optimal synthesis of linear antenna arrays using a harmony search algorithm

8
Optimal synthesis of linear antenna arrays using a harmony search algorithm K. Guney a,, M. Onay b a Department of Electrical and Electronics Engineering, Faculty of Engineering, Nuh Naci Yazgan University, Kayseri, Turkey b Department of Aircraft Electrical and Electronics, Civil Aviation School, Erciyes University, 38039 Kayseri, Turkey article info Keywords: Antenna array Interference suppression Harmony search algorithm Array synthesis abstract A method based on harmony search algorithm (HSA) for the pattern synthesis of linear antenna arrays with the prescribed nulls is presented. Nulling of the pattern is achieved by controlling the amplitude- only, the phase-only, and the position-only. The HSA is conceptualized using the musical process of searching for a perfect state of harmony. To show the effectiveness of the proposed HSA, several examples of linear antenna array patterns with the imposed single, multiple and broad nulls are given. It is found that the nulling method based on HSA is capable of steering the array nulls precisely to the undesired interference directions. The results of HSA are compared with the results of the modified touring ant col- ony optimization (MTACO), the bees algorithm (BA), the bacterial foraging algorithm (BFA), the plant growth simulation algorithm (PGSA), the clonal selection algorithm (CLONALG), the particle swarm opti- mization (PSO), the quadratic programming method (QPM), the tabu search algorithm (TSA), the genetic algorithm (GA), the memetic algorithm (MA), the nondominated sorting genetic algorithm 2 (NSGA-2), the multiobjective differential evolution (MODE), and the multiobjective evolutionary algorithms based on decomposition with differential evolution (MOEA/D-DE). Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Antenna arrays have been widely used in mobile, wireless, sa- tellite and radar communications systems to improve signal qual- ity, thereby increasing system coverage, capacity and link quality (Mailloux, 1994). The performance of these systems depends firmly on the antenna array design. The process of determining the parameters of an antenna array to obtain the required antenna radiation pattern is known as pattern synthesis. Pattern nulling methods that achieve suppression of interfering signals from pre- scribed directions while receiving the desired signal from a chosen look direction are important in radar, sonar and many communica- tion systems. Pattern nulling methods, including controlling the complex weights (both the amplitude and phase), amplitude-only, phase-only, and position-only, have been extensively studied in the literature (Abu-Al-Nadi, Ismail, & Mismar, 2006; Akdagli & Gu- ney, 2004; Akdagli, Guney, & Karaboga, 2002; Ares, Rodriguez, Villanueva, & Rengarajan, 1999; Babayigit, Akdagli, & Guney, 2006; Chung & Haupt, 2000; Er, 1990; Guney & Akdagli, 2001; Gu- ney & Babayigit, 2008; Guney, Babayigit, & Akdagli, 2007, 2008; Guney & Basbug, 2008a, 2008b; Guney, Durmus, & Basbug, 2009; Guney & Onay, 2007a, 2007b, 2010; Hamici & Ismail, 2009; Haupt, 1997; Hejres, 2004; Ibrahim, 1991; Ismail & Dawoud, 1991; Karab- oga, Guney, & Akdagli 2002, 2004; Khodier & Christodoulou, 2005; Liao & Chu, 1997, 1999; Lu & Yeo, 2000; Mouhamadou, Armand, Vaudon, & Rammal, 2006; Mouhamadou, Vaudon, & Rammal, 2006; Pal, Qu, Das, & Suganthan, 2010; Shore, 1984; Singh, Kumar, & Kamal, 2010; Steyskal, Shore, & Haupt 1986; Tennant, Dawoud, & Anderson, 1994; Yang, Gan, & Qing, 2004). Interference suppression with complex weights is the most effective since it has the larger solution alternatives. However, it is also the most expensive considering the cost of the controllers used for phase shifters and variable attenuators for each array ele- ment. Moreover, when the number of elements in the array in- creases, the computational time to find the values of element amplitudes and phases will also increase. The amplitude-only con- trol uses a set of variable attenuators to adjust the element ampli- tudes. If the element amplitudes have even symmetry about the center of the array, the number of attenuators and the computa- tional time are halved. The problem of phase-only and position- only nulling is inherently nonlinear and it cannot be solved directly by analytical methods without any approximation. By assuming that the phase or position perturbations are small, the nulling equations can be linearized. The phase-only control utilizes the phase shifters while the position-only control needs a mechanical driving system such as servomotors to move the array elements. Phase-only null synthesizing is less complicated and attractive for the phased arrays since the required controls are available at no extra cost, but it has still common problem. In this paper, an alternative method based on HSA (Geem, 2009; Geem, Kim, & Loganathan, 2001; Lee & Geem, 2005) is presented for linear antenna array pattern synthesis with prescribed nulls. 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.06.015 Corresponding author. Tel.: +90 352 437 57 55; fax: +90 352 437 57 84. E-mail addresses: [email protected] (K. Guney), [email protected] (M. Onay). Expert Systems with Applications 38 (2011) 15455–15462 Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa

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Page 1: Optimal synthesis of linear antenna arrays using a harmony search algorithm

Expert Systems with Applications 38 (2011) 15455–15462

Contents lists available at ScienceDirect

Expert Systems with Applications

journal homepage: www.elsevier .com/locate /eswa

Optimal synthesis of linear antenna arrays using a harmony search algorithm

K. Guney a,⇑, M. Onay b

a Department of Electrical and Electronics Engineering, Faculty of Engineering, Nuh Naci Yazgan University, Kayseri, Turkeyb Department of Aircraft Electrical and Electronics, Civil Aviation School, Erciyes University, 38039 Kayseri, Turkey

a r t i c l e i n f o a b s t r a c t

Keywords:Antenna arrayInterference suppressionHarmony search algorithmArray synthesis

0957-4174/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.eswa.2011.06.015

⇑ Corresponding author. Tel.: +90 352 437 57 55; fE-mail addresses: [email protected] (K. Guney), m

Onay).

A method based on harmony search algorithm (HSA) for the pattern synthesis of linear antenna arrayswith the prescribed nulls is presented. Nulling of the pattern is achieved by controlling the amplitude-only, the phase-only, and the position-only. The HSA is conceptualized using the musical process ofsearching for a perfect state of harmony. To show the effectiveness of the proposed HSA, several examplesof linear antenna array patterns with the imposed single, multiple and broad nulls are given. It is foundthat the nulling method based on HSA is capable of steering the array nulls precisely to the undesiredinterference directions. The results of HSA are compared with the results of the modified touring ant col-ony optimization (MTACO), the bees algorithm (BA), the bacterial foraging algorithm (BFA), the plantgrowth simulation algorithm (PGSA), the clonal selection algorithm (CLONALG), the particle swarm opti-mization (PSO), the quadratic programming method (QPM), the tabu search algorithm (TSA), the geneticalgorithm (GA), the memetic algorithm (MA), the nondominated sorting genetic algorithm 2 (NSGA-2),the multiobjective differential evolution (MODE), and the multiobjective evolutionary algorithms basedon decomposition with differential evolution (MOEA/D-DE).

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction Liao & Chu, 1997, 1999; Lu & Yeo, 2000; Mouhamadou, Armand,

Antenna arrays have been widely used in mobile, wireless, sa-tellite and radar communications systems to improve signal qual-ity, thereby increasing system coverage, capacity and link quality(Mailloux, 1994). The performance of these systems dependsfirmly on the antenna array design. The process of determiningthe parameters of an antenna array to obtain the required antennaradiation pattern is known as pattern synthesis. Pattern nullingmethods that achieve suppression of interfering signals from pre-scribed directions while receiving the desired signal from a chosenlook direction are important in radar, sonar and many communica-tion systems. Pattern nulling methods, including controlling thecomplex weights (both the amplitude and phase), amplitude-only,phase-only, and position-only, have been extensively studied inthe literature (Abu-Al-Nadi, Ismail, & Mismar, 2006; Akdagli & Gu-ney, 2004; Akdagli, Guney, & Karaboga, 2002; Ares, Rodriguez,Villanueva, & Rengarajan, 1999; Babayigit, Akdagli, & Guney,2006; Chung & Haupt, 2000; Er, 1990; Guney & Akdagli, 2001; Gu-ney & Babayigit, 2008; Guney, Babayigit, & Akdagli, 2007, 2008;Guney & Basbug, 2008a, 2008b; Guney, Durmus, & Basbug, 2009;Guney & Onay, 2007a, 2007b, 2010; Hamici & Ismail, 2009; Haupt,1997; Hejres, 2004; Ibrahim, 1991; Ismail & Dawoud, 1991; Karab-oga, Guney, & Akdagli 2002, 2004; Khodier & Christodoulou, 2005;

ll rights reserved.

ax: +90 352 437 57 [email protected] (M.

Vaudon, & Rammal, 2006; Mouhamadou, Vaudon, & Rammal,2006; Pal, Qu, Das, & Suganthan, 2010; Shore, 1984; Singh, Kumar,& Kamal, 2010; Steyskal, Shore, & Haupt 1986; Tennant, Dawoud, &Anderson, 1994; Yang, Gan, & Qing, 2004).

Interference suppression with complex weights is the mosteffective since it has the larger solution alternatives. However, itis also the most expensive considering the cost of the controllersused for phase shifters and variable attenuators for each array ele-ment. Moreover, when the number of elements in the array in-creases, the computational time to find the values of elementamplitudes and phases will also increase. The amplitude-only con-trol uses a set of variable attenuators to adjust the element ampli-tudes. If the element amplitudes have even symmetry about thecenter of the array, the number of attenuators and the computa-tional time are halved. The problem of phase-only and position-only nulling is inherently nonlinear and it cannot be solved directlyby analytical methods without any approximation. By assumingthat the phase or position perturbations are small, the nullingequations can be linearized. The phase-only control utilizes thephase shifters while the position-only control needs a mechanicaldriving system such as servomotors to move the array elements.Phase-only null synthesizing is less complicated and attractivefor the phased arrays since the required controls are available atno extra cost, but it has still common problem.

In this paper, an alternative method based on HSA (Geem, 2009;Geem, Kim, & Loganathan, 2001; Lee & Geem, 2005) is presentedfor linear antenna array pattern synthesis with prescribed nulls.

Page 2: Optimal synthesis of linear antenna arrays using a harmony search algorithm

15456 K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462

The HSA conceptualizes a behavioral phenomenon of musicians inthe improvisation process whereby each musician searches to im-prove the tune in order to produce a better state of harmony. Theharmony in music is analogous to the optimization solution vector,and the musician’s improvisations are analogous to local and glo-bal search schemes in optimization techniques. HSA does not re-quire initial values for the decision variables, thus it may escapelocal optima. Furthermore, instead of a gradient search, the HSAuses a stochastic random search that is based on the harmonymemory considering rate and the pitch adjusting rate so that deriv-ative information is unnecessary. Compared to earlier meta-heuris-tic optimization algorithms, the HSA imposes fewer mathematicalrequirements and can be easily adopted for various types of engi-neering optimization problems (Geem, 2009; Geem, Kim, & Loga-nathan, 2001; Lee and Geem, 2005). The HSA used here employsan adaptive mechanism for pitch adjustment. Therefore, the pitchadjustment mechanism of the proposed HSA is different from thatof the basic HSA presented by Geem (2009).

2. Problem formulation

Let us assume that the array elements are symmetrically placedand conjugate-symmetrically excited about the center of a lineararray. The far field array factor of such an array with an even num-ber of isotropic elements (2N) is defined as

FðhÞ ¼ 2XN

k¼1

ak cos2pk

dk sin hþ dk

� �ð1Þ

where h is the angle from broadside, dk is the distance between po-sition of the kth element and the array center, and ak and dk areamplitude and phase weights of the kth element, respectively. Themain purpose of this work is to find an optimal set of element exci-tation or position values such that obtained pattern achieves the de-sired nulling performance. Therefore, the following cost functionwill be minimized with the use of HSA.

g ¼X90�

h¼�90�WðhÞjFoðhÞ � FdðhÞj ð2Þ

where Fo(h) and Fd(h) are, respectively, the pattern obtained byusing HSA and the desired pattern. The value of W(h) should be se-lected by experience such that the cost function is capable of guid-ing potential solutions to obtain satisfactory array patternperformance with desired properties. The factor W(h) gives the an-tenna designer greater flexibility and control over the actual pat-tern. The trade-off of the relative importance between null depthand sidelobe level can easily be obtained by changing the value ofthe factor W(h).

3. Harmony search algorithm (HSA)

The HSA (Geem, 2009; Geem, Kim, & Loganathan, 2001; Lee andGeem, 2005) is an optimization algorithm which simulates theimprovisation process while musicians searching for a perfect stateof harmony. There is an analogy between music and optimization:each musical instrument corresponds to each decision variable;musical note corresponds to a variable value; and harmony corre-sponds to a solution vector. Just like musicians in a jazz improvisa-tion playing notes randomly or based on experiences to find thebest harmony, variables in the HSA have random values or previ-ously-memorized good values to find optimum solution. A jazzmusician’s improvisation process can be compared to the searchprocess in optimization. The perfectly pleasing harmony is deter-mined by the audio esthetic standard. A musician always intendsto produce a piece of music with perfect harmony, which can be

provided by numerous practices. The pitches of the instrumentsare adjusted after each practice. Each musician improvises thepitches of his/her instrument to produce a better state of harmony.An optimal solution to an optimization problem should be the bestsolution available to the problem under the given objectives andconstraints. Both processes intend to produce the best or optimum(Geem, 2009). The steps of the HSA are described below for anamplitude-only control synthesis problem:

Step 1. Initialize the optimization problem and algorithmparameters.Step 2. Initialize the harmony memory (HM).Step 3. Improvise a new harmony from the HM.Step 4. Update the HM.Step 5. Repeat Steps 3 and 4 until the termination criterion issatisfied.

Step 1. The first step is to specify the problem and initialize theparameter values. The optimization problem is defined as mini-mize g such that Lak 6 ak 6 Uak, where g is the cost function, a isa solution vector consisting of N amplitudes (ak), and Lak and Uak

are the lower and upper bounds of each amplitude, respectively.The HSA parameters that are required to solve the optimizationproblem are also specified in this step: harmony memory size(HMS), harmony memory considering rate (HMCR), pitch adjustingrate (PAR), distance bandwidth (BW), and termination criterion.

Step 2. The second step is harmony memory initialization. TheHSA has memory storage, named harmony memory (HM). The ini-tial HM consists of a certain number of randomly generated solu-tions for the optimization problem under consideration. For an Ndimension amplitude-only synthesis problem, an HM with the sizeof HMS can be represented as follows:

HM ¼

a11; a1

2; . . . ; a1N

a21; a2

2; . . . ; a2N

:

aHMS1 ; aHMS

2 ; . . . ; aHMSN

26664

37775; ð3Þ

where ðaj1; a

j2; . . . ; aj

NÞ, j e(1, 2, . . . , HMS), is a candidate solution.HMS is typically set to be between 10 and 100.

Step 3. A new harmony vector anew ¼ ðanew1 ; anew

2 ; . . . ; anewN Þ is pro-

duced by using the process called improvisation. anewk is produced

based on three rules: memory consideration; pitch adjustment;and random selection. If rand() is less than HMCR, anew

k is producedby the memory consideration; otherwise, anew

k is determined by arandom selection. rand() is a random number between 0 and 1.Each anew

k will undergo a pitch adjustment with a probability ofPAR if it is updated by the memory consideration. The memoryconsideration, pitch adjustment and random selection are givenin the following Eqs. (4)–(6), respectively.

anewk ¼ aj

k where j�ð1;2; . . . ;HMSÞ ð4Þ

anewk ¼ anew

k � randðÞBW ð5Þ

and

anewk ¼ Lak þ randðÞðUak � LakÞ ð6Þ

where BW is an arbitrary distance bandwidth.The HSA used here employs an adaptive mechanism for pitch

adjustment. In this mechanism, the BW value is determined by

BW ¼ Cf ðtÞBWiv ð7Þ

Cf ðt þ 1Þ ¼Cf ðtÞ; LIN þ 5 P t

0:5Cf ðtÞ; LIN þ 5 < t

�ð8Þ

Page 3: Optimal synthesis of linear antenna arrays using a harmony search algorithm

Fig. 1. Structure of harmony memory (figure adopted from Lee and Geem (2005)).

θ (degree)-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Fig. 2. The initial 30-dB Chebyshev pattern.

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-150

-140-130

-120-110

-100

-90-80

-70-60

-50

-40

-30-20-10

0

Fig. 3. Radiation pattern obtained by amplitude-only control with one imposed nullat 14�.

K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462 15457

where t is the iteration number, Cf(t) is an arbitrary positive coeffi-cient, BWiv is the initial value of BW, and LIN is the iteration numberat which the latest improvement was obtained. The value of Cf(1) isdetermined by experience on the HSA. In this paper, the initial valueof Cf(1) is determined as 2 and then iteratively updated using (8).

Step 4. In this step, the harmony memory is updated. If the gen-erated harmony vector anew ¼ ðanew

1 ; anew2 ; . . . ; anew

N Þ yields a betterfitness than that of the worst member in the HM, it replaces thatone. Otherwise, it is eliminated.

Step 5. The steps 3 and 4 are repeated until a termination crite-rion is satisfied.

Fig. 1 shows the structure of the HM that is the core part of theHSA. Consider a jazz trio consisted of saxophone, double bass, andguitar. There exists a certain amount of preferable pitches in eachmusician’s memory: the saxophonist, {Si, Mi, Fa}; the double bass-ist, {Sol, Re, Fa}; and the guitarist, {Do, Mi, La}. If saxophonist ran-domly plays {Fa} out of his memory {Si, Mi, Fa}, the doublebassist {Sol} out of {Sol, Re, Fa}, and the guitarist {La} out of {Do, -Mi, La}, this harmony {Fa, Sol, La} creates another harmony. And ifthis new harmony is better than existing worst harmony in theHM, the new harmony is included in the HM and the worst har-mony is excluded from the HM. Similarly in the antenna array syn-thesis, if first amplitude chooses {0.92124} out of {1.00000,0.97365, 0.92124}, second {0.56120} out of {0.56120, 0.48120, 0.92124}, and third {0.38303} out of {0.91365, 0.97365, 0.38303},these values {0.92124, 0.56120, 0.38303} makes another solutionvector. And if this new vector is better than existing worst vectorin the HM, the new vector is included in the HM and the worst vec-tor is excluded from the HM. This procedure is repeated until fan-tastic harmony is found. In the antenna array synthesis, thisprocedure is repeated until maximum iteration number isachieved.

The steps of HSA for the phase-only control and position-onlycontrol synthesis problems can also be explained like amplitude-only control synthesis problem. The details on the HSA can befound in the literature (Geem, 2009; Geem, Kim, & Loganathan,2001; Lee and Geem, 2005).

4. Numerical results

To show the performance of the proposed HSA for steering sin-gle, multiple and broad nulls with the imposed directions by con-

trolling the amplitude-only, the phase-only, and the position-only,nine examples of a linear array with 20, 22 and 32 isotropic ele-ments have been performed. In the first seven examples, a 30-dBChebyshev pattern given in Fig. 2 for 20 equispaced elements withk/2 interelement spacing is utilized as the initial pattern. In theeighth and ninth examples, a uniform array pattern for 32 and22 equispaced elements with k/2 interelement spacing is utilizedas the initial pattern. In the first five examples, the pattern nullingis achieved by controlling only the element amplitudes. In the sixthand seventh examples, the HSA is used for interference suppres-sion with phase-only control. In the eighth and ninth examples,forming null in the pattern is achieved by controlling only the ele-ment positions. In the last example, the sidelobe suppression withno prescribed nulls is achieved by controlling only the elementpositions for 10-element uniform linear array.

It should be noted that each musical instrument corresponds toeach array element amplitude; musical note corresponds to ampli-tude, phase or position value; and harmony corresponds to solu-tion vector. The simulation results are obtained within 30–60 son a personal computer with a Pentium IV processor running at3 GHz. This is sufficient to produce satisfactory patterns with thedesired performance on the average.

The values of the cost function parameters given in (2) are se-lected as follows:

Page 4: Optimal synthesis of linear antenna arrays using a harmony search algorithm

Number of Iterations0 100 200 300 400 500 600 700 800 900 100011001200130014001500

Cos

t Fun

ctio

n

0

10

20

30

40

50

60HSA used in this paperBasic HSA

Fig. 4. Convergence curve of the nulling pattern achieved by the basic HSA (dottedline) and the HSA used in this paper (solid line).

Table 1NDL, MSL, and DRR of the patterns obtained by HSA, MTACO (Karaboga, Guney, &Akdagli 2002), BA (Guney & Onay, 2007a), BFA (Guney & Basbug, 2008a), and PGSA(Guney, Durmus, & Basbug, 2009) for one null imposed at 14�.

HSA (Fig. 3) MTACO BA BFA PGSA

NDL (dB) �122.5 �88.1 �100.4 �113.6 �118.3MSL (dB) �29.14 �29.1 �28.8 �28.11 �28.03DRR 3.87 4.36 4.09 3.89 3.89

θ (degree)-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

0

Fig. 6. Radiation pattern obtained by amplitude-only control with double imposednull at 14� and 26�.

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

0

Fig. 7. Radiation pattern obtained by amplitude-only control with triple imposednull at 14�, 26�, and 33�.

15458 K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462

FdðhÞ ¼0; for h ¼ hi

Initial pattern; elsewhere

�ð9Þ

and

WðhÞ ¼100; for h ¼ hi

1; elsewhere

�ð10Þ

where hi is the direction of interference.In the first example, the Chebyshev pattern with a single null

imposed at the direction of the second peak from main beam(hi = 14�) is considered. The resultant pattern is illustrated inFig. 3. As it can be seen from this figure, the pattern closely resem-bles to the initial Chebyshev pattern except for the nulling direc-

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-50

-40

-30

-20

-10

0

Fig. 8. Radiation pattern obtained by amplitude-only control with a broad nullsector centered 30� with Dh = 5�.

θ (degree)-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-160-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

0

Fig. 5. Radiation pattern obtained by amplitude-only control with a null depth leveldeeper than that of the first example having one imposed null at 14�.

Page 5: Optimal synthesis of linear antenna arrays using a harmony search algorithm

Table 3NDL and MSL of the patterns obtained by HSA, CLONALG (Guney, Babayigit, & Akdaglı,2008), BFA (Guney & Basbug, 2008b), and BA (Guney & Onay, 2010) for one nullimposed at �10�.

HSA (Fig. 9) CLONALG BFA BA

NDL (dB) �202 �175 �161 �185MSL (dB) �24.23 �24.0 �24.0 �24.22

θ (degree)-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-200-190-180-170-160-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

0

Fig. 9. Radiation pattern obtained by phase-only control with one imposed null at�10�.

K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462 15459

tion. It is also clear from Fig. 3 that the HSA can obtain the patternwith satisfactory null depth level and maximum sidelobe level.

Fig. 4 shows the convergence curves of the basic HSA and HSAused in this paper. It is clear from Fig. 4 that the HSA used in thispaper leads to better convergence and that 1000 iterations areneeded to find the optimal solutions.

In Table 1, the null depth level (NDL), maximum sidelobe level(MSL), and dynamic range ratio (DRR = |amax/amin|) of the patternsobtained by using HSA are compared with the results of the MTA-CO (see Fig. 6 in Karaboga, Guney, and Akdagli (2002)), the BA (seeFig. 4 in Guney and Onay 2007a), the BFA (see Fig. 8 in Guney andBasbug (2008a)), and the PGSA (see Fig. 3 in Guney, Durmus, andBasbug (2009)) for radiation pattern with one null imposed at14�. As can be seen, our results are better than those of MTACO,BA, BFA, and PGSA.

In order to show the flexibility of the HSA, in the second exam-ple, the value of W(h) is selected as given below while the values ofother design parameters are the same as those of the first example.

WðhÞ ¼150; for h ¼ hi

1; elsewhere

�ð11Þ

The corresponding pattern is shown in Fig. 5. The NDL of the patternin Fig. 5 is �160.68 dB while the NDL of the pattern in Fig. 3 is�122.5 dB. But in response to such an improvement of the NDL,the MSL of the pattern in Fig. 5 is �28.83 dB whereas that of the pat-tern in Fig. 3 is �29.14 dB. There is a trade-off between the NDL andMSL; usually, performance cannot be improved significantly for onewithout sacrificing the other. The antenna designer should make atrade-off between the achievable and the desired pattern.

The proposed HSA is also capable of synthesizing the patternwith multiple nulls at any desired directions. For this purpose, inthe third and fourth examples, the Chebyshev patterns with doublenulls imposed at the directions of the second and the fourth peaksfrom main beam (14� and 26�) and with triple nulls imposed at thedirections of the second, fourth and fifth peaks from main beam(14�, 26�, and 33�) are considered. The patterns with multiple nullsare illustrated in Figs. 6 and 7. It can be seen from Figs. 6 and 7 thatall desired nulls are deeper than �120 dB. The results depicted inFigs. 6 and 7 show the ability of the proposed technique for the ar-ray pattern synthesis with multiple nulls steered in the prescribedinterference directions.

In the fifth example, the pattern having a broad null located at30� with Dh = 5� is achieved and is shown in Fig. 8. From the figure,a null depth level deeper than �55 dB is obtained over the spatialregion of interest. This computer simulation example clearly showsthe capacity of HSA to synthesize the array pattern with broad nullimposed at the direction of interference.

The element amplitude values normalized according to centerelements for the patterns illustrated in Figs. 2, 3 and 5–8 are givenin Table 2. It is apparent from Figs. 3 and 5–8 that the patterns aresymmetric with respect to the main beam. This is because the sym-

Table 2The element amplitudes (ak) for the array patterns given in Figs. 2, 3 and 5–8, and the ph

k Initial Chebyshev pattern Computed with the HSA

Fig. 2 (ak) Fig. 3 (ak) Fig. 5 (ak) Fi

±1 1.00000 1.00000 1.00000 1.±2 0.97010 0.99615 0.97821 0.±3 0.91243 0.94121 0.98283 1.±4 0.83102 0.88716 0.87167 0.±5 0.73147 0.77110 0.75479 0.±6 0.62034 0.63527 0.62960 0.±7 0.50461 0.48282 0.50008 0.±8 0.39104 0.33416 0.34250 0.±9 0.28558 0.25863 0.24274 0.±10 0.32561 0.28238 0.27130 0.

metry property of the element amplitudes around the array centerresults in a pattern that is symmetric about the main beam. There-fore, when a null imposed at the one side of the main beam, an im-age null occurs at the other side of the main beam.

In order to show the performance of the proposed method, inthe sixth and seventh examples, forming nulls in the pattern isachieved by controlling only the phase of each array element.The amplitude weights (ak) of these examples are the same asthose of the initial Chebyshev array. In the sixth example, theChebyshev pattern with a single null imposed at the direction ofthe first peak from main beam, which occurs about �10�, is consid-ered. The pattern is then obtained by HSA and illustrated in Fig. 9.In Table 3, the NDL and MSL of the pattern obtained by using HSAare compared with the results of the CLONALG (see Fig. 5 in Guney,Babayigit, and Akdaglı (2008)), the BFA (see Fig. 6 in Guney andBasbug (2008b)), and the BA (see Fig. 6 in Guney and Onay(2010)). It is clear from Table 3 that NDL and MSL of the patternproduced by using HSA are better than those of CLONALG, BFA,and BA.

ases (dk) in degree for the array patterns given in Figs. 9 and 10.

g. 6 (ak) Fig. 7 (ak) Fig. 8 (ak) Fig. 9 (dk) Fig. 10 (dk)

00000 1.00000 1.00000 �0.53377 �1.5048598743 0.97295 0.96846 �0.80616 �3.4356902394 0.97027 0.93047 �2.44447 �3.2506188130 0.91062 0.84284 �4.41257 �3.5120472227 0.73885 0.69631 �6.55748 �3.3468565103 0.58262 0.55467 �9.17606 �7.9940450916 0.53047 0.52939 �11.25927 �13.841838055 0.39214 0.43959 �15.21435 �21.8980120706 0.21401 0.22926 �26.32926 �24.5406326944 0.22650 0.21624 �24.10166 �15.74381

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θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-210-200-190-180-170-160-150-140-130-120-110-100-90-80-70-60-50-40-30-20-10

0

Fig. 10. Radiation pattern obtained by phase-only control with double imposed nullat �10� and 20�.

Table 4NDL and MSL of the patterns obtained by HSA, CLONALG (Guney, Babayigit, & Akdaglı,2008), BFA (Guney & Basbug, 2008b), and BA (Guney & Onay, 2010) for double nullimposed at �10� and 20�.

HSA (Fig. 10) CLONALG BFA BA

NDL at �10� (dB) �202.4 �154.2 �156.3 �190.7NDL at 20� (dB) �201.0 �156.5 187.6 �191.3MSL (dB) �24.48 �24.05 �24.28 �24.4

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Fig. 11. Radiation pattern obtained by position-only control with one imposed nullat 9� for 32-element linear array.

Table 5NDL and MSL of the patterns obtained from PSO and QPM presented by Khodier andChristodoulou (2005), and HSA for 32-element linear array with one null imposed at9�.

HSA (Fig. 11) PSO QPM

NDL (dB) �88.08 �62.06 �34.74MSL (dB) �19.51 �18.71 �17.73

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Fig. 12. Radiation pattern obtained by position-only control with one imposed nullat 9� for 22-element linear array.

15460 K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462

In the seventh example, the pattern having double nulls im-posed at the peaks of the first and the third sidelobes (hi1 = �10�and hi2 = 20�) are obtained by controlling only the phase of arrayelements. The resulting pattern is shown in Fig. 10. In Table 4,the NDL and MSL of the pattern obtained by using HSA are com-pared with the results of the CLONALG (see Fig. 7 in Guney, Babayi-git, & Akdaglı (2008)), the BFA (see Fig. 7 in Guney and Basbug(2008b)), and the BA (see Fig. 7 in Guney and Onay (2010)). Itcan be seen from Table 4 that the results of HSA are better thanthose of CLONALG, BFA, and BA.

The element phase values obtained by HSA for the patterns inFigs. 9 and 10 have odd symmetry about the center of the arrayand are given in Table 2. It is clear from Figs. 9 and 10 that the pat-terns are not symmetric with respect to the main beam. This is aconsequence of the odd symmetry of the element phases aroundthe array center which, coupled with the even symmetry of the ele-ment amplitudes, results in a pattern that is not symmetric aboutthe main beam peak at 0�. It should also be noted that since theelement phases have odd symmetry about the center of the array,the number of phase shifters to be used is 2N, but the number ofcontrollers for the phase shifters is N for an array with 2Nelements.

The eighth example is that of 32-element array with a desirednull at the direction of 9� as in Khodier and Christodoulou (2005)for uniform linear array, which has a uniform spacing of k/2 be-tween neighboring elements. The array pattern obtained by posi-

Table 6The element positions (dk) normalized with respect to k/2 for Fig. 11.

k ±1 ±2 ±3 ±4dk 0.51296 1.24909 2.17223 3.06560

k ±9 ±10 ±11 ±12dk 6.69505 7.87192 8.66334 9.82255

tion-only control using HSA is shown in Fig. 11. In Table 5, theNDL and MSL of the pattern obtained by using HSA are comparedwith those of the patterns obtained by using PSO and QPM in Kho-dier and Christodoulou (2005). It is apparent from Table 5 that NDL

±5 ±6 ±7 ±83.81140 4.82113 5.16274 6.31201

±13 ±14 ±15 ±1611.10770 12.37757 14.16282 15.72252

Page 7: Optimal synthesis of linear antenna arrays using a harmony search algorithm

Table 9MSL of the patterns obtained from PSO and QPM presented by Khodier andChristodoulou (2005), and HSA for 10-element linear array with no prescribed nulls.

HSA (Fig. 13) PSO QPM

MSL (dB) �19.4 �17.4 �17.09

Table 10The element positions (dk) normalized with respect to k/2 for Fig. 13.

k ±1 ±2 ±3 ±4 ±5dk 0.497 1.122 2.088 3.000 4.350

Table 7NDL and MSL of the patterns obtained from TSA, GA, MA, PSO, NSGA-2, MODE, and MOEA/D-DE presented by Pal et al. (2010), and HSA for 22-element linear array with one nullimposed at 9�.

HSA (Fig. 12) TSA GA MA PSO NSGA-2 MODE MOEA/D-DE

NDL (dB) �103.30 �67.94 �54.29 �73.92 �49.94 �58.79 �78.78 �69.64MSL (dB) �23.28 �17.17 �15.73 �18.11 �20.68 �17.27 �18.05 �20.93

Table 8The element positions (dk) normalized with respect to k/2 for Fig. 12.

k ±1 ±2 ±3 ±4 ±5 ±6 ±7 ±8 ±9 ±10 ±11dk 0.39948 0.77596 1.36271 1.7992 2.67384 3.01344 4.06108 4.33007 5.51764 6.4005 7.63163

θ (degree)

-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80

Arra

y Pa

ttern

(dB)

-40

-30

-20

-10

0

Fig. 13. Radiation pattern obtained by position-only control for the sidelobesuppression with no prescribed nulls.

K. Guney, M. Onay / Expert Systems with Applications 38 (2011) 15455–15462 15461

and MSL of the pattern produced by using HSA are better thanthose of PSO and QPM. The element position values calculated bythe HSA for the pattern given in Fig. 11 are given in Table 6.

In the ninth example, the uniform array pattern with a singlenull imposed at the direction of 9� as in Pal et al. (2010) is consid-ered for 22-element array. The pattern obtained by position-onlycontrol is shown in Fig. 12. In Table 7, the NDL and MSL of the pat-tern obtained by using HSA are compared with the NDL and MSL ofthe patterns produced by using TSA, GA, MA, PSO, NSGA-2, MODE,and MOEA/D-DE in Pal et al. (2010). It can be seen from Table 7that NDL and MSL of the pattern obtained by using HSA are betterthan those of TSA, GA, MA, PSO, NSGA-2, MODE, and MOEA/D-DE.The element position values calculated by the HSA for the patterngiven in Fig. 12 are given in Table 8.

The results depicted in Figs. 3 and 5–12 confirm that the HSAproposed in this paper can accurately produce the nulling patternsby controlling the amplitude-only, the phase-only, and the posi-tion-only. From the null depth and the maximum sidelobe levelpoints of view, the performances of the patterns are very good.

Further to inspect the versatility of the HSA on the pattern syn-thesis, in the last example, the sidelobe suppression with no pre-scribed nulls is achieved by controlling only the elementpositions for 10-element uniform array. The resultant pattern thatmakes use of position control by the HSA is illustrated in Fig. 13. InTable 9, the MSL of the pattern obtained by using HSA is comparedwith that of PSO and QPM presented by Khodier and Christodoulou(2005). It is evident from Table 9 that MSL of the pattern producedby using HSA is better than that of PSO and QPM. The element po-sition values calculated by the HSA for the pattern given in Fig. 13are given in Table 10.

5. Conclusion

In this paper, the HSA has been used for interference suppres-sion of a linear antenna array by controlling the amplitude-only,the phase-only, and the position-only. Numerical results show thatthe HSA is capable of synthesizing the array patterns with single,multiple and broad nulls imposed at the directions of interferences.By using the HSA, the maximum sidelobe level and null depth levelof the nulling pattern can easily be controlled. Successful applica-tions show that the proposed technique is simple and easy toimplement. The MSL and NDL results of the patterns obtained byHSA are also compared with the results of 13 optimization algo-rithms available in the literature. It is observed that the results ofHSA are better than those of 13 optimization algorithms. The appli-cations in this paper concern nulling, but a wide variety of otherapplications in antenna theory are possible.

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