UNIVERSITY OF TRENTO COMPROMISE PATTERN SYNTHESIS BY MEANS OF OPTIMIZED TIME-MODULATED ARRAY SOLUTIONS. P. Rocca, L. Poli, L. Manica, and A.

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1 UNIVERSITY OF TRENTO DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL INFORMAZIONE 3823 Povo Treto (Italy), Via Sommarive 4 COMPROMISE PATTERN SYNTHESIS BY MEANS OF OPTIMIZED TIME-MODULATED ARRAY SOLUTIONS P. Rocca, L. Poli, L. Maica, ad A. Massa Jauary 20 Techical Report # DISI--90

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3 Compromise Patter Sythesis by meas of Optimized Time Modulated Array Solutios P. Rocca*, L. Poli, L. Maica, ad A. Massa ELEDIA Group Departmet of Iformatio Egieerig ad Computer Sciece, Uiversity of Treto, Via Sommarive, 4,I Treto, Italy E mail: adrea.massa@ig.uit.it, Web page: Itroductio Origially, the use of time modulated arrays was maily devoted to the sythesis of low ad ultra low sidelobes arrays for the detectio of radar sigals []. Successively, oly a few works have cosidered the use of time modulated arrays for other applicatios (e.g., wireless commuicatios [2]), although the first work proposig the use of time as a additioal degree of freedom dates back to the 950s [3]. Such a evet has bee probably caused by the mai drawback of time modulatio that is the uavoidable presece of udesired sidebad radiatios (SRs) i the radiated field due to the periodic commutatio of the RF switches betwee the o ad off states. A detailed aalysis o such a topic has bee recetly preseted i [4], where a close form relatioship uatifyig the power wasted i SR has bee obtaied. I the last decades, thaks to the growig computatioal capabilities of moder PC, some approaches based o evolutioary algorithms have bee proposed for the reductio of the losses due to SRs i time modulated architectures. More specifically, the differetial evolutio (DE) algorithm [5] ad the simulated aealig (SA) [6] have bee cosidered. Despite the successful results, the potetialities of time modulated array have bee oly partially ivestigated. As a matter of fact, a few works have bee devoted to exted the possibilities of applyig the time modulatio to array atea sythesis. I [7], the problem cocered with the sythesis of sum ad differece patters has bee discussed. Moreover, differet switchig strategies ad their effects o SR have bee show i [8]. I this paper, the potetialities of time modulatio are further ivestigated i the framework of compromise array sythesis as a alterative solutio to stadard compromise methods (see [9] ad the referece therei). More i detail, startig from a set of static excitatios aimed at sythesizig a userdefied sum patter at the carrier freuecy, a compromise differece beam is geerated through a suitable sub arrayig patter matchig procedure [9] by optimizig the pulse duratios at the iput ports of each sub array. Successively, the uavoidable SR at the harmoic freuecies is miimized by applyig a particle swarm optimizatio to set the switch o time istats of each time seuece.

4 Mathematical Formulatio Let us cosider a liear array of N eui spaced elemets layig o the z axis. The static array excitatios α, = 0, K,N- are fixed to geerate a sum patter whose array factor is expressed as N ( θ, ) α exp( Σ = F α jkd cosθ) () = 0 2π where k = is the backgroud wave umber, d is the iter elemet spacig, λ0 ad θ is the agular directio with respect to the array axis. The compromise differece beam is the geerated by aggregatig the array elemets ito Q sub arrays ad eforcig a periodic o off seuece U ( t), =,..., Q to the output sigal of each sub array. The rectagular pulse fuctio has period U () t 2 2 eual to T p ad it holds true that U ( t) = for t t t (with 0 t t Tp ) 2π ad U () t = 0. Moreover, T P >> T 0 =. Sice U ( t) ω0, =,...,Q, is periodic, it ca be represeted through its Fourier series where ω p = 2π T p () t = = U ω (2) uh exp( jh pt ) h. Accordigly, the array factor of the differece patter turs out beig the summatio of a ifiite umber of harmoics [4] where the patter at the carrier agular freuecy ω 0 is eual to where F N Q Δ = = 0 = ( 0 ) ( θ,c, ) α δ τ c exp( jkd cosθ) τ (3) Tp () t 2 t t Tp τ = u = U =. (4) 0 T p 0 ad δ is the Kroecker delta, beig δ = if c =, ad δ = 0 otherwise, c [ ] where c ;Q, = 0,...,N, are iteger values idetifyig the sub array membership of the array elemets. Differetly, the patter of the harmoic radiatios (i.e., h 0 ) is give by c c F N Q ( h ) ( θ,c,uh ) = exp[ j( h p ω0 )] α δc uh exp( jkd cosθ) Δ + h= h 0 ω. = 0 = (5)

5 where the Fourier coefficiets u h are fuctio of the switch o ad switch off 2 istats ad or euivaletly of the switch o istats ad the pulse t t duratio τ thaks to (4). It is worth otig from (3), that it is eough to defie the values τ, =,...,Q ad c, = 0,...,N, to sythesized the compromise patter at the carrier freuecy. Hece, accordig to the guidelies of [9], the followig cost fuctio is miimized (a) Compromise ad referece patter. (b) Elemet switch o time. Figure. (a) No optimized SBL. (b) Optimized SBL. Figure 2 Normalized power patter. Ψ Q β τ δc τ. (5) N = 0 α =, is set of referece excitatio coefficiets computed 0 N ( ) ( ) cm, = α where β, = 0,...,N accordig to classical methods (e.g., Bayliss). The, i order to reduce the iterfereces due to SR, oly the values t, 2 =,..., Q, are optimized sice the

6 τ, =,...,Q are fixed to those computed i (5). Accordigly, a PSO based strategy is used ad a proper fuctioal ( h ) ( t ) Ψ is miimized measurig the mismatch betwee the actual sidebad level ad the desired oe. Numerical Results For represetative purposes, let us cosider the sythesis of a N = 30 array atea. Startig from a set of static excitatios affordig a Villeeuve sum patter with SLL = 20 db ad = 3, the compromise differece patter sythesized through the proposed approach is show i Fig. (a) together with the referece differece patter (Modified Zolotarev differece patter) whe Q = 0. The values of τ, =,..., Q ad c, = 0,...,N are show i Fig. (b), as well. The harmoic freuecies before ad after the optimizatio by meas of the PSO strategy are show i Fig. 2(a) ad Fig. 2(b), respectively. It is worth otig how the level of the higher harmoics is reduced. Refereces [] W. H. Kummer, A. T. Villeeuve, T. S. Fog, ad F. G. Terrio, Ultra low sidelobes from time modulated arrays, IEEE Tras. Ateas Propag., vol., o. 6, pp , Nov [2] R. W. Bickmore, Time versus space i atea theory, i Microwave Scaig Ateas, R. C. Hase, Ed. Los Altos, CA: Peisula, 985, vol. III, ch. 4. [3] H. E. Shaks ad R. W. Bickmore, Four dimesioal electromagetic radiators, Caad. J. Phys., vol. 37, pp , Mar [4] J. C. Brégais, J. Fodevila Gómez, G. Fraceschetti, ad F. Ares, Sigal radiatio ad power losses of time modulated arrays, IEEE Tras. Ateas Propag., vol. 56, o. 6, pp , Ju [5] S. Yag, Y. B. Ga, ad A. Qig, Sidebad suppressio i time modulated liear arrays by the differetial evolutio algorithm, IEEE Ateas Wireless Propag. Lett., vol., pp , [6] J. Fodevila, J. C. Brégais, F. Ares, ad E. Moreo, Optimizig uiformly excited liear arrays through time modulatio, IEEE Ateas Wireless Propag. Lett., vol. 3, pp , [7] J. Fodevila, J. C. Brégais, F. Ares, ad E. Moreo, Applicatio of time modulatio i the sythesis of sum ad differece patters by usig liear arrays, Microw. Opt. Techol. Lett., vol. 48, pp , May [8] A. Teat ad B. Chambers, Cotrol of the harmoic radiatio patters of time modulated atea arrays, Proc IEEE AP S Iteratioal Symp., S. Diego, Califoria, USA, July 5 2, [9] L. Maica, P. Rocca, A. Martii, ad A. Massa, A iovative approach based o a tree searchig algorithm for the optimal matchig of idepedetly optimum sum ad differece excitatios," IEEE Tras. Ateas Propag., vol. 56, o., pp , Ja