Controlled Random Search Optimization For Linear Antenna Arrays

Transcription

1 L. MERAD, F. T. BENDIMERAD, S. M. MERIAH, CONTROLLED RANDOM SEARCH OPTIMIZATION FOR LINEAR Controlled Random Search Optmzaton For Lnear Antenna Arrays Lotf MERAD, Feth Tar BENDIMERAD, Sd Mohammed MERIAH Telecommuncatons Laboratory, Electroncs Departmen Faculty of Engneerng Unversty of Abou-Ber Belaïd Tlemcen, PO 3, Chetouane, 3 Tlemcen, Algera merad77@yahoo.fr Abstract. An optmzaton problem for desgnng nonunformly spaced, lnear antenna s arrays s formulated and solved by means of a controlled random search algorthm. The proposed teratve method ams at a lnear array and the optmzaton of element postons and exctatons coeffcents by mzng the sde-lobes level and respectng a beam pattern shape. Selected examples are ncluded, whch demonstrate the effectveness and the desgn flexblty of the proposed method n the framewor of the electromagnetc synthess of lnear antenna arrays. Keywords Antenna array, pattern shape, synthess, optmzaton, controlled random search.. Introducton An antenna array wth certan radaton characterstcs s often ased to be desgned. Necessarly, the nulls have to be n a certan drecton [], or the man lobe has to be drected n a certan drecton; also other requrements for the drecton and the level of the sde lobes [] mght be stated. The global synthess of antenna arrays that generate a desred radaton pattern s a hghly nonlnear optmzaton problem. Many analytcal methods have been proposed for ts soluton. Examples of analytcal technques nclude the well-nown Taylor method and the Chebyshev method [3]. In many applcatons, the synthess problem of an antenna array consst of fndng an approprate set of ampltude and phase weghts that wll yeld the desred far-feld pattern wth an equally spaced lnear array [4]. However, t s well nown that the antenna performance related to the beam wdth and sde lobes levels can be mproved by choosng both the best poston and the best set of the ampltude and phase for each element of an unequally spaced array [5]. The paper s amed to present a modular method, based on a controlled random search (CRS) algorthm, whch s able to smultaneously optmze the exctaton coeffcents (ampltude and phase) and the best poston, accordng to dfferent constrants, such as sde lobes pea mzaton, and beam pattern (BP) shape modelng. Although the CRS has proven to be very robust n many applcatons [6], [7], t s somewhat less nown to the engneerng communty. The CRS s characterstc by fndng good near-optmal solutons early n the optmzaton run. The CRS does not use dervatves, and s also ndependent on the complexty of the obectve functon under consderaton. Two examples are used to demonstrate the effectveness of the proposed controlled random search based procedure for the antenna array optmzaton. The CRSsmulated results are also compared wth those obtaned by genetc algorthms n [4].. Antenna Array Pattern Formulaton The far feld factor of a lnear array wth an even number of unformly spaced sotropc (N) can be wrtten n the form: N ( ) = a cos( d snθ + ψ ) F θ, () = where = π/λ s the wave number, θ denotes the angular drecton, a and ψ are the ampltude and the phase of the exctaton complex feed. The poston d can be computed from the nter-element spacng (Fg. ), accordng to the followng formula: d = m= m. () In order to synthesze the desred pattern, element exctatons (a and ψ ) and postons d are be optmally detered by the CRS. In order to control the actual pattern both n the shapng regon and n the predetered sde lobe regons, we have adopted the cost functon gven as: ( ) C cost = L θ, (3) θ where

2 RADIOENGINEERING, VOL. 5, NO. 3, SEPTEMBER 6 L ( θ ) ( θ ) + ( θ ) =, (4) [ ][ SR ( θ ) F( θ ) ] ( θ ) SR ( θ ) F( θ ) =. (5) Here, SR and SR represent the mum and the mum shapng regon, respectvely (Fg. ), and L(θ) does not equal to only f F(θ) s stuated nsde the shapng regon. antenna element θ / d d 3 Fg.. Non-equdstant symmetrcal lnear array. In order to obtan the desred pattern, the cost functon gven by (3) s mzed by the CRS, whch s descrbed n the followng secton. 3. Control Random Search Algorthm The dea of the Controlled Random Search (CRS) algorthm was developed by W. L. Prce [8]. The algorthm requres a mum preparaton of data to operate, and can be appled to constraned as well as to unconstraned optmzaton problems where the gradent of the obectve functon s unavalable. In the prelary stage, a pool of randomly selected ponts s generated and the functon values are calculated for each pont. Then the man routne starts whch computes subsequent tral ponts and updates pool accordngly drectvty [db] SR SR Fg.. Desred pattern shape. N CRS, although smple, has several dsadvantages n ts ntal form. Prmarly, the convergence rate worsens whle approachng neghborhood of the soluton. Secondary, the uncondtonal reecton of a tral pont that falls off the constrants maes t dffcult to fnd optmal ponts that are located on the optmzaton doman boundares. The CRS algorthm used n ths paper s presented below: Step :Let be the step counter; set =. Randomly choose M ponts from the optmzaton doman D. The chosen ponts consttute the ntal pool P = {X,..., X M }. The pool sze should be suffcently large, usually M = ( n+). Evaluate the obectve functon for the chosen ponts f( X ), =... M. Here, X s the array parameters beng searched, D s the optmzaton doman (e.g., ampltude a [, ], phase ψ [, π] and nter- spacng [.5 λ,.5 λ], where λ =.6 m correspondng to the frequency 5 GHz), and n s the dmenson of the cost functon defned n (3), n = 3N. Step : Fnd the pont X,l n the current pool P that provdes the lowest cost functon value, and pont X,h provdng the hghest cost functon value. Step : Create n+ dmensonal subset of P called the smplex S. The smplex has to contan the pont X,l. The remanng n ponts are randomly chosen from P (wthout duplcatons; choosng X,h s allowed). Compute the centre of the smplex: c = X x S n +. (6) Step 3: Compute a tral pont X t by reflectng the current worst smplex element X,h from the smplex centre S : X t = c X,h. Step 4: If any coordnate of X t volates constrants, bounce the pont bac nto the doman usng the followng scheme: X X f X < X X = X X f X > X. (7) X else Step 5: If the test pont s better than the worst pont n the pool,.e. f f(x t ) < f(x,h ), then create a new pool P + that s P wth X,h replaced by X t, and ncrease the step counter. Otherwse, go to the step n order to repeat the creaton of a smplex S once more. Step 6: Stop f the stop crteron s satsfed,.e. f the obectve functon value has not decreased below ε n a prescrbed number of steps h: f f ( X h, l ) f ( X, h ) < ε, ( X ) = f ( X ) f h <. h, l, h Step 7: Go to the step. The total number of functon evaluatons made can be appled as an addtonal stop crteron. There are two reasons for ntroducng ths crteron: (8)

3 L. MERAD, F. T. BENDIMERAD, S. M. MERIAH, CONTROLLED RANDOM SEARCH OPTIMIZATION FOR LINEAR We obtan a general control of costly functon evaluatons (n terms of the CPU-tme used); We can prevent the algorthm from fallng accdentally nto nfnte loops. The mplementaton of CRS concerns two ts parts: a pool ntalzaton and a tral pont computaton. In the ntalzaton phase, the ndex of the frst pont n the ntal pool P that has not been yet processed s avalable for all the threads. A sngle thread starts to process the pont (.e. starts to compute the cost functon value) and ncrements the ndex. Therefore, the pool s not dvded between threads n advance, but s processed accordngly to a thread computaton capablty. When the ntalzaton s done, each thread enters ts own tral pont computaton loop. There are several possbltes of nteractons between loops n order to mprove convergence. W. L. Prce n [8] proposes all threads to operate on the same pool that s updated as soon as a better pont s found. In ths scheme, all threads reflect the same pont X h, but wth regard to varous smplex centers. 4. Numercal Results In order to llustrate the capabltes of the CRS for the shaped beam pattern synthess of a lnear array, two examples are consdered. In the frst example, an unequally spaced lnear array consstng of sotropc s used, and n the second one, we consder an unequally spaced lnear array consstng of 8 sotropc. In the optmzaton process, the value of n s 8 and for the two examples, respectvely. The total number of functon evaluatons s fxed to 3 and ε to -4, whch s found to be suffcent to obtan satsfactory patterns wth a desred performance. The calculatons are performed on a personal computer wth Pentum IV processor runnng at.6 GHz, and for the two examples consdered here, the optmzaton results are obtaned wthn 485 s and 35 s, respectvely. - - drectvty [db] As the frst example, a pattern wth a man beam wdth defned n the sector θ [, ] and a sde lobe level lower than db were chosen as the desred pattern. The mas of the desred pattern s shown n Fg. 3. In order to obtan the desred pattern, the values of the cost functon parameters gven n (5) are used as follows: db for θ (9) db elsewhere, 5dB for θ () 5dB elsewhere. In Fg. 3, the pattern obtaned from the CRS by deterng both the ampltude, the phase and the poston of each element of the array s llustrated. Clearly, the pattern shows a good performance n the shape regon, and there are no sde lobes exceedng the specfed value db. a) b) ampltude phase [rad] c) Fg. 3. The mas of desred pattern (dashed lne) and the radaton pattern obtaned by the CRS usng (sold lne) Fg. 4. Optmzed parameters of array: a) optmzed ampltude exctaton, b) optmzed phase exctaton, c) poston of each element n the array. The requred element ampltude, the phase and the poston of each array element for the pattern gven n Fg. 3 are depcted n Fg. 4.

4 RADIOENGINEERING, VOL. 5, NO. 3, SEPTEMBER 6 3 In the second example, we are gong to produce the array pattern wth a large man beam wdth defned n the sector θ [, 6 ]. The SR and SR are gven below: db for θ 6 () db elsewhere, 5dB for θ 5 () 5dB elsewhere. Fg. 5 shows the CRS s capable to synthesze an unequally spaced lnear array producng a shaped beam pattern wth the good performance both n the shaped regon and n the sde lobe regon. The algorthm can easly consder any parameter of nterest (sde lobe level, beam wdth, etc.) by ncludng t n the cost functon. drectvty [db] Fg. 5. The mas of desred pattern (dashed lne) and the radaton pattern obtaned by the CRS usng 8 (sold lne). The ampltude, phase and postons obtaned by the CRS to produce the pattern depcted n Fg. 5 are shown n Fg Comparatve Study In order to evaluate the performance of the proposed CRS algorthm, ths secton compares the numercal results calculated by CRS and the Genetc Algorthm (GA). For comparson, a lnear array of 3 sotropc at half wavelength spacng s consdered. The array exctaton ampltude s symmetrc wth null phase, the number of teratons (generatons) s set to [4]. Fg. 7 shows the comparson of the far-feld patterns among the CRS-smulated results, and the GA-smulated results n [4]. Obvously, the sde lobes close to the man beam are lowered, and the levelled sde lobes n Fg. 7 ndcate that the result s close to the optmum soluton for that partcular beam wdth. Ths fact can be confrmed by comparng the sold lne GA pattern [4] and the dashed lne CRS one. Although the CRS sde lobe level s 35.5 db, ths result remans comparable to the GA one: 36. db. a) b) c) ampltude phase [rad] Fg. 6. Optmzed parameters of array: a) optmzed ampltude exctatons, b) optmzed phase exctatons, c) poston of each element n the array. For the smulaton speed comparson between CRS and GA [4], the GA smulaton taes about s on 486/33 MHz PC [4]. The CRS smulaton taes 4 s on Pentum IV processor runnng at.6 GHz. Obvously, the GA smulaton s much faster than CRS for array-pattern synthess. drectvty [db] CRS GA Fg. 7. Comparson of the radaton patterns of the broad sde lnear array wth exctaton coeffcents by the CRS (dashed lne) and the GA method (sold lne).

5 4 L. MERAD, F. T. BENDIMERAD, S. M. MERIAH, CONTROLLED RANDOM SEARCH OPTIMIZATION FOR LINEAR However, the GA requres a good adustment of these parameters n order to obtan good results. Fg. 8 shows the ampltude exctatons generated by the CRS algorthm and GA method. All ampltude exctatons are almost dentcal. 6. Concluson A new optmzaton method for the synthess of lnear array pattern functons has been proposed and assessed. The shaped beam pattern, the constraned sde-lobes level, and the man-lobe wdth are contemporarly taen nto the account by mzng a cost functon by means of an nnovatve mproved Controlled Random Search procedure. ampltude CRS GA Fg. 8. Comparson of the optmzed ampltude exctatons computed by CRS (dashed lne) and GA (sold lne). The dfferent results show the great flexblty of the proposed approach. Many addtonal extensons of the CRS method could be also easly mplemented both n term of the cost functon defnton, the optmzaton methodology and applcatons, and antenna types. References [] YANG, S., BENG GAN, Y., QING, A. Antenna-array pattern nullng usng a dfferental evoluton algorthm. Internatonal Journal on RF and Mcrowave.4, vol. CAE 4, p [] LUE, W. C., HSU, F. Use of B-splne curves and genetc algorthms to reduce the sdelobe level n array-patterns. Mcrowave and Optcal Technology Letters, 3, vol. 38, no. 4, p [3] BALANIS, C. Antenna Theory Analyss and Desgn. New Yor: Wley, 98. [4] YAN, K., LU, Y. Sdelobe reducton n array-pattern synthess usng genetc algorthms. IEEE Transactons on Antennas and Propagaton, 997, vol. 45, no. 7, p. 7. [5] AKDAGLI, A., GUNEY, K. Shaped beam pattern synthess of equally and unequally spaced lnear antenna arrays usng modfed tabu search algorthm. Mcrowave and Optcal Technology Letters, 3, vol. 36, No., p. 6. [6] ALI, M. Some Modfed Global Optmzaton Algorthms wth Applcatons. Loughborough Unversty of Technology, PH.D Thess, 994. [7] ALI, M. M., STOREY, C., TORN, A. Applcaton of some stochastc global optmzaton algorthms to practcal problems. Journal of Optmzaton Theory and Applcatons, 997, vol. 95, no. 3, p. 545 to 563. [8] PRICE, W.L. Global optmzaton by controlled random search. Journal of Optmzaton Theory and Applcatons, 983, vol. 4, p About Authors... Lotf MERAD was born n Tlemcen (Algera) n 977, he receved the dploma of state engneer n Electroncs (998), Magster degree n Electroncs () and doctorate degree from Tlemcen Unversty (5). He s a Junor Researcher wthn the Telecommuncatons Laboratory. Hs feld of research s centered on the optmzaton process and ther applcaton on antenna array synthess. Feth Tar BENDIMERAD was born n Sd Belabbes (Algera) n 959, he receved the dploma of state engneer n Electroncs (983) from Oran Unversty, Master degree n Electroncs (984) and doctorate degree from Nce Sopha-Antpols Unversty (989). He wored as a Senor Lecturer n Tlemcen Unversty and he s the Drector of the Telecommuncatons Laboratory. Hs man area of research s the mcrowave technques and radaton and he s responsble of the antenna secton. Sd Mohammed MERIAH was born n Tlemcen (Algera) n 97, he receved the dploma of state engneer n Electroncs (99) from Natonal Polytechnc College, Magster degree n Electroncs (997) and doctorate degree from Tlemcen Unversty (5). He s a Senor Lecturer n Tlemcen Unversty and Researcher wthn the Telecommuncatons Laboratory. He wored on smart array antennas.