved in[3], [4] to provide a pure-real distribution of the excitation.A different approach exploits an adaptive array theory in thesynthesis array pattern. Olen and Compton [5] developed anumerical pattern synthesis algorithm based on the adaptivearray theory. Unlike the classical methods, the algorithmcan be used for both USA and non-uniformly spaced array(NUSA) and can provide an arbitrary side-lobe level (SLL).While Olen and Compton’s method considers only the SLL,the new algorithm presented in [6] can achieve the desiredmain-lobe shape as well as low SLL, simultaneously.

It should be noted however, that most of the methods inthe first category inevitably have one disadvantage, namely,an amplitude tapering. The amplitude tapering of the exci-tation may require a complicated feed system and also in-creases the main beamwidth. In addition, if the amplitudetapering is large, then the mutual coupling effects may causeappreciable changes in the small antenna current [7].

In the second category, Unz [8] originally analyzed alinear array for arbitrarily distributed elements. From theinitial concept of Unz, Harrington [9] developed a methodfor reducing the first SLL of a linear array with non-uniformelement spacing. This synthesis method is based on theFourier transform formula and can reduce the inner side-lobes (nearby side-lobes from the main beam) to about 2/Ntimes the field intensity of the main lobe, where N is thenumber of elements of linear array. Hodjat and Hovanes-sian [10] suggested an iterative method to adjust positionsof a non-uniformly spaced linear array (NUSLA), whichprovides symmetrical non-uniform arrangements with re-spect to the array center. In addition, to design severalnon-uniformly spaced planar arrays (NUSPAs), they ex-tended their NUSLA structures to 2-D planar array struc-tures. Other array pattern synthesis approaches for NUSAhave been studied and presented in [11]–[16].

Another major improvement for the second category isthe use of thinned array theory. It is illustrated in [17] thatthinning an array means turning off some elements in a uni-formly spaced or periodic array to create a desired amplitudedensity across the aperture. Thinned arrays have been in-vestigated for several decades in many array antenna fieldssince Skolnik et al. [18] applied dynamic programming tothe design of thinned array. Recently, derivative-free opti-mization methods, such as simulated annealing and genetic

Copyright c© 2005 The Institute of Electronics, Information and Communication Engineers

346IEICE TRANS. COMMUN., VOL.E88–B, NO.1 JANUARY 2005

algorithm, have drawn great attention in this area. Simu-lated annealing (SA) was derived from the physical charac-teristic of spin glasses and is performed by the mechanismof an annealing or cooling schedule [19]. While SA usesa single agent in search for an optimal solution, genetic al-gorithm (GA) has a multi-agent system. GA is composedof the natural evolution mechanism of SA, random searchmechanism, and biological mechanism. SA was used to de-sign the thinned arrays with low SLL [20]. In [21], J ele-ments are optimally placed on a K-linear lattice with uni-form spacing under some design constraints to synthesize adesired linear array pattern, where the size of J is less thanthat of K. Ultimately, optimally thinning some elements lo-cated on the uniformly spaced linear grid can cause the arrayto have a suitable non-uniform spacing with low SLL, whichcorresponds to the NUSLA theory above. In the case of 2-Dplanar arrays, thinning some elements may change the ele-ment density, which can give an amplitude variation to thearrays where the amplitude at each element is presented byone bit [17]. Therefore, it is possible to achieve low SLL forthe planar array if some elements are optimally turned off,resulting in the effect of the quantized amplitude taper. Notethat in this secondary synthesis field, all the amplitudes andphases of the excited array elements have uniform valuesor fixed values during optimization procedure. The attrac-tive aspect of the NUSPA is that a radiation pattern with lowSLL can be determined by only the proper array structure,maintaining uniform excitations of given array antennas.

The maximum SLL (MSLL) of an array radiation pat-tern in the second category can be reduced by adjustingthe inter-element spacing appropriately, namely, by apply-ing the NUSA theory. However, it should be noted that theNUSA geometry may cause the outer SLL to increase espe-cially when the main beam direction is scanned [9]. Becauseof this phenomenon, the undesirable large side-lobes whichare greater than the first SLL can be seen within the visibleregion of −90 ≤ θ ≤ 90, when the NUSA geometry is ex-ploited. Therefore, in this study, we consider the outer SLLas well as the inner SLL, resulting in a simultaneous reduc-tion of MSLL and pattern distortion of the steerable linearand planar array.

On the basis of NUSA theory, we propose linear andplanar array pattern synthesis methods for side-lobe reduc-tion. The purpose of this paper is to find an optimal NUSAstructure, maintaining a low SLL, without pattern distortionduring beam steering. In our synthesis method for an op-timal NUSLA with low SLL, first, the non-optimized andnon-uniform element positions are calculated by the Fouriertransform based formula, and they are used as initial ele-ment positions. This smart initial guess can facilitate a fastconvergence of the iterative optimization process. Next, weoptimize the inter-element spacing from the initial elementpositions using the Gauss-Newton method, which is one ofthe derivative-based optimization techniques. The resultingNUSLA method can reduce both the inner side-lobes andthe large outer side-lobes, simultaneously.

We also propose our design scheme for an optimal

NUSPA with low SLL. Our approach makes use of thethinned array theory combined with our proposed NUSLAtechnique. To generate a 2-D planar grid of non-uniformspacing, the resulting optimized NUSLA is extended to a 2-D rectangular array lattice. This NUSPA is used as an initialarray geometry. Next, the GA is applied to implement thethinned array theory in order to adjust the arrangements ofthe initial NUSPA. The resulting NUSPA, under some con-straints, can accomplish low SLL without pattern distortion,although the beam direction is steered.

This paper is organized as follows. In Sects. 2 and4, we formulate the problem of interest for a NUSLA andNUSPA. In Sects. 3 and 5, we describe in detail the pat-tern synthesis methods of linear and planar array with non-uniform spacing. In Sect. 6, we show some simulation ex-amples. Finally, we draw our conclusions in Sect. 7.

2. Problem Formulation for Non-uniform Linear Ar-ray

For an odd number of elements, if isotropic array elementsare uniformly distributed along the x-axis and are assumedto be symmetric about the array center, the radiation fieldpattern over the set of angles θ1, θ2, · · · θLcan be describedas follows:

p1D(θi) =1N

N−1∑n=0

exp[ jκndx(sin θi − sin θ0)]

=2N

M∑n=1

cos[κndx(sin θi − sin θ0)] +1N

(1)

where N is the number of element antennas, κ is the freespace propagation constant, dx is the inter-element spac-ing, θ0 is the maximum radiation angle, and M is given by(N − 1)/2. As shown in Fig. 1, if the uniform array elementpositions are perturbed by the fractional change, ex

n, the re-sulting element spacing can be expressed as follows:

dxn = (n + ex

n)dx [9]. (2)

Then, the normalized pattern of the NUSLA is given by

p1Dnu (θi) =

2N

M∑n=1

cos[κ(n + exn)dx(sin θi − sin θ0)]

Fig. 1 Structure of non-uniform linear antenna array.

BAE et al.: DESIGN OF STEERABLE LINEAR AND PLANAR ARRAY GEOMETRY347

+1N

(3)

In the following section, a pattern synthesis method tofind the most suitable element positions is derived from thebasic formula of Eq. (3).

3. Non-uniform Linear Array Pattern Synthesis

In this section, the Gauss-Newton algorithm is used to ex-tract the optimal parameter, ex

n from Eq. (3). This algorithmis based on the derivative-based optimization technique andimplements the minimization of a cost function that is ex-pressed as the sum of error squares [19].

If we expand the cosine term of Eq. (3) and assume avery small ex

n, Eq. (3) can be transformed into the followingform:

p1Dnu (θi) ≈ p1D(θi) − 2

N

M∑n=1

exn sin[κdx(sin θi − sin θ0)n]

× κdx(sin θi − sin θ0) (4)

Equation (4) can be further simplified into the follow-ing form:

g(zi) = p1D(zi) − p1Dnu (zi)

=

M∑n=1

exn

[2(zi − z0) sin(n · (zi − z0))

N

](5)

where zi = κdx sin θi, z0 = κdx sin θ0, and p1D(zi) is thenormalized pattern of the uniform linear array.

When the cost function is defined as C1 =

12

L∑i=1

[g(zi) − gdes(zi)]2 = 12

L∑i=1ε2(zi), the Gauss-Newton al-

gorithm can be used to minimize this cost function, whereg(zi) means the difference between the radiation pattern withnon-uniform spacing and the original radiation pattern withuniform spacing, and gdes(zi) represents the desired differ-ence pattern between the original pattern and the desiredreference pattern with low SLL. In our case, to obtain thedesired reference pattern, several inner side-lobes includingthe first side-lobe are set up to a predefined low SLL, andthe other SLLs are fixed to each value slightly smaller thanthe outer SLLs of the original pattern. At the same time,the main lobe of the desired pattern is identical with that ofthe original USLA pattern. In order to estimate the prede-termined low SLL in an optimal way, we performed manysimulations varying the predefined SLL, and found that thepredetermined SLL between −25 dB and −30 dB can guar-antee convergence of the proposed algorithm in most casesas well as sufficiently low SLL without pattern distortion. Inaddition, ε(zi) is a function of the adjustable variable, ex

n. Ifwe apply the Talyor series expansion to ε(zi), then the fol-lowing equation holds:

∂ε(k)(zi)

∂ex(k)n

[ex(k+1)n − ex(k)

n ] = ε(k+1)(zi) − ε(k)(zi),

i = 1, 2, · · · , L, n = 1, 2, · · · ,M (6)

where k denotes an iteration number. The matrix form ofEq. (6) is given by

Ω(k+1) = Ω(k) + J · (E(k+1) − E(k)) (7)

where Ω = [ε(z1), ε(z2), · · · , ε(zL)]T , E = [ex1, e

x2, · · · , ex

M]T

and J is a Jacobian matrix, given by

J =

∂ε(z1)∂ex

1

∂ε(z2)∂ex

1· · · · · · ∂ε(zL)

∂ex1

∂ε(z1)∂ex

2

∂ε(z2)∂ex

2· · · · · · ∂ε(zL)

∂ex2

......

......

......

∂ε(z1)∂ex

M

∂ε(z2)∂ex

M· · · · · · ∂ε(zL)

∂exM

T

=

2(z1−z0)·sin(z1−z0)N

2(z1−z0)·sin(2(z1−z0))N · · · 2(z1−z0)·sin(M·(z1−z0))

N2(z2−z0)·sin(z2−z0)

N2(z2−z0)·sin(2(z2−z0))

N · · · 2(z2−z0)·sin(M·(z2−z0))N

...... · · · ...

...... · · · ...

2(zL−z0)·sin(zL−z0)N

2(zL−z0)·sin(2(zL−z0))N · · · 2(zL−z0)·sin(M·(zL−z0))

N

.

Therefore, the desired small perturbation vector, E(k+1)canbe obtained by minimizing the following nonlinear least-squares (NLS) criterion:

C2 = arg minE(k)

12

∥∥∥Ω(k) + J · (E(k+1) − E(k))∥∥∥2 (8)

where ‖·‖ denotes the Euclidean norm. ∂C2

∂E(k) = 0 yields thefollowing NLS solution:

E(k+1) = E(k) − [(JT J)−1JT ]Ω(k). (9)

For fast and stable convergence, Eq. (9) can be representedin a slightly modified form as follows:

E(k+1) = E(k) − η[(δ2I + JT J)−1JT ]Ω(k) (10)

where I is an L × M identity matrix, η = η0 exp(−r · k) is aniteration gain, and η0 and r are all fixed constants. A smallquantity δ2 is added to each diagonal element of the Jaco-bian matrix to prevent it from being ill conditioned [6]. Forfast and stable convergence of Eq. (10), it is essential to as-sign proper values to the η0 and r. For small value of r, theconvergence of Eq. (10) may be guaranteed, but its rate ofconvergence becomes too slow. In contrast, for large valueof r, the convergence may not be guaranteed due to the fluc-tuation phenomenon of the convergence curve around theoptimal solution. From many simulation results, we foundthat η0 between 1.2 and 1.6, and r between 0.2 and 0.25 canprovide a suitable performance for our algorithm. The ini-tial small distance vector E(0) can be obtained by the formuladescribed in [9] to prevent the algorithm from reaching a lo-cal minimum and provide a quick convergence to the globalminimum.

The procedure for the pattern synthesis of the NUSLAusing the proposed algorithm is summarized as follows:

Step 1: The initial array element positions are calcu-lated by the following formula based on Fourier coefficients,


ex(0)n =

Nπ

∫ π0

n(p1D(z) − p1Dnu (z))S a(n · z)dz,

n = 1, 2, · · · ,M (11)

where S a(n · z) = sin(n·z)n·z .

Step 2: To improve the radiation pattern of the steerablelinear array, new array element positions are obtained byEq. (10).

Step 3: Iterate the Step 2 until the following relativeerror (RE) is less than a predefined small quantity, γ.

RE =1

2N

L∑i=1

∣∣∣p1D(k+1)nu (θi) − p1D(k)

nu (θi)∣∣∣2 < γ. (12)

4. Problem Formulation for Non-uniform Planar Ar-ray

To carry out the design of an optimal planar array, with non-uniform spacing, we use the optimized linear array geom-etry introduced in Sect. 3 as an initial NUSPA. The result-ing linear array is extended to a 2-D rectangular array latticealong the row and column directions respectively; accordingto the non-uniformly distributed positions of the optimizedlinear array elements. Thus, the initial NUSPA pattern canbe described as follows:

p2Dnu (u, v)

=1

N2

2M∑

n=1

cos(κ(n + exn)dx · (u − u0)) + 1

·2

M∑m=1

cos(κ(m + eym)dy · (v − v0)) + 1

=1

N2

2M∑

n=1

cos(κdxn · (u − u0)) + 1

·2

M∑m=1

cos(κdym · (v − v0)) + 1

(13)

where u = sin θ cosφ, v = sin θ sin φ, u0 = sin θ0 cosφ0,v0 = sin θ0 sin φ0, dx

n = (n + exn)dx, dym = (m + eym)dy, dy=dx,

and eym = exn.

Furthermore, this rectangular array geometry can bemodified to achieve further reduction of the MSLL. In thefollowing section, a pattern synthesis method to find themost suitable planar array structure for the maximum side-lobe reduction is derived from the formula of Eq. (13).

5. Non-uniform Planar Array Pattern Synthesis

In this section, to accomplish lower SLL from the initialNUSPA of Eq. (13), the GA and thinned array concept areexploited to modify the initial NUSPA structure. Considerthe linear array pattern of Eq. (1). The linear array patternin Eq. (1) is similar to the Fourier series expression for anarbitrary real-valued function in that the array pattern canbe expressed as the sum of cosine terms. If we define an

array frequency, ωn = 2πn · dx sin θ, the lowest array fre-quency can be associated with the center array element, andhigher order array frequencies with the outer array elements[12]. In addition, when an arbitrary real-valued function iscomposed of slowly as well as rapidly varying functions,the higher frequencies may determine the higher variationsof the function. Therefore, SLL of the linear array may bemore sensitive to the adjustment of components associatedwith the high order array frequencies, which physically cor-respond to the outer array elements far from the array center.A concept of the array frequency of the linear array can beextended to a 2-D planar array problem. It is shown in [10]that the elimination of some elements of the four corner ofa rectangular array, with non-uniform spacing, can give acircular radiation pattern and provide greater reduction ofthe SLL. In this study, to achieve lower SLL from the initialNUSPA, we find the unnecessary elements around the outerregions of the 2-D rectangular array using a GA, and thenthey are removed from that array geometry.

GA is a stochastic search procedure modeled on theDarwinian concepts of natural selection and evolution [22].It is the highest merit of the GA to provide a global optimalsolution for complex electromagnetic (EM) problems. Thebasic process, general concepts and applications of the GAin EM problems have been presented in [17], [21]–[24]. Inour case, to formulate a pattern synthesis method from theinitial NUSPA, we start with Fig. 2. Fig. 2 shows the ini-tial N × N, non-uniformly spaced rectangular array arrange-ment. As shown in Fig. 2, each array element is symmetri-cally positioned along the non-uniformly spaced rectangulargrid with respect to x-axis and y-axis. Due to this symmetry,we consider merely the array elements in the first quadrant,not whole spaces, for the optimization procedure. In ad-dition, among the elements belonging to the first quadrant,only the outer elements contained in (A), (B), and (C) ofRegion I are optimized through the GA. The outer elementswithin the Regions, II, III, and IV adopt the same geome-try as in Region I. Therefore, the number of parameters thatmust be optimized in the GA can be significantly reduced.The NUSPA pattern of Eq. (13) can be written as follows:

p2Dnu =

1N2

2R∑

n=1

cos(κdxn · (u − u0)) + 1

·2

Q∑m=1

cos(κdym · (v − v0)) + 1

+1

N2

4M∑

m=Q+1

R∑n=1

Wa

mn · cos(κdym · (v − v0))

· cos(κdxn · (u − u0))

+ 2

M∑m=Q+1

Wam0 · cos(κdym · (v − v0))

+1

N2

4Q∑

m=1

M∑n=R+1

Wb

mn · cos(κdym · (v − v0))


Fig. 2 NUSPA geometry for the formulation of planar array pattern synthesis.


+ 2

M∑n=R+1

Wb0n · cos(κdx

n · (u − u0))

+1

N2

4M∑

m=Q+1

M∑n=R+1

Wcmn · cos(κdym · (v − v0))


](14)

where (Wamn,Wa

m0), (Wbmn,Wb

0n), and Wcmn are the amplitude

weights of elements = 1 or 0 corresponding to region(A), (B) and (C), respectively, and M, Q, and R are definedin Fig. 2. We assume that Wa

m0 = Wb0n=1. In addition, we fur-

ther assume that Wbmn has the same symmetrical arrangement

with Wamn. This can lead to a further reduction of the compu-

tational complexity during the optimization process. Wamn=1

represents the element status as “on,” whereas Wamn=0 rep-

resents the element status as “off.” Wcmn is also expressed

in the same manner as Wamn. Values for the parameters of

the GA can be represented by a binary string or real-valuedstring. In this paper, we adopt a binary string because Wa

mn,Wb

mn, and Wcmn have discrete values, namely, 1 or 0.

When the main lobe region of the NUSPA pattern isUL ≤ u ≤ UR and VL ≤ v ≤ VR, the cost function, F, toevaluate the fitness value of given individuals is defined asfollows:

F = maxS LL

20 log(

∣∣∣p2Dnu (u, v)

∣∣∣) (15)

in side-lobe regions subjected to

−1 ≤ u ≤ UL and UR ≤ u ≤ 1 for −1 ≤ v ≤ 1−1 ≤ v ≤ VL and VR ≤ v ≤ 1 for UL ≤ u ≤ UR

.

The procedure for the pattern synthesis of the NUSPAusing the GA is summarized as follows:

Step 1: The optimized linear array in Sect. 3 is ex-tended to a 2-D rectangular array to obtain an initialNUSPA geometry. (Note that a uniformly spaced planar ar-ray (USPA), extended by a uniformly spaced linear array(USLA) instead of the optimized NUSLA, can also be usedas an initial planar array in Step 1 for the design of an opti-mal USPA.)

Step 2: Randomly generate an initial population forWmn= [Wa

mn Wcmn] which represent a chromosome consist-

ing of binary string.Step 3: Calculate the MSLL using Eq. (15).Step 4: Rank chromosomes from best to worst, accord-

ing to their fitness values obtained by Step 3, and discard thebottom 50%.

Step 5: Create new offspring settings from the selectedtop 50% using the crossover operator.

Step 6: The best individual from the present generationis saved, but it will not take part in Step 7 of the mutationprocess.

Step 7: Mutate the new offspring based on the proba-bility of mutation.

Step 8: Iterate Step 2–Step 7 until there is no improve-ment about the best fitness value Fbest during K successivegenerations as follows:


C =k+K∑

k

(F(k+1)best − F(k)

best) = 0

during K successive generations (16)

Meanwhile, to finish the optimization procedure, thedesired low SLL can be also used to the Step 8. Althoughwe cannot expect the lowest MSLL over a given array sizein advance, we can roughly set a desired low SLL before theoptimization is implemented. Therefore, if the best fitnessvalue Fbest satisfies the desired low SLL in the process of theoptimization, the algorithm is stopped. In addition, althoughthe best fitness value Fbest does not satisfy the predeterminedlow SLL, the algorithm is terminated when the condition inEq. (16) is achieved. From many simulation results, in orderto guarantee both a robust performance for the MSLL reduc-tion and computational efficiency, approximately, the valueof K over five times the array size N (=2M) was needed forthe pattern synthesis algorithm.

6. Simulation Results

In this section, we will show a few examples for the purposeof demonstrating the performance of the linear and planararray pattern synthesis methods in Sects. 2–5.

6.1 Linear Array

As the first example, suppose we have a 13-element lineararray of isotropic elements spaced every half-wavelength.First, this uniformly spaced linear array (USLA) is synthe-sized to reduce the SLL using the Fourier transform basedformula described in [9]. The resulting NUSLA has thearray pattern shown in Fig. 3. In comparison with theUSLA, an amount of about 5 dB reduction of the first SLL isachieved with the NUSLA.

However, when the main beam of NUSLA is steered to30, some large outer side-lobes greater than the first side-lobe are observed within the visible region as shown in Case

Fig. 3 Radiation pattern of non-uniform linear array using the Fouriertransform based formula: the maximum radiation angle is 0.

1 of Fig. 4. Furthermore, their levels in the vicinity of an an-gle of −60 are also higher than the first SLL of the USLAin Case 3 of Fig. 4. In order to reduce the undesirable largeSLLs shown above, we now apply the proposed algorithm topattern synthesis of the scannable NUSLA with beam steer-ing. For the pattern synthesis algorithm, we set the iterationgain to η = 1.5 exp(−0.225 · k), the maximum allowable er-ror, γ to 10−4, and δ2 to 0.001, respectively. The range of−π ≤ z ≤ π with 0.04π steps corresponding to the observa-tion angle (−90 ≤ θ ≤ 90) was used for the optimization,resulting in L=51. Considering convergence time and per-formances, approximately, L was chosen as 4 times the arraysize for the optimization. For the desired reference pattern,the first and second side-lobes with respect to the main lobepositioned at θ0 = 30 were set up to −25 dB and the otherSLLs were reduced to 0.5 times the outer SLLs of the origi-nal pattern. The resulting radiation pattern is shown in Case2 of Fig. 4. As shown in this figure, the proposed algorithmcauses the undesirable large side-lobes near −60 to be sig-nificantly reduced. They have even smaller levels than thefirst SLLs of Case 1 and Case 3.

As a second example, we consider an equally excited17-element linear array with the same uniform spacing andscanning angle, as in the previous example. For this ex-ample, we set the iteration gain to η = 1.45 exp(−0.25 · k),the maximum allowable error, γ to10−4, and δ2 to 0.001, re-spectively. In case of the 17-element NUSLA, the range of−π ≤ z ≤ π with 0.03π increments was considered for thisoptimization, resulting in L=67. The desired reference pat-tern was composed of three inner side-lobes of −27 dB andthe other outer side-lobes which were 0.5 times the outerones of the original 17-element USLA. The resulting opti-mized positions of the NUSLA are given in Table 1 and thebeam pattern for the positions of Table 1 is plotted in Fig. 5.Case 1 represents the beam pattern of the NUSLA with theFourier transform based formula, Case 2 represents the op-timized NUSLA with the proposed technique, and Case 3that of the USLA, respectively. It is shown in Fig. 5 that the

Fig. 4 Comarison of the optimized and the non-optimized radiation pat-terns for the 13-element linear array when the maximum radiation angle issteered to 30.


Table 1 Position of the optimized non-uniform array elements.

Fig. 5 Comarison of the optimized and the non-optimized radiation pat-terns for the 17-element linear array when the maximum radiation angle issteered to 30.

proposed method can reduce not only the first SLL, but alsopattern distortions in the outer side-lobes, although the mainbeam direction is steered to 30.

The MSLLs and 3 dB main-lobe beamwidths of severallinear array arrangements are compared and summarized inTable 2, when the main beam direction is steered to the fivedifferent angles. These linear array arrangements are ex-plained as follows:

Case 1: linear array geometry (N = 13) + uniformspacing

Case 2: linear array geometry (N = 13) + non-uniformspacing (Fourier transform based method)

Case 3: linear array geometry (N = 13) + non-uniformspacing (proposed method)

Case 4: linear array geometry (N = 17) + uniformspacing

Case 5: linear array geometry (N = 17) + non-uniformspacing (Fourier transform based method)

Case 6: linear array geometry (N = 17) + non-uniformspacing (proposed method)

From Table 2, the optimized beam patterns of Case 3and Case 6, with the proposed method, can maintain lowSLLs for the scanning range of −30 ≤ θ0 ≤ 30, whilethe non-optimized beam patterns of Case 2 and Case 5 cannot. In addition, we observe that the main-lobe beamwidthbroadening in the optimized arrays is very small comparedto that of the USLA. As shown in Table 2, it should bepointed out that the optimized NUSLA geometry with theproposed method can provide low SLLs without pattern dis-tortion over the wide scan angles.

6.2 Planar Array

As design examples of an optimal planar array, we considertwo planar arrays, namely, USPA and NUSPA.

The optimized USPA and NUSPA structure, using theplanar array pattern synthesis method in Sect. 5, are pre-sented and are also compared.

6.2.1 Optimized Planar Array from the Uniformly SpacedPlanar Array

In our planar array pattern synthesis method, it is necessaryto determine a design parameter, R or Q, in Fig. 2, in ad-vance. It determines the search region of the GA for theoptimal solution. In our case, the size of R is equal to thatof Q to maintain a symmetric array structure. Fig. 6 showsthe relative amount of MSLL reduction (RMSLLR) for sev-eral USPAs, when the ratio of R to M (R/M) is varied overseveral array sizes. The RMSLLR is defined as follows:RMSLLR [dB] = MSLL of uniform planar array (i.e. about−13 dB)—MSLL of optimized planar array. The result ofFig. 6 is based on the proposed technique when the USPA isused as an initial array geometry in Step 1, rather than theNUSPA. Each initial USPA was obtained by expanding eachUSLA with a half-wavelength spacing to each 2-D rectan-gular lattice. As shown in Fig. 6, RMSLLRs of all the US-PAs, with various array sizes, are respectively significantlyincreased as the R/M decreases. However, there is no notice-able increase of RMSLLRs, although the R/M decreases to avalue smaller than 0.6. It should be pointed out that, as R/Mdecreases to a value less than 0.6, the number of adjustablearray elements during the GA optimization increases. Atthe same time, there is no further reduction of MSLL whenthe R/M is less than 0.6. In contrast, for R/M values largerthan about 0.6, the MSLL of the designed NUSPA gradu-ally increases, although the computational complexity can


Table 2 Comparison of several linear array arrangements.

be reduced. Therefore, it is reasonable to choose R/M as0.6, since the proposed method with R/M ≈ 0.6 is efficientin the context of MSLL and computational complexity. Thevalue of R/M ≈0.6 will also be used to determine the properboundary of the search region in the next NUSPA case ex-ample.

Although the R/M =0.6 is a proper choice in terms ofMSLL and computation time, as shown in Fig. 7, the opti-mized USPA with R/M =0.6 yields an amount of about 3 dBreduction of relative power level at boresight, compared tothe initial USPA case. This is because, through the thinningof the array geometry, the number of array elements in thedesigned USPA with the proposed method and R/M =0.6 isabout 70–75%, compared to the 100% initial filled USPAgeometry.

For example, we consider a 13 × 13 USPA for the op-timization of an array geometry. A 13 × 13 USPA extendedby a 13-element USLA is used as an initial array structure.The MSLL of the initial 13 × 13 USPA is −13.08 dB. Inorder to design an optimal USPA with low SLL from theinitial 13 × 13 USPA, the GA parameters were determinedas follows: population size for the planar array was set tothree times the length of each chromosome, a probability

Case 1: RMSLLR vs. R/M with the initial 11 × 11 USPACase 2: RMSLLR vs. R/M with the initial 13 × 13 USPACase 3: RMSLLR vs. R/M with the initial 15 × 15 USPACase 4: RMSLLR vs. R/M with the initial 17 × 17 USPACase 5: RMSLLR vs. R/M with the initial 19 × 19 USPACase 6: RMSLLR vs. R/M with the initial 21 × 21 USPA

Fig. 6 Relative amount of MS LL reduction for several USPAs.

of crossover to 0.8 and that of mutation to 0.025 for theUSPA. The main lobe region of −0.18 ≤ u ≤ 0.18 and−0.18 ≤ v ≤ 0.18 corresponding to (θ0, φ0) = (0, 0) was


Case 1: Relative power vs. R/M with the initial 11×11 USPACase 2: Relative power vs. R/M with the initial 13×13 USPACase 3: Relative power vs. R/M with the initial 15×15 USPACase 4: Relative power vs. R/M with the initial 17×17 USPACase 5: Relative power vs. R/M with the initial 19×19 USPACase 6: Relative power vs. R/M with the initial 21×21 USPA

Fig. 7 R/M versus relative power at boresight.

considered for the optimization. K in step 8 was chosen as70 to terminate the iterative generations. Fig. 8 shows theoptimized 13 × 13 USPA with the MSLL of −19.73 dB forthe value of R = Q = 3. Fig. 8(c) shows the side view ofthe planar array pattern when the main beam is scanned toθ0 = 25 and φ0 = 0. In comparison with the initial USPA,the optimized USPA structure can provide more reduction ofthe MSLL than the initial USPA structure. Note that gener-ally there is no occurrence of pattern distortion in the USPAcase while the main beam direction is steered, as in [17].

6.2.2 Optimized Planar Array from the Non-uniformlySpaced Planar Array

To generate an optimal NUSPA geometry in terms of MSLLand pattern distortion, the optimized 17-element NUSLAdesigned by the proposed method in Sect. 6.1 is extendedto a 17 × 17 rectangular array with non-uniform spacing.Next, we further modify the initial NUSPA to achieve anoptimal planar array geometry. As stated above, in order todetermine the genetic search boundary under the design ofthe optimal NUSPA structure, we apply the same result ofR/M ≈0.6 obtained in Sect. 6.2.1 to the initial NUSPA, re-sulting in R = Q = 5 for the 17×17 NUSPA. Population sizefor each planar array is three times the length of each chro-mosome. A probability of crossover is set to 0.82 and thatof mutation to 0.02 for the NUSPA. The main lobe regionof 0.36 ≤ u ≤ 0.63 and −0.13 ≤ v ≤ 0.13 correspondingto (θ0, φ0) = (30, 0) was considered for the calculationof the cost function F in Eq. (14). K was selected as 90 toterminate the optimization. Fig. 9(a) shows the optimizedNUSPA obtained from the initial 17 × 17 NUSPA. Fig. 9(b)is a 2-D planar array pattern as a function of u = sin θ cosφand v = sin θ sinφ for the array lattice given in Fig. 9(a).Fig. 9(c) is the side view of the array pattern when the main

(a) Optimized USPA structure.

(b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ.

(c) Side view of the radiation pattern when the main beam is scannedto θ0 = 25 and φ0 = 0.

Fig. 8 Optimized USPA for the initial 13 × 13 USPA.

beam is scanned to θ0 = 25 and φ0 = 0. The MSLL of−22.86 dB is achieved in all the side-lobe regions exceptthe main-lobe region for the optimized 17 × 17 NUSPA.Meanwhile, Fig. 10 shows the non-optimized NUSPA struc-ture with MSLL of about −17.4 dB and its associated beampatterns. This NUSPA was generated from a 17-elementNUSLA which has been obtained by using the Fourier trans-form based formula of Eq. (11). When the main beam angle


(a) Optimized NUSPA structure.

(b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sinφ.


Fig. 9 Optimized NUSPA geometry for the initial 17 × 17 NUSPA.

is scanned to the same direction as in Fig. 9(c), the arraystructure can not maintain a low SLL in the visible region,leading to a pattern distortion as shown in Fig. 10(c). In con-trast, the optimized NUSPA structure can provide low SLLwithout pattern distortions, although the main beam direc-tion is steered.

Next, Fig. 11(a) shows the optimized USPA geometry

(a) Non-optimized NUSPA structure.

(b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sin φ.


Fig. 10 Non-optimized NUSPA geometry for the initial 17×17 NUSPA.

when a 17× 17 USPA is used as an initial array structure. Inthis case, the MSLL of −20.94 dB in all the side-lobe regionsis accomplished for the value of R = Q = 5. Comparingthe optimized NUSPA in Fig. 9 with the optimized USPA inFig. 11, we observe that the optimized NUSPA structure canachieve more reduction of MSLL without pattern distortionsthan the optimized USPA geometry. In addition, we observe


(a) Optimized USPA structure.

(b) Radiation pattern as a function of u = sin θ cos φ and v = sin θ sinφ.


Fig. 11 Optimized USPA geometry for the initial 17 × 17 USPA.

that the two classes of the optimized arrays, namely NUSPAand USPA, are quite different in shape and the optimizedNUSPA structure rather than the optimized USPA geometryis very similar to a circular array shape.

From the above described results of Sects. 6.2.1 and6.2.2, the MSLLs and 3 dB main-lobe beamwidths of severalplanar array for the five different main beam directions are

(a) Radiation pattern in the plane of φ = 0.

(b) Radiation pattern in the plane of φ = 90.

(c) Steered radiation pattern in the plane of φ = 0.

Fig. 12 Comparison of normalized radiation patterns when the opti-mized NUSPA was obtained from the initial 11 × 11 NUSPA.

compared and summarized in Table 3. These planar arrayarrangements are explained as follows:

Case 1: Planar array geometry (17 × 17) with uniformspacing

Case 2: Non-optimized planar array geometry (17×17)with non-uniform spacing (Fourier transform based for-mula)

Case 3: Optimized planar array geometry with uniformspacing from initial 17 × 17 USPA

Case 4: Optimized planar array geometry with non-uniform spacing from initial 17 × 17 NUSPA

Case 5: Optimized planar array geometry with non-uniform spacing from initial 17 × 17 NUSPA

In Case 4, to obtain an initial 17 × 17 NUSPA, the


Table 3 Comparison of several planar array arrangements.

non-optimized 17-element NUSLA was extended to the 2-D NUSPA. In contrast, the optimized NUSLA was used toconstruct an initial NUSPA, in Case 5. From Table 3, we ob-serve that the optimized beam patterns except Case 2 main-tain a low SLL, for the scanning range of −30 ≤ θ0 ≤ 30,without beam pattern distortion. Comparing the results ofCase 4 and Case 5, we observe that the proper initial arraystructure is quite important to obtain efficient and reliableperformances. In addition, beam broadening of Case 4 ismore noticeable than any other cases. Next, the optimizedNUSPA structure of Case 5 shows a lower SLL than the op-timized USPA of Case 3. Meanwhile, when the whole ar-ray without boundary condition (R/M ≈0.6) was thinned,the same result was obtained like Case 5, after 89 genera-tion. However, the result of Case 5 was driven after only21 generation. The result of considering the whole arraywithout the boundary condition also cost much more timethan Case 5. Contrary to the 1-D linear array case, as thenumber of elements are increased in 2-D planar array, thecomputational complexity for the optimization can be sig-nificantly increased and a fast convergence may not be guar-anteed. Therefore, it is very important to determine theproper boundary condition for the 2-D array case, althoughthe global optimization technique is applied. As a result, itcan be indicated that the optimized NUSPA geometry underproper constraints is superior to the other array structuresfrom the viewpoint of MSLL and pattern distortion.

Finally, we also simulated the array performance us-ing a commercial full-wave analysis software tool (CST Mi-crowave Studio 4.2) based on FDTD algorithm, when an ini-tial 11 × 11 NUSPA with R = Q = 3 was considered for theconstruction of an optimal NUSPA. A wire antenna operat-ing at 2 GHz was used as a radiating element. The radiusof this element is 4.6 mm and the physical length, 61.4 mm.The antennas were positioned at the optimized NUSPA lat-tice, such that the boresight of the wire antenna was underthe direction of z-axis. The PML (Perfect Matched Layer)

which operates like free space is used as an open boundary.Fig. 12(a) and Fig. 12(b) show the normalized radiation pat-terns in the plane of φ = 0 and φ = 90, respectively. Inaddition, Fig. 12(c) represents the scanned radiation patternin the plane of φ = 0 when the main beam direction is po-sitioned at θ0 = 30 and φ0 = 0.

While the result of Case 1 was obtained by using theproposed method, that of Case 2 was acquired by using thefull-wave analysis software. From the result of Fig. 12, weobserve that, although there is a slight difference of MSLLbetween Case 1 and Case 2, the pattern shapes of the twocases are very similar each other, maintaining low SLLswithout pattern distortion.

7. Conclusion

In this paper, noble design schemes for the optimal NUSLAand NUSPA in the context of MSLL reduction, has been pre-sented using optimization techniques. In the NUSLA case,the Gauss-Newton method was applied to optimally adjustpositions of the initial non-uniform array elements whichwere obtained by the Fourier transform based formula. Theresults show that the optimized scannable NUSLA struc-ture designed by the proposed synthesis method can reduceouter SLL as well as inner SLL simultaneously without pat-tern distortion. In the NUSPA case, on the basis of the pro-posed NUSLA, the thinned array theory combined with thegenetic algorithm was applied to the design of an optimalNUSPA geometry, in terms of maximum side-lobe reduc-tion. First, the initial NUSPA is obtained by expanding theabove-optimized NUSLA to the 2-D rectangular array lat-tice along the row and column directions, respectively. Next,some elements in the outer regions of the array are opti-mally turned off from the initial NUSPA. During the pro-cess of genetic optimization, the genetic search boundaryfor the design of the optimal NUSPA was experimentallyderived considering requirements, MSLL and computationalcomplexity. The results show that the optimized NUSPA ge-


ometry using the proper boundary and initial conditions cansignificantly achieve low SLL without pattern distortion dur-ing the main beam steering and can accomplish lower SLLthan the optimized USPA structure.

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Ji-Hoon Bae received the B.S. degreein electronic engineering from Kyungpook Na-tional University in 2000, and the M.S. degreein electrical and computer engineering fromPohang University of Science and Technology(POSTECH) in 2002. Since 2002, he has beenwith Electronics and Telecommunications Re-search Institute (ETRI) as a researcher. His re-search interests include array antenna, radar tar-get imaging, and RFID system.

Kyung-Tae Kim received the B.S., M.T.S.,and Ph.D. degrees in electrical engineering fromthe Department of Electrical Engineering, Po-hang University of Science and Technology(POSTECH), Pohang, Kyungbuk, Korea, in1994, 1996, and 1999, respectively. DuringMarch 1999 and March 2001, he was withthe Electromagnetics Technology Laboratory,POSTECH, as a Research Fellow. During April2001 and February 2002, he was a Research As-sistant Professor, Electrical and Computer En-

gineering Division, POSTECH. He joined the Faculty of the Departmentof Electrical Engineering and Computer Science, Yeungnam University,Kyongsan, Kyungbuk. His primary research interests include radar targetrecognition and imaging, array signal processing, spectral estimation, pat-tern recognition, neural networks, and RCS measurement and prediction.

Cheol-Sig Pyo received the B.S. degree inelectronic engineering from Yonsei Universityin 1991, the M.S. degree in electrical engineer-ing from Korea Advanced institute of Scienceand Technology (KAIST) in 1999. Since 1991,he has been a senior engineer at Electronics andTelecommunications Research Institute (ETRI).At present, he is a team leader of RFID tech-nology research team in ETRI. His research in-terests include antenna, radio system, and RFIDsystem.

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