The Application of Sparse Antenna Array Synthesis Based on Improved Mind Evolutionary Algorithm

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1 I.J. Itelliget Systems ad Applicatios Published Olie May i MECS ( The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm a Li Hea Zhumadia Power Supply Compay Zhumadia Chia liawd@63.com Abstract Mid Evolutioary Algorithm (MEA imitates the huma mid evolutio by usig similartaxis ad dissimilatio operatios which overcomes the prematurity ad improves searchig efficiecy. But the geeratio of the iitial populatio is blid ad the additio of aturally washed out temporary subpopulatios is radom. This paper improved MEA by itroducig chaos ad differece ito it which brought adequate diversity to the iitial populatio ad saved the excellet gees i the evolutio. The the improved MEA is used i the sythesis of sparse atea arrays. The excellet results of computer simulatio show the advatage of array atea patters sythesis usig the improved MEA. Idex Terms-MEA; DEA; sparse atea array; circular atea array; side lobe level I. ITRODUCTIO With the rapid developmet of the telecommuicatios idustry the electromagetic eviromet of space is icreasigly deterioratig electromagetic iterferece is ehacig ad the quality of commuicatio is decliig. To solve these problems smart atea which has the ability of low side-lobes strog directioal ad ati-iterferece has bee a great deal of cocer. The atea patter sythesis problem becomes a hot research. Atea array sythesis meas i a give atea radiatio patter or atea performace desig atea array elemet umber elemet spacig elemets curret amplitude ad phase distributio. The traditioal umerical optimizatio method based o gradiet searchig techology is ot effective eough for satisfactory results i project because the objective fuctio of atea optimizatio problem is multiparameter oliear odifferetiable ad eve discotiuous. Itelliget Algorithm has become a powerful optimizatio tool for its search does ot deped o the gradiet ad search space iformatio of specific problem. Mid Evolutioary Algorithm (MEA is brought forward by Professor K. M. Xie based o the aalysis of huma mid developmet ad the imitatio the similartaxis ad dissimilatio pheomea i huma society []. But it also has some shortcomigs. or example the additio of aturally washed out temporary subpopulatios is mootoous ad radom; the existig searchig modes are easily to fall ito local covergece. This paper itegrated MEA with Differetial Evolutioary Algorithm (DEA ad proposed Differetial Mid Evolutioary Algorithm (DMEA. It itroduced the thought of differece ito the additio of aturally washed out temporary subpopulatios. DMEA saved the excellet gees i the evolutio ad brought adequate diversity to the populatio. Compared with MEA DMEA is more quickly exactly ad effectively avoidig local covergece. II. THE IMPROVED MID EVOLUTIOARY ALGORITHM(IMEA A. Mid Evolutioary Algorithm (MEA The aggregatio of all idividuals i every geeratio i MEA is called a populatio; a populatio is divided ito some subpopulatios. There are two kids i subpopulatios: superior subpopulatios ad temporary subpopulatios. Superior subpopulatios record the wiers iformatio i global competitio; temporary subpopulatios record the middle process i global competitio. The billboard provides the chace for the commuicatio of the idividuals ad the subpopulatios. There are three basic kids of iformatio i billboard: the sequece umber actio ad score of the idividual or the subpopulatio. The score is the valuatio that the eviromet evaluates to the actio of the idividual or subpopulatio. The idividuals i subpopulatios paste their iformatio o local billboard. Ad the global billboard is used to paste the subpopulatios iformatio. MEA iherited the idea of populatio ad evolutio from GA but i other aspects MEA has big iovatios. MEA divided a populatio ito two kids of subpopulatios superior subpopulatios ad temporary subpopulatios. Superior subpopulatios ote the wiers iformatio i global temporary subpopulatios ote the process of the global competitive. irstly MEA takes similartax operatio o all the subpopulatios ad search optimal value i the local quickly by comparig fitess fuctios. The dissimilatio operatio is used to search i the whole solutio space choice the better idividuals as the ceters ad create ew temporary subpopulatios. Similartax ad dissimilatio both iclude Copyright MECS I.J. Itelliget Systems ad Applicatios

The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm 4 choice operatio. They repeat i tur i the evolutioary process.

The chaos state of improved Tet mappig K x K x K xj igure. ramework of MEA B. Chaos Mappig The chaos pheomeo is the commo pheomeo i the oliear dyamic systems.

The chaos optimizatio algorithm is sesitively to the iitial value easily to jump out the local miimum ad quickly to search out the global optimizatio.

It is ca be expressed as follow: xk xk.5 x k+ = ( ( xk.

2 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm 4 choice operatio. They repeat i tur i the evolutioary process. The idividuals i the subpopulatios post their iformatio o the local bulleti board. The global bulleti board is used to post each subpopulatio s iformatio. G G igure. The chaos state of improved Tet mappig K x K x K xj igure. ramework of MEA B. Chaos Mappig The chaos pheomeo is the commo pheomeo i the oliear dyamic systems. The chaos s behaviors are complex ad similar to the radom process but have the iheret property of regularity. The chaos optimizatio algorithm is sesitively to the iitial value easily to jump out the local miimum ad quickly to search out the global optimizatio. It has the property of global asymptotical covergece [-4]. Tet mappig is a importat model of Chaos Dyamics with uiform probability desity power spectral desity ad ideal related characteristics. It is ca be expressed as follow: xk xk.5 x k+ = ( ( xk.5 < xk < Its probability desity fuctio is: ρ ( x = ( Tet mappig has simple structure ad good ergodic uiformity more suitable for a large umber of data processig sequeces. But there are small iterative cycle ad ustable periodic poit i Tet mappig. It will make the iteratio to the fixed poit. I order to avoid iterates to fixed poit this paper uses the followig method to improve Tet mappig: x( k /.4 x( k.4 x ( k + = (3 ( x( k /.6 x( k >.4 We ca see that from ig. the poits geerated by improved Tet mappig are scattered uiformly i [.]. It overcomes its ow lacks such as small cycle ad istability cycle poits. G x C. Differetial Mid Evolutioary Algorithm(DMEA I dissimilatio operatio of MEA if the mature temporary subpopulatio s score is lower tha ay subpopulatio it will be abado. Ad the search exploits ew poit to create ew temporary subpopulatio. But the creatio of ew temporary subpopulatio is radom ad blid lost the excellet gees i the evolutio. This paper improved MEA for this shortcomig. DEA is proposed by Raier Stor ad Keth Price accordig to the evolutioary law of ature [5]. It is a real codig algorithm with simple cofiguratio quick covergece rate ad strog stability. There are three kids of operatios i DEA: mutatio crossover ad selectio operatios. Ad mutatio operatio is the key operatio i DEA. I mutatio operatio firstly select ay two idividuals i the populatio ad compute their differece the weight sum the differece ad aother idividual to create ew idividual. Let X i G i =... is the idividual i geeratio G G is the geeratio time is the populatio size. The ew idividual ca be created by followig: V i G + = df ( X r G X r G + X r3 G (4 where r r ad r3 are three differet itegers selected radomly i [ ]. df [ ] is the mutatio factor to cotrol the scalig of X. X r G r G D. The Improved Mid Evolutioary Algorithm(IMEA MEA simulated the similartaxis ad dissimilatio pheomeo i huma society ad resolved the problem of prematurity ad low covergece speed of traditioal Itelliget Algorithm to a certai extet. But it also has some defects. or istace the geeratio of the iitial populatio is blid ad the additio of aturally washed out temporary subpopulatios is radom. This paper itroduced improved Tet mappig ad the mutatio operatio of DEA ito MEA ad proposed the improved MEA which brought adequate diversity to the iitial populatio ad saved the excellet gees i the evolutio. The improved MEA is described as followig: Step Set evolutioary parameters: populatio size subpopulatio size ad coditios for ed; Copyright MECS I.J. Itelliget Systems ad Applicatios

3 4 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm Step Iitializatio: scatter idividuals i terms of (3 i the whole solutio space; Step3 Similartax: idividuals are produced by ormal distributio with variace aroud each wier ad the idividual with highest score is the ew wier replacig the old oe i followig steps; Step4 Dissimilatio: realize global optimizatio some with lower score are washed out ad replaced by ew oes scattered accordig to (4 i the solutio space; Step5 Coditios for ed: if the ed coditios are filled tur to step6; else repeat step3 ad step 4; Step6 Output evolutioary result algorithm eds. To assess the performace of IMEA this paper selects three typical fuctios that commoly used to test optimizatio algorithm to experimet which has multipeaks o-raised ad so o. The objective fuctio is 3 = = ( x x + ( x.48 x x.48 4 ( 4.x + x 3 x x x + ( 4 + 4x + - x x = si ( x + x.5 [. +.( x + x ] - x x x.5 is Rosebrock fuctio which has a global miimum. The global miimum positio is ( ad the value is. It is used to test the premature covergece of the algorithm. is Camel fuctio which has six local miimums. The two global miimums are ad their positios are ( ad ( is Schaffer fuctio which has ifiite local maximum. The oly oe global is maximum ad its positio is (. The parameters of the experimet are: MEA ad IMEA have the same large iitial populatios the populatio size is 3 the subpopulatio size is 8 the temporary subpopulatio is ad termiatio time is. TABLE I. igure 4. The covergece curve of Camel igure 5. The covergece curve of Schaffer COMPARISO O THREE UCTIOS OPTIMIZATIO RESULTS fuctio algorithm X X solutio MEA e-6 Rosebrock IMEA e-7 GA e-3 MEA Camel IMEA GA MEA Shaffer IMEA GA rom ig.4-6 ad Table it ca be see i the three kids of fuctios optimizatio MEA ad IMEA are effective. They are both able to fid the optimal solutio. rom the solutios accuracy the optimal values accuracy gaied by IMEA is higher tha those gaied by MEA ad the solutios positios are more accurately. This is because i differet stages of the optimizatio process adoptig differet chaotic sequeces to geerate subpopulatios. It s ot oly to esure the uiform distributio of the subpopulatios ad also improves the quality of the idividuals. Thereby ehace the covergece of the algorithm ad the accuracy of the optimal value. igure 3. The covergece curve of Rosebrock III. SPARSE ATEA ARRAY SYTHESIS Atea array is a kid of atea cosisted of some array elemets to obtai special radiatio characteristic. Copyright MECS I.J. Itelliget Systems ad Applicatios

4 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm 43 Accordig to the geometry structure it ca be divided ito several categories: liear atea array rectagle atea array ad circular atea array. A. Uiform Liear Atea Array r r p r igure 6. The model of uiform liear atea or a liear atea array with give elemets ad spacig arraged i axis as it shows i ig.6 its patter formula is: = I exp[ j( kdcosθ + ϕ ] θ (5 = where I is the th elemet s amplitude; ϕ is the th elemet s phase; θ is the agle betwee the array axis ad the ray; d is the space betwee the adjacet elemets; k = π λ is wave umber. ϕ Let the atea array s mai-lobe poit at θ the = kd cosθ (5 ca be writte as followig: = I exp[ j( kd( cosθ cosθ ] θ (6 = B. Uiform Circular Atea Array Atea array is a kid of atea cosisted of some array elemets to obtai special radiatio characteristic for example liear atea array rectagle atea array ad circular atea array. Compared with liear atea array ad rectagle atea array uiform circular atea array has bigger scaig rage reached to 36 ad couteractig mutual couplig effect with the symmetric uiform distributio. As ig. 7 shows there are elemets distributig o a circle R is the circle s radius. θ [ π ] is the agle betwee sigal icidet directio ad axis z φ [ π ] is the agle betwee the directio that the sigal icidet directio projected o the xy plae. r = ( siθ cosφsiθ siφcosθ is the directio vector r ad p = ( cosφsiφ is the positio vector. φ = π / is the th elemet s azimuth agle. is: igure 7. Uiform circular atea array The uiform circular atea array s patter fuctio = [ siθ ] j β kr cos ( φ φ θ φ = I e where I is the th elemet s amplitude ad β is the th elemet s phase. If let the atea array s mai lobe poit at ( θ φ the β = krcos( φ φ siθ (7 ca be writte as followig: jkr cos ( φ φ si θ cos( φ φ θ φ; θ φ = I e (7 [ si θ ] (8 = If oly cosiderig the atea patter o the xy plae the θ = θ = 9 the patter fuctio ca be writte as followig: [ φ φ ] jkr cos ( φ φ cos( φ;φ = I e = C. Circular Sparse Atea Array I the egieerig applicatio of atea array arrow mai lobe is ofte used to improve spatial resolutio. To gai arrow mai lobe i a full matrix atea array we have to elarge the atea array s aperture. But it ot oly adds the cost but also icreases the complexity of the atea system. Distributig elemets sparsely is aother method to gai mai lobe which saved the cost improved the spatial resolutio ad avoided gratig lobe. Circular sparse atea array has the big scaig rage ad high precisio characteristics of circular array ad we ca gai required atea patter by oly optimizig the atea array s distributio. The patter fuctio is decided by the elemets positio parameter whe optimize the atea array s distributio: [ cos θ ϕ cos θ ϕ ] jkr ( ( θ = e = (9 ( where θ is the mai lobe s positio ϕ is the th elemet s azimuth agle R = mλ is the circular array s Copyright MECS I.J. Itelliget Systems ad Applicatios

8 is the patter of 4 elemets circular sparse array ad elemets circular sparse array.

5 44 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm radius k = π λ is wave umber. I this paper m= ( ca be writte as: 4 [ ] 4π ( cosθ cosθ = j + e ( j π cos ( θ ϕ cos( θ ϕ θ e = or a uiform circular full array its elemet umber is 4. ig. 8 is the patter of 4 elemets circular sparse array ad elemets circular sparse array. d x + dc dc d x dc dc D 3 + X = = = + = X + DC M M M d x + ( dc ( dc (5 or a liear sparse array with L = 9λ = d c = λ LZ = 9λ D c = [ λ λ3λ λ5λ 3λ7λ 4λ9λ ]. We request the mai lobe s width is 5 ad the highest side lobe level is -5dB. Differet methods GA ad IMEA were ivestigated ad compared with simulatio solutios i order to assess the effectiveess ad the flexibility of IMEA. The experimet parameters of GA are: p c =.6 p m =.. The experimet parameters of IMEA are: df=.5 the subpopulatio size is 8 the temporary subpopulatio is. I two algorithms the evolutioary geeratio is ad the populatio size is 3. The simulatio results are as followig: igure 8. The circular radom sparse atea array As ig. 8 shows the mai lobe s width of elemets array is similar to 4 elemets array. It attests a circular sparse array with fixed radius ca gai similar mai lobe s width with less elemet umber savig the cost. But the highest side lobe level of circular sparse array is high the followig optimizatio aimed at reducig the highest side lobe level of the circular sparse array. IV. SIMULATIO RESULTS A. Liear Sparse Atea Array A liear sparse atea array i fact is a uequal liear array with uequal space betwee the adjacet elemets. Its patter fuctio ca be writte as followig: ( θ = exp( jkd cosθ = ( where d is the space betwee the first elemet ad the th elemet. Before optimize the liear sparse array we ca pretreat the positio variables to reduce the search rage ad improve the optimizatio efficiecy [6]. Let the first elemet s positio variable is d = d c is the positio variable s differece betwee two adjacet elemets: mi { di d j} dc j i (3 The the left solutio space is: LZ = L d c d c ( 3 = L ( d c ( 4 Scatter - idividuals composig iitial populatio i the left solutio space ad sort them ascedig to T compose X = [ x x L x ]. The the - positio variables i the liear sparse array ca be writte as: igure 9. Optimizatio results of elemets sparse array for low SLL TABLE II. OPTIMIZATIO RESULTS O LIEAR SPARSE ARRAY ELEMETS POSITIOS GA IMEA X D X D.44λ.744λ.376λ.876λ.783λ.783λ.6793λ.6793λ λ.3357λ.95λ.795λ 4.96λ.96λ.586λ 3.586λ 5.97λ 3.47λ 3.3λ 5.53λ 6.6λ 4.6λ 3.885λ 6.885λ 7.47λ 4.547λ 3.77λ 7.7λ 8.75λ 5.75λ 4.85λ 8.85λ rom ig.9 it ca be see that the highest side lobe level of GA is -db it does ot satisfy the experimet request while the highest side lobe level of IMEA reached -5dB. rom Table we ca see that the last elemet s positio variable is 5.73λ i GA while the last positio variable is 8.83λ i IMEA whe the solutio space is [ 8.5λ]. It shows the distributio optimized by Copyright MECS I.J. Itelliget Systems ad Applicatios

The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm 45 GA does ot give full play to the advatage of sparse atea array. It is most likely to get ito local optimum.

6 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm 45 GA does ot give full play to the advatage of sparse atea array. It is most likely to get ito local optimum. B. Circular Sparse Atea Array or a circular sparse array we ca pre-treat the positio variables to reduce the search rage ad improve the optimizatio efficiecy. The first elemet s azimuth agle is ϕ = ϕ c is the azimuth agle s differece betwee two adjacet elemets: mi { ϕi ϕ j} ϕc j i (6 The the left solutio space is: LZ = π ϕc (7 Scatter - idividuals composig iitial populatio i the left solutio space ad sort them ascedig to T compose X = [ x x L x ]. The the - positio variables i the circular sparse array ca be writte as: ϕ x + ϕc ϕ c ϕ x ϕc ϕc ψ 3 + X (8 = = = + = X + ψ C M M M ϕ x + ( ϕc ( ϕc or a circular sparse atea array with R = λ = ϕ c = π θ = the LZ = π ( π = π. We request the mai lobe s width is ad the highest side lobe level is -db. Differet methods GA ad IMEA were ivestigated ad compared with simulatio solutios i order to assess the effectiveess ad the flexibility of IMEA. The experimet parameters of GA are: p c =.6 p m =.. The experimet parameters of IMEA are: df=.5 the subpopulatio size is 8 the temporary subpopulatio is. I two algorithms the evolutioary geeratio is ad the populatio size is 3. The simulatio results are as followig: igure. The polar patter gaied by IMEA igure. The polar patter gaied by GA ig. ad ig. are the patters of the circular sparse array i polar coordiate optimized by IMEA ad GA. We ca see that there are oly three side lobes whose ormalized values exceeded. i ig. while there are eight side lobes whose ormalized values exceeded. i ig.. rom Table 3 it ca be see that the last elemet s azimuth agle is 8.64 i GA ad the last azimuth agle is 345. i IMEA while the solutio space is [ 36 ]. It shows the distributio optimized by GA does ot give full play to the advatage of sparse atea array. It is most likely to get ito local optimum. TABLE III. OPTIMIZATIO RESULTS O CIRCULAR SPARSE ARRAY ELEMETS POSITIOS igure. Optimizatio results of elemets sparse circular array for low SLL rom ig. it ca be see that the highest side lobe level of GA has bee reduced 6dB compared with ig. 9. but it does ot reach -db while the highest side lobe level of IMEA reached the experimet request. GA IMEA ψ c ψ ψ c ψ Copyright MECS I.J. Itelliget Systems ad Applicatios

7 46 The Applicatio of Sparse Atea Array Sythesis Based o Improved Mid Evolutioary Algorithm V. COCLUSIO Poit at the defects of MEA - the geeratio of the iitial populatio is blid ad the additio of aturally washed out temporary subpopulatios is radom. We itroduced improved Tet mappig ad mutatio operatio of DEA ito MEA ad proposed the improved MEA which brought adequate diversity to the iitial populatio ad saved the excellet gees i the evolutio. The IMEA is adopted to optimize the liear sparse array ad circular sparse array. There is a good agreemet betwee the desired ad calculated radiatio patters. Ad the optimizig result is better tha that of GA. REERECES [] K. M. Xie Y. G. Du ad C. Y. Su Applicatio of the Mid- Evolutio-Based Machie Learig i Mixture-Ratio Calculatio of Raw Materials Cemet Proceedigs of the 3rd World Cogress o Itelliget Cotrol ad Automatio pp (i Chiese [] K. Keiji Stability exteded delayed-feedback cotrol for discrete time chaotic systems IEEE Tras. O Circuits ad Systems vol. 46 pp Oct 999. [3] L. Che G. R. Che uzzy modelig predictio ad cotrol of ucertai chaotic systems based o time series IEEE Tras. Circuits ad Systems-I: udametal Theory ad Applicatio vol. 47 pp Oct. [4] R. L. Devaey A Itroductio to Chaotic Dyamical Systems Addiso-Wesley d ed 989 ew York. [5] R. Stor ad K. Price Differetial Evolutio - A Simple ad Efficiet Adaptive Scheme for Global Optimizatio over Cotiuous Spaces Techical Report Iteratioal Computer Sciece Istitute vol. 8 pp. -5 Aug 995. [6] K. S. Che C. L. Ha ad Z. S. He A Sythesis Techique for Liear Sparse Arrays with Optimizatio Costrait of Miimum Elemet Spacig Chiese Joural of Radio Sciece vol. pp. 7-3 eb 7. (i Chiese [7] V. Murio A. Trucco ad C. S. Regazzoi Sythesis of Uequally Spaced Arrays by Simulated Aealig IEEE Tras. Ateas Sigal Processig vol. 44. pp [8] B. P. Kumar ad G. R. Braer Desig of Uequally Spaced Arrays for Performace Improvemet IEEE Tras. o Atea ad Propagatio. vol. 3 pp [9] K. S. Che Z. S. He ad C. L. Ha Desig of Uequally Spaced Arrays for Performace Improvemet Sesor Array ad Multichael Sigal Processig IEEE Workshop 6 pp (i Chiese [] E. M. Thomas ad M. P. Krishma Patter Sythesis of Coformal Arrays for Airbore Vehicles IEEE Aerospace Coferece Proceedig vol. pp [] G. Caille E. Vourch ad M. J.Marti Coformal Array Atea for Observatio platforms i low Earth Orbit IEEE Atea ad Propagatio Magazie vol. 44 pp a Li was bor i Chia i May 985. She got her Master of Egieerig i electroic circuit ad system from Taiyua Uiversity of Techology Chia i. She is ow workig as egieer with Hea Zhumadia Power Supply Compay. Her fields of iterest iclude microwave techology ateea ad computor sciese. Copyright MECS I.J. Itelliget Systems ad Applicatios

Application of Improved Genetic Algorithm to Two-side Assembly Line Balancing

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