Multiple Beam Array Pattern Synthesis by Amplitude and Phase with the Use of a Modified Particle Swarm Optimisation Algorithm

Transcription

1 SETIT 29 5 th Internatonal Conference: Scences of Electronc, Technologes of Informaton and Telecommuncatons March 22-26, 29 TUNISIA Multple Beam Array Pattern Synthess by Ampltude and wth the Use of a Modfed Partcle Swarm Optmsaton Algorthm Hchem CHAKER*, S Mohamed MERIAH* and Feth Tark BENDIMERAD* *Telecom Laboratory Department of Electronc Engneerng Unversty of Tlemcen, ALGERIA mh_chaker25@yahoo.fr merah_m@yahoo.fr ftbendmerad@mal.unv-tlemcem.dz Abstract: Multple beam antenna arrays have mportant applcatons n communcatons and radar. In ths paper, an adaptve partcle swarm optmzaton algorthm s proposed to synthess multple beam antenna arrays. These modern heurstc algorthms are consered as practcal tools for nonlnear optmzaton problems, whch do not requre that the objectve functon to be dfferentable or be contnuous. The partcle swarm optmzaton algorthm s an evolutonary computaton technque, whch s nspred by socal behavour of swarms. The ampltude dstrbuton common to all of the exctaton dstrbutons can be optmzed jontly wth the phase dstrbutons. The examples are smulated to demonstrate the effectveness and the desgn flexblty of adaptve partcle swarm optmzaton algorthm n the framework of electromagnetc synthess of lnear arrays. The proposed method allows the successful synthess of multple beam pattern lnear antenna arrays. The partcle swarm-optmzed reconfgurable array desgns were found to be resstant to smulated varatons n the exctaton coeffcents. Results have show good agreement between the desred specfcatons and the syntheszed one. Ths demonstrates the effectveness of the proposed procedure. Key words: Adaptve partcle swarm optmzaton, antenna arrays, multple beam, synthess. INTRODUCTION Reconfgurable antenna arrays that are capable of radatng wth multple power patterns usng a sngle power dvng network are desrable for many applcatons. In the doman of antenna arrays, several methods of synthess exst such as stochastc and determnst method [GHA ] [BAH 2]. Conserng the dversty of ams searched for by users, we don't fnd a general method of synthess whch s applcable synthess to all cases, but rather an mportant number of methods to every type of problem. Ths dversty of solutons can be exploted to consttute a useful tool for a general approach of synthess of a multple beam array. Recently, evolutonary algorthms, such as genetc algorthms and smulated annealng, have been successfully appled to antenna array synthess problems lke null steerng n phased arrays by postonal perturbatons [ALP 3], reconfgurable phase dfferentated array desgn [GIE 4] [ROB 5], the corrugated horn antenna desgn [ROB 6]. Recently, adaptve partcle swarm optmzaton algorthm s proposed for solvng global numercal optmzaton problem. The result shows that the adaptve partcle swarm optmzaton algorthm can fnd a hgh qualty soluton even for a very hgh dmensonal problem. In ths paper, we are nterest to present the adaptve swarm optmzaton method that wll be appled to perform multple beam antennas array synthess. A bg flexblty between features of the antennas array: ampltude and phase of feedng, undulaton doman, and secondary lobe level s ntroduced.. Problem formulaton An array can form multple narrow beams towards dfferent drectons. For example, suppose t s desred to form two or three beams towards the steerng angles, 2, and 3. The desgn of a lnear array antenna s based on fndng both magntudes and phases exctaton that can generate the desred patterns. We conser a lnear array of 2N sotropc antenna elements, whch are assumed, uncoupled, symmetrcally and equally spaced wth half wavelength. Its array pattern can be descrbed as - -

2 SETIT29 follows [MER 7]: N 2 F( ) 2a k cos d k sn( ) k k Where N = element number; = wavelength; k = phases of the elements ( 8 8 ); a k d k = ampltude of the elements; = dstance between poston of th element and the array center; = scannng angle. () In order to generate a beam pattern fulfllng some constrants, SLL lower than a fxed threshold or reproducng a desred shape, an array confguraton must be syntheszed. Frst of all, t s necessary to defne the objectve functon that measures the dfference between desred and syntheszed beam pattern. Let us defne a functon called ftness functon as follows: Ftness Max (2) The ftness functon defned n (2) represents the general form for antenna pattern synthess, and the desred mask shape can be defned as follow and plotted n fgure : - Of three prncpal lobes, we defne all the angular zones T to T5. - Of two prncpal lobes, we have T=T2=T 3=T4=T5=T6. -9 f ) () (2) (3) (4) (5) (6) (7) (8) (9) ()() (2) (3) (4) (5) Fgure. The shaped mult beam mask. d ( f ( ) d 2. Adaptve partcle swarm optmzaton +9 Modern heurstc algorthms are consered as practcal tools for nonlnear optmzaton problems, whch do not requre that the objectve functon to be dfferentable or be contnuous. The partcle swarm optmzaton algorthm [XIE 8] s an evolutonary computaton technque, whch s nspred by socal behavour of swarms. The partcle swarm optmzaton algorthm s smlar to the other evolutonary algorthms n that the system s ntalzed wth a populaton of random solutons. Each potental soluton, call partcles, fles n the D-dmensonal problem space wth a velocty whch s dynamcally adjusted accordng to the flyng experences of ts own and ts colleagues. The locaton of the th partcle s represented as X ( x,..., x,... x D ). The best prevous poston (whch gvng the best ftness value) of the th partcle s recorded and represented as P p,..., p,..., p ), whch s also ( D called pbest. The ndex of the best pbest among all the partcles s represented by the symbol g. the locaton P s also called gbest. The velocty for the th partcle g s represented as V ( v,..., v,..., v D ). The partcle swarm optmzaton conssts of, at each tme step, changng the velocty and locaton of each partcle toward ts pbest and gbest locatons accordng to the equatons (3) and (4) respectvely: v w v c rand() ( p x ) c2 rand() ( p x gd (3) x x v (4) Where w s nerta weght, c and c 2 are acceleraton constants [EBE 9], and rand() s a random functon n the range [ ]. For equaton 3, the frst part represents the nerta of prevous velocty; the second part s the cognton part, whch represents the prvate thnkng by tself; the thrd part s the socal part, whch represents the cooperaton among the partcle [Ken ]. V s clamped to a maxmum velocty Vmax ( vmax,,..., vmax, d,..., vmax, D). V max determnes the resoluton wth whch regons between the present and the target poston are searched [EBE 9]. The process for mplementaton PSO s as follows: a). Set current teraton generaton G c. Intalze a populaton whch ncludng m partcles, for the th partcle, t has random locaton X n specfed space and for the d th dmenson of V, v rand vmax, d, 2() where rand2 () s a random value n the range of [- ]; b). Evaluate the ftness for each partcle; c). Compare the evaluated ftness value of each partcle wth ts pbest.f the current value s better than pbest, and then set the current locaton as the pbest locaton. Furthermore, f current value s better than gbest, then reset gbest to the current ndex n partcle array; d). change the velocty and locaton of the partcle accordng to the equatons () and (2), respectvely; e). G c G c, loop to step b) untl a stop crteron s met, usually a suffcently good ftness value or G c s acheve a predefned maxmum generaton G. The parameters of partcle swarm optmzaton ncludes: number of partcles m, nerta weght w, max ) - 2 -

3 SETIT29 acceleraton constants c and c 2, maxmum velocty V max. As evoluton goes on, the swarm mght undergo an undesred process of dversty loss. Some partcles becomes nactvely whle lost both the global and local search capablty n the next generatons. For a partcle, the loss of global search capablty means that t wll be only flyng wthn a qute small space, whch wll be occurs when ts locaton and pbest s close to gbest (f the gbest has not sgnfcant change) and ts velocty s close to zero for all dmensons; the loss of local search capablty means that the possble flyng cannot lead perceptble effect on ts ftness. From the theory of self-organzaton [NIC ], f the system s gong to be n equlbrum, the evoluton process wll be stagnated. If gbest s located n a local optmum, then the swarm becomes premature convergence as all the partcles become nactvely. To stmulate the swarm wth sustanable development, the nactve partcle should be replaced by a fresh one adaptvely so as to keepng the nonlnear relatons of feedback n equaton (3) effcently by mantanng the socal dversty of swarm. However t s hard to entfy the nactve partcles, snce the local search capablty of a partcle s hghly depended on the specfc locaton n the complex ftness landscape for dfferent problems. Fortunately, the precson requrement for ftness value s more easly to de deced for specfed problem. The adaptve partcle swarm optmzaton s executed by substtutng the step d) of standard partcle swarm optmzaton process, as the pseudo code of adaptve partcle swarm optmzaton that s shown n fgure 2. F s the ftness of the th partcle, gbest F s the ftness of gbest. F f F, F ), where ( gbest f (x) s an error functon. The s a predefned crtcal constant accordng to the precson requrement. T c s the count constant. The replace () functon s employed to replace the th partcle, where the X and V s rentalzed by followng the process n step a) of standard partcle swarm optmzaton, and ts pbest s equal to X. The array smlar Count s employed to store the counts whch are satsfyng the condton F n successvely for the th partcle whch s not gbest. The nactve partcle s natural to satsfy the replace condton; however, f the partcle s not nactvely, t has less chance to be replaced as T c ncreases. smlar Count new ntm; // at ntalzaton stage nt // Next codes employed to replacestep d) // n stan dard PSO process For ( ; m; ) IF( g & & F IF ( smlarcount Fgure 2. Inserted pseudo code of adaptve PSO For APSO, F s set as a relatve error functon, whch s F F )/ Mn( abs( F ), abs( F )), where -9 THEN smlarcount[ ] ; -6 ( gbest gbest abs(x) gets the absolute value of x, Mn ( x, x 2) gets the mnmum value between x and x 2.The crtcal constant s set as e-4, and the count constant T c s set as Results -3 ELSE smlarcount ; THEN replace( the ) T ) partcle); ELSE execute( step d) n s tandard PSO th c deg -3 // for each partcle // add // reset // predefnedcount In ths secton, we conser an array of sotropc elements spaced.5 λ apart n order to generate two beams towards the steerng angles -2, 4 wth ampltude-phase synthess. Because of symmetry, here only fve phases and fve ampltudes are to be optmzed. Acceptable se lobe level (SLL) should be equal to or less than the desred value - 25dB. Fgures 3 shows the normalzed absolute power pattern n db, the maxmum se lobes level reach db, there s a very good agreement between desred and obtaned results. The optmzed exctaton magntudes and phases (degree) elements s shown n the fgure 4, and values are presented n the table. For desgn specfcatons of ampltude-phase synthess, the adaptve partcle swarm optmzaton s run for 5 generatons db Fgure 3. Mult-beam arrays wth maxmum selobes level equal to db n polar coordnates

4 Exctaton Magntude Magntude SETIT29-3 deg db Fgure 5. Mult-beam arrays wth maxmum selobes level equal to db n polar coordnates Fgure 4. The element exctaton requred to acheve the desred pattern. 5 5 Element N Magntude Table. Ampltude and phase dstrbutons In order to llustrate the capabltes of adaptve partcle swarm optmzaton for solvng the array confguraton for desred pattern synthess by varyng the ampltude and phase of the elements feed, we ntroduce the case of an array wth 2 equspaced sotropc elements wth λ/2 nterelement spacng, whch s supposed to generate three beam steered towards the three angles θ =-3, θ 2 = and θ 3 =2, fgures 5 shows the output pattern, the relatve ampltudes of the three beams were equal to unty, after 49 teratons maxmum se lobes level of dB was acheved ampltude and phase dstrbutons are shown n fgure 6, and presented n the table Fgure 6. The element exctaton requred to acheve the desred pattern. Wth the same array as the second case, and the same type of synthess, we present synthess results of multple beam array as ndcated n the fgures 7. For desgn specfcatons of ampltude-phase synthess, adaptve partcle swarm optmzaton s run for generatons; the elements exctaton requred to acheve the desred pattern are shown n fgure 8. Se lobes level obtaned for desred pattern s -2dB. Smulated results are shown n table deg db Fgure 7. Mult-beam arrays wth maxmum selobes level equal to 2 db selobes n polar coordnates

5 Exctaton Magntude SETIT Concluson In ths paper, an adaptve partcle swarm algorthm s developed. The proposed method allows the successful synthess of multple beam pattern lnear antenna arrays. The partcle swarm-optmzed reconfgurable array desgns were found to be resstant to smulated varatons n the exctaton coeffcents. Results have show good agreement between the desred specfcatons and the syntheszed one. Ths demonstrates the effectveness of the proposed procedure. REFERENCES Fgure 8. The element exctaton requred to acheve the desred pattern. Element N Ampltude Fg.8 Ampltude Fg Table 2. Ampltude and phase dstrbutons [ALP 3] A. ALPHONES and V. PASSOUPATHI, "Null steerng n phased arrays by postonal perturbatons: a genetc algorthm approach", Proc. IEEE d arrays, 996, pp [GIE 4] D. GIES and Y. RAHMAT-SAMII, "Partcle swarm optmzaton for reconfgurable phase dfferentated array desgn", Mcrowave and Optcal Technol. Lett., Vol.38, No.3, August 23, pp [NIC 77] G. NICOLIS, I. PRIGOGINE, "Self-organzaton n no equlbrum systems: from dsspatve systems to order through fluctuatons", John Wley, NY, 977. [BAH 2] I. J. BAHL, P. BHARTIA, "Mcrostrp antennas", Artech House, USA, 98. [KEN ] J. KENNEDY, "The partcle swarm: socal adaptaton of knowledge", IEEE Int. Conf. on Evolutonary Computaton, 997: [ROB 6] J. ROBINSON, S. SINTON, and Y. RAHMAT- SAMII, "Partcle swarm, genetc algorthm, and ther hybrs: Optmzaton of a profled corrugated horn antenna", IEEE AP-S, 22, pp [ROB 5] J. ROBINSON, and Y. RAHMAT-SAMII, "Partcle swarm optmzaton n Electromagnetc", IEEE Trans. Antennas & Propagaton, Vol.52, No.2, Feb.24, pp [EBE 9] R. EBERHART, Y. SHI, "Partcle swarm optmzaton: developments, applcatons and resources", IEEE Int. Conf. on Evolutonary Computaton, 2: [GHA ] R. GHAYOULA, M. TRAII and A. GHARSALLAH, "Applcaton of the Neural Network to the Synthess of Multple beam Antennas Arrays". Transacton on engneerng, computng and technology volume 4 august 26 ISSN [MER 7] S.M. MERIAH, F.T. BENDIMERAD, "Low level se lobes n thnned antennas arrays by swtched supply law", Optmzaton by genetc algorthm. Annals of telecommuncatons. 55, N -2, 2, pp [XIE 8] X-F XIE, W-J ZHANG and Z-L YANG, "Adaptve Partcle Swarm Optmzaton on Indvual Level", Internatonal Conference on Sgnal Processng (ICSP), Bejng, Chna, 22: