Pattern Search optimization with applications on synthesis of linear antenna arrays

Expert Systems with Applications 37 (2010) 4698–4705

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Pattern Search optimization with applications on synthesis of linear antenna arrays Filiz Günesß *, Fikret Tokan Department of Electronics and Communication Engineering, Yıldız Technical University, Besßiktasß, 34349 Istanbul, Turkey

a r t i c l e

i n f o

Keywords: Optimization Pattern Search (PSearch) Objectives Fitness function Convergence

a b s t r a c t In this work, Pattern Search (PSearch) method is introduced as a direct, efficient and derivative-free optimization tool with the applications on the antenna array synthesis in the antenna engineering. PSearch is a nonrandom method which can be exploited as a direct searching tool for minimization of a function which is not necessarily differentiable, stochastic, or even continuous. Thus, firstly antenna array synthesis is defined as a multi-objective optimization problem with its feasible variable and target spaces. For this aim, maximum amount of the side-lobe suppressions and broad/narrow null generations in any desired directions are simultaneously expressed as objectives in the target space while ensuring maximization in the gain performance of the antenna array. At the same time, the inter-element spacings and excitation amplitudes are considered as optimization variables that results in determination of the physical layout and feeding network of the array. In the optimization procedure, a fitness function is defined based on the target and synthesis variable spaces that can be applied into various antenna array designs, combining part by part with the different requirements. Besides convergence is made fast by a seeding process which consists of running ‘‘genetic” algorithm once with the random initial values. Finally, the whole PSearch synthesis method is verified by applying into the many linear antenna arrays synthesizes with various multi-objective requirements, and all of the optimized arrays are observed to outperform uniform arrays and representative designs. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In this work, the Pattern Search (PSearch) algorithm is exploited as an optimization tool with applications on the antenna array synthesis in the antenna engineering. The PSearch is a nonrandom method for searching minima of a function which is not necessarily differentiable, stochastic, or even continuous without requiring the gradient information. Thus, firstly antenna array synthesis is defined as a multi-objective optimization problem gathering simultaneously all the main requirements expected forming such an antenna array. These requirements are well-known in the antenna engineering: Since in many communication systems one is interested in point-to-point communication, thus much more highly directive beam of radiation can be used to advantage. Hence, by arranging elementary radiators into an array, a more directive beam of radiation can be obtained. A more directive beam means that the antenna will also have a higher gain. The other important requirement for a communication system is high ratio of the signal to interference. This is achieved by the suppression of interference and multipath signals and obtaining nulls in the directions of inter-

* Corresponding author. Tel.: +90 212 2613989; fax: +90 212 2599321. E-mail address: [email protected] (F. Günesß). 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2009.11.012

fering signals. Thus all of these simultaneous requirements necessitate synthesis of an antenna array system by a multi-objective optimization process. However, currently many synthesis methods are concerned with suppressing the side-lobe level (SLL) whereas preserving the gain of the main beam. Other methods deal with only the null control to reduce the effects of interference and jamming (Babayigit, Akdagli, & Guney, 2006; Boeringer & Werner, 2004; Guney & Basbug, 2008; Guney & Onay, 2007; Khodier & Christodoulou, 2005; Mahmoud, Eladawy, Bansal, Zainud-Deen, & Ibrahem, 2008; Murino, Trucco, & Regazzoni, 1996; Yan & Lu, 1997). On the other hand, variables that can be used in synthesis of an antenna array may be grouped in the categories: Array geometry and feeding network. In this work, it is also verified it is possible to synthesize highly good quality radiation patterns by optimizing inter-element spacing d(i)s of elementary radiators placed simply on a line and the excitation amplitudes A(i) i = 1, . . .. N where N is number of the radiators. In practical terminology, determination of d(i) and A(i) i = 1, . . .. N can define the physical layout and feeding network of the linear antenna array. As it is well known that the exploited algorithm itself also acts an important role as the optimization methodology to determine the fitness of the antenna array system during the optimization process. In the recent years, the evolutionary algorithms such as

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genetic algorithms (GA), simulated annealing (SA), particle swarm optimization (PSO), the clonal selection algorithm (CLONALG), bacterial foraging algorithm (BFA), bees algorithm (BA) become popular to synthesize either the physical layout or the feeding network of antenna array. The pioneering works for the evolutionary antenna array synthesizes can be found in Boeringer and Werner (2004), Babayigit et al. (2006), Guney and Onay (2007), Guney and Basbug (2008), Khodier and Christodoulou (2005), Mahmoud et al. (2008), Murino et al. (1996), Yan and Lu (1997). Here we have chosen PSearch algorithm as the optimization methodology that results in a comparatively fast and repeatable performance in the antenna synthesis process with no need for gradients of the objective function. Moreover, to initialize PSearch a seeding process is employed using genetic algorithm with its random parameters to have relatively faster convergence rate. The paper is organized as follows: Next section is devoted to explanation of the PSearch algorithm. In the third section the multi-objective function used in the work is presented in terms of the array factor defining antenna radiation pattern. The fourth section explains a typical initialization process. Optimization (synthesis) results will be discussed in the fifth section and finally the last section presents conclusions. 2. PSearch algorithm PSearch is a direct method for searching minima of a function which is not necessarily differentiable, stochastic, or even continuous. Thus it can be exploited efficiently in solving optimization problems without requiring any information about the gradient of the fitness function. As opposed to more traditional optimization methods that use information about the gradient or higher derivatives to search for an optimal point, a PSearch algorithm searches a set of points around the current point, looking for one where the value of the fitness function is lower than the value at the current point. PSearch algorithm can briefly be explained as follows: It computes a sequence of points gets closer and closer to the optimal point. At each step, the algorithm searches a set of points, called a mesh, around the current point-until finding a point in the mesh where value of the fitness function decreases compared to the value at the current point. This new point becomes the current point at the next step of the algorithm. The mesh is formed by adding a scalar multiple of a fixed set of vectors called ‘‘Pattern Vector” to the current point. Flow chart of the PSearch algorithm is given in Fig. 1. According to this flow chart, the algorithm can be considered in the three main steps: definitions, poll process, and stopping. Step 1. Definitions: This step consists of the following substeps: Step 1.1. Definition of the solution space consisting of optimization variables X j ; j ¼ 1; . . . N together with their lower X j‘ and upper X ju limitations; j ¼ 1; . . . N and fitness function. Since an optimization problem is under discussion, then optimization variables X j ; j ¼ 1; . . . N and a fitness function should be defined so that they can provide the interface between the physical problem and the optimization algorithm. Furthermore, if the fitness is considered as a ‘‘cost”, thus, the smaller the fitness value, the better the solution. Step 1.2. Definition of the working parameters: DX j ; j ¼ 1; . . . N; gexp , and gcont . In this sub-step, the N-dimensional pattern vector is defined with together its expansion gexp and contraction gcont factors to form mesh around the current working point. Step1.3. Definition of the starting point P start ðX j0 ; j ¼ 1; . . . NÞ: A starting point in the solution space should be defined for the algorithm to start. This process can also be called as initializa-

tion. Initialization set may be obtained by running different algorithms which may be either stochastic or deterministic. Step 2. Poll process: In this second main step, the following processes are performed: Step 2.1. Evaluate the fitness at the mesh points Mj ; j ¼ 1; . . . N; Step 2.2. Compare the fitnesses at the current Pcurrent and the mesh points Mj ; j ¼ 1; . . . N; Step 2.3. If one of the Mj ; j ¼ 1; . . . N is fitter than the current point Pcurrent , firstly expand the pattern vector, () gexp DX j ; j ¼ 1; . . . N then add it to the current vector X j ; j ¼ 1; . . . N represented by the P current (X jcurrent ; j ¼ 1; ::::N) to update the current point and finally build a new mesh on this current point M j () X jcurrent þ DX j ; j ¼ 1; . . . N and go to Step 2.1; Step 2.4. If the P current is fitter than the Mj ; j ¼ 1; . . . N; check if the stopping criteria are met. If it is satisfied, then the Psolution and its fitness value are outputted. Otherwise, follow the next step; Step 2.5. In this step, firstly contract the pattern vector () gcont DX j ; j ¼ 1; . . . N; then add it to the current vector Pcurrent () X jcurrent ; j ¼ 1; . . . N to update the current point and finally build a new mesh on this current point Mj () X jcurrent þ DX j ; j ¼ 1; . . . N and go to Step 2.1; Step 3. The following criteria can be used to stop the algorithm: (i) Mesh tolerance which is the minimum tolerance for mesh size; (ii) maximum iteration which is the maximum number of iterations the algorithm performs; (iii) maximum function evaluations which is the maximum number of evaluations of the objective and constraint functions; (iv) time limit which is the maximum time in seconds the Pattern Search algorithm runs before stopping; and finally (v) function tolerance which is the termination tolerance for the objective function value.

3. Objective function The Pattern Search algorithm presented in the previous section is applied to synthesize the linear antenna array in Fig. 2 having highly directive radiation pattern with high side-lobe suppression in the desired directions. Our solution space X j ; j ¼ 1; . . . N consists of the inter-element spacing and the excitation amplitudes which are dðiÞ and AðiÞ, i ¼ 1; . . . N, respectively. For this aim, our objectives are to meet the minimum SLL, maximization of the mainbeam in the desired direction () maximum directivity/gain, to obtain nulls in the directions of interfering signals; hence the fitness (objective) function is defined in the sense of ‘‘cost” as follows:

Fitness ¼ w1 20 log

( XZ hl

l

þ w2 20 log

( X i

þ w3 20 log

hul

( X

) jAFðhÞjdh

l

hu maxfjAFðhÞjghl i

)

i

) jAFðhk Þj

 w4 20 log D

ð1Þ

k

where AFðhÞ is the array factor of the antenna array having even number of elements (Balanis, 1997; Elliott, 1981; Stutzman & Thiele, 1998):

AFðhÞ ¼ 2

N X n¼1

AðnÞ cos

  2p dðnÞ cos h þ bðnÞ k

ð2Þ

where k is the free-wave length and bðnÞ is the feeding phase angle of the nth element which is taken as zero in our case. Furthermore directivity D in (1) can be given as follows (Balanis, 1997; Elliott, 1981; Stutzman & Thiele, 1998):

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Fig. 1. Flow chart for the PSearch algorithm.

D¼

2ðaÞ2 ; 2ðb1 þ b2 Þ

b1 ,

N X N X

AðnÞAðmÞ

n¼1 m¼1

 b2 ,

N P N P

n¼1 m¼1 N P N P n¼1 m¼1

a,

N X

AðnÞ;

n¼1

sin½2pðdðnÞ þ dðmÞÞ ; 2pðdðnÞ þ dðmÞÞ

AðnÞAðmÞ 

pðdðnÞdðmÞÞ AðnÞAðmÞ sin½2 2pðdðnÞdðmÞÞ

if

n ¼ m;

if

n–m:

ð3Þ

If the antenna array is constituted by odd number of radiators by adding an extra element to the origin, the contribution of this extra element is added to radiation and directivity into (2) and (3), respectively (Balanis, 1997; Elliott, 1981). In the fitness function given by (1), hul and hll are the upper and lower angles of the regions l ¼ 1; :::L where areas of the SLLs are aimed at suppressing and hk , k = 1, . . . K are the directions where the nulls are required hui ; hli are the boundaries of the spatial regions where the maxima are intended to be suppressed. Besides, wi ; i ¼ 1; . . . 4 are the weighting coefficients which should be adapted to the problem, in our case they are adopted as unity.

Fig. 2. The linear antenna array geometry of sum pattern formed with even number of elements.

Thus, in the objective function given by (1), the first two terms are employed to minimize the SLL between the desired angles whereas the third one is for having nulls in the desired directions.

F. Günesß, F. Tokan / Expert Systems with Applications 37 (2010) 4698–4705

Moreover the integral term in (1) can also be used to obtain broad nulls. However, the final term is simultaneously exploited to guar-

0 PSearch Uniform Dolph-Cheb.

-10

Gain(dB)

-20

antee maximization of the electromagnetic energy towards the intended user. The objective function given by (1) includes all the main requirements; however this function can also be partly employed, depending on the demanded specifications from the radiation pattern. 4. PSearch optimization In order to explain operation mechanism of PSearch optimizer with the seeding process, the fitness function given by (1) is applied to synthesis of a linear array of 24 radiators with having broad null and side-lobe suppression within the regions at (20°– 60°) and (0°–83°), respectively:

-30 -40 -50

Fitness ¼

-60 -70 0

4701

20

40

60

80 100 120 theta(degree)

140

160

180

Fig. 3. Normalized patterns of the 24-element linear array obtained from the threestage initialization. The regions of suppressed SLL are [0°, 83°] and [97°, 180°], and broad nulling effects within the regions of [20°, 60°] and [120°, 160°] while high directivity and narrow beamwidth are achieved (features of the patterns and their solution spaces are given in Table 1).

(Z ) hu ¼60 1 20 log jAFðhÞjdh Dh u hl ¼20 n n oo ¼83  20 log D þ 20 log max jAFðhÞjhhul ¼0 

ð4Þ

In this optimization process with the fitness given by (4), the following parameter suite is utilized:  The pattern vector has 2N dimensions with each having unity amplitude.  The expansion factor gexp ¼ 2.  The Contraction factor gcont ¼ 0:5:

Fig. 4. Convergence curves for: (a) genetic (b) PSearch 1 (c) PSearch 2 for the radiation patterns in Fig. 3.

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Table 1 (a) Fitnesses and features of the radiation patterns in Fig. 3. (b) Normalized solution spaces for the radiation patterns in Fig. 3. Optimization algorithm (a) Genetic PSearch1 PSearch2 (b) Genetic

Fitness (dB)

D

HP (°)

|AF(h)|max (dB) for 0° 6 h 6 83°

|AF(h)|max (dB) for 20° 6 h 6 60°

|AF(h)|max (dB) for 97° 6 h 6 180°

|AF(h)|max (dB) for 120° 6 h 6 160°

54 79 100

4.57 22.31 21.23

26 5.8 6.2

3 22 20

42 29 49

3 22 20

42 29 49

A1(A) d1(k) A2(A) d2(k) A2(A) d3(k)

PSearch1 PSearch1

0.159 0.035 0.882 0.349 0.768 0.349

0.803 0.159 0.478 0.552 0.649 0.552

0.702 0.211 0.401 1.079 0371 1.079

1.000 0.482 0.783 1370 1.000 1370

0.474 0.489 0.703 2.119 0.687 2.119

Starting point in the solution space is determined by a seeding process which consists of running ‘‘genetic” algorithm once adopted with the following parameter suite:    

Population size: pop = 2N = 2  12 = number of the unknowns. Crossover probability: pc = 0.75. Mutation probability: pm = 0.01. Maximum number of generations: MaxG = 40.

To speed up the PSearch algorithm, we have utilized an initialization process consisting of single trial of Genetic algorithm with the above parameter suite where the parameters pc and pm are set

0 PSearch Uniform Dolph-Cheb.

-10 Desired null directions

Desired null directions

Gain(dB)

-20 -30

-40 -50

-60

-70 0

20

40

60

80 100 theta(degree)

120

140

160

180

Fig. 5. Normalized patterns of the 10-element linear array obtained using three different methods. The regions of suppressed SLL are [0°, 83°] and [97°, 180°], and prescribed nulls at: 20°, 40°, 50°, 130°, 140°, 160° (features of the patterns and their solution spaces are given in Table 2).

0.461 0.535 0.659 2.618 0.445 2.618

0.047 0.622 1.000 3.285 0.997 3.285

0.434 0.931 0.882 4.032 0.967 4.072

0.380 0.988 0.641 4.903 0.608 4.911

0.770 1.099 0.493 5.671 0.337 5.671

0.093 1.135 0.255 6.255 0.194 6.056

0.476 1.646 0.156 7.985 0.165 6.635

to their default values while population size and maximum number of generations are taken equal to the number of unknowns and a small value, respectively. Here three stage initialization process is considered, whose resulted radiation patterns are given in Fig. 3, and can be described as follows: (1) Pattern 1 is resulted from a single trial of the Genetic algorithm with the above parameter suite; (2) Pattern 2 is obtained from the PSearch optimization initialized by the solution space of the Pattern 1; (3) Pattern 3 is obtained using again PSearch optimization initialized by the solution space of the Pattern 2. The resulted solution spaces and fitness values belonging to these optimization processes are given in Tables 1 and 2, respectively. Moreover, the corresponding convergence characteristics are taken place in Figs. 4a–c. As seen from the convergence curves in Fig. 4, the starting point Pstart1 in the solution space for the PSearch1 optimization process is fixed by running only 30 generations (iterations) with the genetic algorithm. Then the PSearch algorithm is utilized searching a fitter solution, starting from the Pstart1, thus at end of a comparatively short run such as 120 iterations, a fitter solution is obtained by changing the feeding amplitudes and inter-element spacing (Pstart2). In the final attempt, PSearch algorithm achieves a good quality of radiation pattern as seen from Fig. 3 and Table 1 starting searching from Pstart2. Thus in the next section, the antenna array synthesis examples with the different pattern requirements will be worked out using the objective function given in Section 4 and the above optimization process. 5. Antenna array synthesis examples In this section, various antenna array synthesis examples are worked out to demonstrate versatilities in performances of the objective function given in (1) and the PSearch optimization process in Section 4. The optimized arrays are compared with the Uniform and Dolph–Chebyshev arrays having the equal electrical

Table 2 Features of the radiation patterns in Fig. 5. (b) Normalized solution spaces for the radiation patterns in Fig. 5.

(a) PSearch Uniform Dolph–Cheb. (b) Uniform Dolph–Cheb. PSearch

D

Peak SLL (dB)

HP (°)

|AF(20°)| (dB)

|AF(40°)| (dB)

|AF(50°)| (dB)

|AF(130°)| (dB)

|AF(140°)| (dB)

|AF(160°)| (dB)

12.14 12.79 11.27

23 13 23

12.2 10.2 12.4

133 29 35

87 43 46

91 25 26

133 29 35

87 43 46

91 25 26

A d A d A d

1.000 0.325 1.000 0.325 0.691 0.209

1.000 0.975 0.886 0.975 1.000 0.739

1.000 1.625 0.689 1.625 0.955 1.438

1.000 2.275 0.458 2.275 0.761 2.228

1.000 2 925 0.303 2.925 0.478 2.944

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lengths, which have well-established regular synthesis procedures, for the directivity and the side-lobe suppression. In the first worked example, the sum pattern configuration of a 10-element linear array is synthesized for SLL suppression within the regions between [0°–80°] and [100°–180°] and obtaining nulls at 20°, 40°, 50°, 130°, 140° and 160° while maintaining high directivity and narrow beamwidth compared with the counterparts. The corresponding fitness function can be expressed using (1) as:

n n oo ¼80 þ 20 log fAFð20 Þg Fitness ¼ 20 log max jAFðhÞjhhul ¼0  þ 20 log fAFð40 Þg þ 20 log fAFð50 Þg  20 log D

ð5Þ

where only minimization of the maxima in (1) within the desired regions is used to obtain the SLL suppression in order to examine the effectiveness of this term. The resulted array pattern from the PSearch algorithm is shown in Fig. 5, together with patterns obtained using the uniform and Dolph–Chebyshev arrays in approximately equal electrical lengths. Moreover, the corresponding features of the patterns and their solution spaces are also given in Table 2. In this worked example, the synthesized antenna has the directivity as high as the uniform array while exhibiting a high SLL suppression as the Dolph-Chebyshev and at the same time having deep nulls at 20°, 40°, 50°, 130°, 140°, 160° directions as seen from Table 2. The second example belongs to synthesis of a 12-element linear array for SLL suppression within the regions between [0°–80°] and

[100°–180°] and obtaining broad nulls within the regions [0°, 20°] and [160°, 180°], while maintaining high directivity and narrow beamwidth. In this case, using (1) the following expression is employed as the fitness function:

n n oo ¼80 Fitness ¼ 20 log max jAFðhÞjhhul ¼0  n n oo ¼20  20 log D þ 20 log max jAFðhÞjhhul ¼0 

where again the fitness terms for minimization of the maxima are used to suppress SLL and generate broad nulls in the required regions. The array pattern using PSearch algorithm is shown in Fig. 6, together with counterpart patterns. In this synthesis example, the SLLs of PSearch and Dolph– Chebyshev arrays are 16 dB lower than uniform array. Besides, PSearch array is easily achieved a broad null region within the angles of [0°, 20°] and [160°, 180°] while having comparable directivity with the uniform array. The corresponding features of the patterns and their solution spaces are given in Table 3. Synthesis of the linear array of sum pattern having odd number of elements is considered in the third example, targeting the SLL suppression in the whole visible region and maintaining as high directivity as it can be achieved, while obtaining nulls at 10°, 170° and having broad nulls within the regions of [30°, 50°] and [130°, 150°]. In this example, in order to obtain the broad nulls,

0

0 PSearch Uniform Dolph-Cheb.

-10

-20 Gain(dB)

Gain(dB)

-30 -40

-40 -50

-60

-60

40

60

80 100 theta(degree)

120

140

160

-70 0

180

Fig. 6. Normalized patterns of the 12-element linear array obtained using three different methods. The regions of suppressed SLL are [0°, 80°] and [100°, 180°], and broad nulling effects within the regions of [0°, 20°] and [160°, 180°] while high directivity and narrow beamwidth are achieved and no prescribed nulls. (Features of the patterns and their solution spaces are given in Table 3).

Desired null direction

Desired null direction

-30

-50

20

PSearch Uniform Dolph-Cheb.

-10

-20

-70 0

ð6Þ

20

40

60

80 100 theta(degree)

120

140

160

180

Fig. 7. Normalized patterns of the 13-element linear array obtained using three different methods. The regions of suppressed SLL are [0°, 82°] and [98°, 180°], and broad nulling effects within the regions of [30°, 50°] and [130°, 150°] while high directivity and narrow beamwidth are achieved and prescribed nulls at: 10°, 170° (features of the patterns and their solution spaces are given in Table 4).

Table 3 Features of the radiation patterns in Fig. 6. (b) Normalized solution spaces for the radiation patterns in Fig. 6.

(a) PSearch Uniform Dolph–Cheb. (b) Uniform Dolph–Cheb. PSearch

D

Peak SLL (dB)

HP (°)

|AF(h)|max (dB) for 0° 6 h 6 20°

|AF(h)|average (dB) for 0° 6 h 6 20°

|AF(h)|max (dB) for 160° 6 h 6 180°

|AF(h)|average (dB) for 160° 6 h 6 180°

11.99 13.87 12.02

28 12 28

12.4 10 12.3

46 22 28

58 (broad nulling effect) 23 30

46 22 28

58 (broad nulling effect) 23 30

A d A d A d

1.000 0.290 1.000 0.290 0.769 0.151

1.000 0.870 0.918 0.870 1.000 0.731

1.000 1.450 0.770 1.450 0.597 1.108

1.000 2.030 0.584 2.030 0.913 1.712

1.000 2.610 0.391 2.610 0.665 2.459

1.000 3.200 0.289 3.200 0.308 3.204

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Table 4 (a) Features of the radiation patterns in Fig. 7. (b) Normalized solution spaces for the radiation patterns in Fig. 7.

(a) PSearch Uniform Dolph–Cheb. (b) Uniform

D

Peak SLL (dB)

HP (°)

|AF(10°)| (dB)

|AF(h)|max (dB) for 30° 6 h 6 50°

|AF(10°)|
|AF(h)|max (dB) for 130° 6 h 6 150°

13.47 16.72 12.16

23 12 23

10.2 8.8 12.2

118 24 24

46 (broad nulling effect) 23 23

118 24 24

46 (broad nulling effect) 23 23

A d A d A d

Dalph–Cheb. PSearch

1.000 0.000 1.000 0.000 0.521 0.000

1.000 0.648 0.973 0.648 1.000 0.455

1.000 1.296 0.895 1.296 0.920 1.091

PSearch Uniform Dolph-Cheb.

Gain(dB)

-20 -30

-40 -50

-60

-70 0

20

40

60

80 100 theta(degree)

120

140

160

180

Fig. 8. Normalized patterns of the 24-element linear array obtained using three different methods. The regions of suppressed SLL are [0°, 84°] and [96°, 180°], and broad nulling effects within the regions of [0°, 70°] and [110°, 180°] while high directivity and narrow beamwidth are achieved and no prescribed nulls (features of the patterns and their solution spaces are given in Table 5).

the integral term in the fitness function in (1) is minimized within the required region:

n o   ¼82o þ 20 log jAFð10o Þj Fitness ¼ 20 log max fjAFðhÞjghhul ¼0 o (Z ) hu ¼50o 1 20 log jAFðhÞjdh  20 log D þ Dh l hl ¼30o

1.000 2.593 0.630 2.593 0.791 2.363

1.000 3.241 0.473 3.241 0.748 3.166

1.000 3.890 0.536 3.890 0.393 3.896

pattern using PSearch algorithm is shown in Fig. 7, together with counterpart patterns. In this synthesis, PSearch array has a more directive pattern than Dolph–Chebyshev array whereas having prescribed nulls at 10°, 170° and broad nulls at [0°, 82°] and [98°, 180°]. Although half power beamwidth of PSearch array is only 1.4° wider than uniform array, its SLL is 11 dB lower than the uniform array. The corresponding features of the patterns and their solution spaces are given in Table 4. The final example is on the sum pattern synthesis of a 24-element linear array. In this synthesis problem, the objective is suppression of the SLL area within the regions of [0°, 84°], [96°, 180°] while having as high directivity as it can be achieved. For this requirement, the following fitness is formed using (1):

0 -10

1.000 1.945 0.776 1.945 0.828 1.667

ð7Þ

In this example, since the array is formed with 13 elements by adding an extra element to the origin into the previous array, thus the array factor given by the Eq. (2) is employed in (3). The array

(Z ) hu ¼70 1 Fitness ¼ 20 log jAFðhÞjdh Dh l hl ¼0 n n oo ¼84  20 log D þ 20 log max jAFðhÞjhhul ¼0 

ð8Þ

where broad nulling effects are generated within the sub-regions at [0°, 70°] and [110°, 180°]. The radiation pattern compared with the designs of uniform and Dolph–Chebyshev methods is given in Fig. 8. It can be observed that very high suppression of average SLL as low as -61 dB in the target bands of [0°, 70°] and [110°, 180°] is achieved whereas maintaining only 2° wider half power beamwidth than uniform array and only 0.6° wider half power beamwidth than Dolph–Chebyshev array. This suppression level between [0°, 70°] and [110°, 180°] angles is generally considered as the ‘‘null” level in the synthesizes given in the literature with the typical examples (Babayigit et al., 2006; Boeringer & Werner, 2004; Guney & Basbug, 2008; Guney & Onay, 2007; Khodier & Christodoulou, 2005; Mahmoud et al., 2008; Murino et al., 1996; Yan & Lu, 1997). The

Table 5 Features of the radiation patterns in Fig. 8. (b) Normalized solution spaces for the radiation patterns in Fig. 8.

(a) PSearch Uniform Dolph–Cheb. (b) Uniform Dolph–Cheb. PSearch

D

Peak SLL (dB)

HP (°)

|AF(h)|max (dB) for 0° 6 h 6 70°

|AF(h)|average (dB) for 0° 6 h 6 70°

|AF(h)|max (dB) for 110° 6 h 6 180°

|AF(h)|average (dB) for 110 6 h 6 180°

23.98 30.80 26.93

30 13 30

6.2 4.2 5.6

46 24 30

61 (broad nulling effect) 31 35

46 24 30

61 (broad nulling effect) 31 35

A d A d A d

1.000 0.323 1.000 0.323 0.224 0.216

1.000 0.969 0.979 0.969 0.690 0.387

1.000 1.615 0.939 1.615 0.588 0.825

1.000 2.261 0.881 2.261 1.000 1.453

1.000 2.907 0.808 2.907 0.989 2.282

1.000 3.553 0.724 3.553 0.878 3.137

1.000 4.199 0.633 4.199 0.693 3.978

1.000 4.845 0.537 4.845 0.377 4.716

1.000 5.491 0.442 5.491 0.286 5.153

1.000 6.137 0.350 6.137 0.331 5.836

1.000 6.783 0.266 6.783 0.189 6.636

1.000 7.429 0.363 7.429 0.059 7.436

F. Günesß, F. Tokan / Expert Systems with Applications 37 (2010) 4698–4705

resulted excitation coefficient A(n)s and spacing d(n)s are given in Table 5. The resulting pattern is given as compared with the counterpart uniform array in Fig. 8. It can be seen from Fig. 8 that the mainbeam of the designed array has an almost equal narrow beam as uniform array, although the SLL of the PSearch array is suppressed 18 dB more than the SLL of the uniform array. The corresponding excitation amplitudes and array configuration of difference pattern is given in Table 5. 6. Conclusions In this work, the Pattern Search method is introduced as a direct, efficient and derivative – free optimization tool with applications on the synthesis of the linear antenna arrays. For this aim, synthesis of linear antenna arrays is presented as a multi-objective optimization problem and a fitness function is defined so that the limitations of all the main radiation features of a pattern can be searched, which is successfully applied into various antenna array synthesizes with the different requirements. Furthermore, convergence is made fast by a seeding process where the ‘‘genetic” algorithm is employed by a very short run. Finally performance of the PSearch optimization is verified by the optimized arrays with their outperforming features to the counterparts.

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