Sideband Suppression Through the Frequency of Sinusoidal Variable Pulse Amplitude Time Scheme in TMAs

Transcription

1 207 Published i 5th Iteratioal Symposium o Iovative Techologies i Egieerig ad Sciece September 207 (ISITES207 Baku - Azerbaija) Sidebad Suppressio Through the Frequecy of Siusoidal Variable Pulse Amplitude Time Scheme i TMAs Yasi Yavuz, *Ertugrul Aksoy ad Mert Karaha Faculty of Egieerig, Departmet of Electrical ad Electroics Egieerig Gazi Uiversity, Akara, Turkey Abstract I time modulated arrays, excitatio strategies affect the characteristics of the array. Istead of usig rectagular pulses, to have more stable respose i frequecy domai, siusoidal sigals may be used as excitatio scheme. I this paper, variable pulse amplitude excitatio strategy is combied with siusoidal sigals ad the frequecy of siusoidal sigal is appeared as a ew degree of freedom. Radiatio patter may easily be shaped via ay optimizer ad sidebad suppressio may be achieved oly cotrollig the frequecy parameter. At the ed of this study, a explaatory example usig Differetial Evolutio Algorithm is give to discuss the results. Key words: Differetial Evolutio, optimizatio, TMLA, sidebad suppressio. Itroductio The time modulatio cocept i atea arrays is achieved by switchig each elemet with high speed RF switches ad, as a ievitable result, switchig causes ifiite umber of harmoic radiatios which are called as sidebads. Ideed, some of the power are beig trasferred to these sidebads []. Sice the received or trasmitted power is limited with regulatory restrictios, i literature, there are may studies to suppress the uwated sidebad radiatios so as to trasfer maximum power to the mai radiatio [2-]. O the other had, istead of suppressig these uwated radiatios, it is asserted that sidebad radiatios may be eligible to use i some applicatios such as directio fidig, secure commuicatio ad steerig the harmoic beams ad use these radiatios as secod, or eve third if there is eough power, receiver which eables commuicatio over sidebads [2-6]. Sice excitatio strategy is directly affects the atea array factor, also there are some studies i order to fid the best scheme which ca easily be produced i time domai ad have more stable frequecy respose allowig some widowig fuctios [7,8]. I this paper, a ew excitatio scheme, which ca be obtaied easily i time domai ad have more stable frequecy respose with respect to rectagular pulse, usig siusoidal sigal with D offset is proposed ad exemplified. The rest of the paper orgaized as follows: ext sectio gives a brief theory about time modulatio ad proposed excitatio strategy. Third ad fourth sectios describe optimizatio process ad array desig with results, respectively. At the ed, the study is cocluded. *orrespodig author: Address: Faculty of Egieerig, Departmet of Electrical ad Electroics Egieerig Gazi Uiversity, 06570, Akara TURKEY. address: ertugrulaksoy@gazi.edu.tr, Phoe:

2 E. AKSOY et al./ ISITES207 Baku - Azerbaija Brief Theory 2.. Time Modulatio Oce the basic structure of time modulated atea array is cosidered, every elemet of array is idividually switched ad, aturally, array factor is beig affected by this switchig operatio. If N isotropic elemets are cosidered to be located alog z- axis, the array factor may be defied as: N AF(θ, t) = I e jβ U (t)e jkz cos θ. () = Here, I, β, z ad θ represets the excitatio amplitude, the excitatio phase, the Euclidia distace to the system origi of th elemet ad the elevatio agle accordig to z- axis, respectively. k = 2π λ deotes the waveumber ad U (t) is the periodic switchig fuctio. The most basic switchig scheme which is amed as variable aperture size (VAS) may be expressed as: U (t) = {, t t 0, otherwise, (2) where t ad deotes the switched o duratio of related elemet ad the switchig period, respectively. Sice (2) is periodic i time ad as every periodic fuctio may be decomposed ito complex Fourier series. After the decompositio ad basic calculatios, the most geeral form of the time-modulated atea array factor may be writte usig () as: N AF(θ, t) = I e jβ m, e jkz cos θ e j(w 0+mw p )t, (3) m= = uder far field ad f 0 f p approximatios. Here, m,, w 0 = 2πf 0 ad w p = 2πf p deotes the complex Fourier coefficiet of th elemet i m th harmoic level, the agular operatig frequecy ad the agular switchig frequecy, respectively Siusoidal Variable Pulse Amplitude (Sie-VPA) Every periodic fuctio may be used as switchig fuctio ad i literature most commo strategies use o-off switchig scheme. However, i order to prevet some disadvatages of ooff switchig such as array silecig which meas all elemets are switched to zero ad o commuicatio at that istat, switchig betwee amplifier or atteuator is proposed by Aksoy i [4]. Switchig fuctio of variable pulse amplitude (VPA) for two elemets ca be defied as:

3 E. AKSOY et al./ ISITES207 Baku - Azerbaija 5 U (t) = { K, 0 < t t K 2, t < t, (4) where K ad K 2 represet th elemet relative pulse amplitudes of adjacet pulses ad t represets the amplitude reversal time istat. Figure. Excitatio Scheme of th elemet. Left Stadard VPA Right Siusoidal VPA ( = 0.) Here, we itroduce a ew excitatio strategy applyig siusoidal sigal with D offset to adjacet pulses ad i Fig., VPA ad Sie-VPA excitatio schemes for th elemet are show. With this approach, frequecy respose of the switchig fuctio becomes more stable [8] ad sidebad radiatio (SR) becomes depedet to frequecy of the sie fuctio which may be used as a ew degree of freedom. Periodic switchig fuctio of Sie-VPA may be writte as: U (t) = { K + cos(2πf t), 0 < t t K 2 + cos(2πf t), t < t, (5) where f deote sie fuctio frequecy of th elemet ad <, R. I order to compute array factor, complex Fourier coefficiets for Sie-VPA should be calculated for all harmoic levels. After required calculatios which ca be see i Appedix A, the coefficiets may be defied as (A.4) ad (A.9):

4 E. AKSOY et al./ ISITES207 Baku - Azerbaija 52 m, = τ + K 2 + { 2π[(f rat ) 2 m 2 ] {jm + f rat 2πf rat mπ si(mπτ ) e jmπτ + [si(2πf rat )], m = 0 si(2πf rat ) jmcos(2πf rat )}, m > 0. (6) Here, = K K 2, τ = t / ad ω = 2πf represet D offset differece of adjacet pulses, ormalized pulse reversal time ad agular sie fuctio frequecy of th elemet. Additioally, = f /f p stads for sie fuctio frequecy to switchig frequecy ratio. f rat 3. Optimizatio As it ca be clearly uderstood from (6), oly f rat may be used by ay optimizer i order to suppress SR. I this study, Differetial Evolutio (DE) usig DE/best//bi scheme is applied with mutatio factor, crossover factor ad populatio size of 0.6, 0.95 ad 50, respectively. The algorithm may be summarized as: Step Radomly create populatio matrix with respect to parameter defiitios. Step 2 alculate cost fuctios for populatio. Step 3 reate a mutat vector usig parameter vectors. Step 4 reate a trial vector with crossover. Step 5 alculate cost fuctio for trial vector. Step 6 ompare cost fuctios for j th elemet of the populatio ad choose the best oe for ew populatio. Step 7 Go to Step 3 util j reaches populatio idex. Step 8 reate ew populatio ad go to Step 3 util iteratio limit ad/or stop criterio is reached. The cost fuctio is calculated accordig to steer mai radiatio to broadside ad to reduce side lobe level (SLL) ad SR below -20dB. A iteratio limit (75 iteratios) is set as the stop coditio. The hardware of P which is used for optimizatio process cosist of Itel i7 M GHz PU, 8GB RAM ad 28GB SSD with Widows 0 ad MATLAB R206b softwares. 4. Array Desig ad Results As a explaatory example, a array whose 20 idetical elemets located alog z- axis with d = 0.5λ spacig may be cosidered. Without loss of geerality, other parameters are selected as follows: I =, β = 0, = 0.5, = 0.65 ad K 2 = 0.2. I additio, ormalized pulse reversal time of each elemet is selected as correspodig hebyshev polyomial coefficiets. After 75 iteratios, accordig to optimized frequecy ratio parameters of each elemet which are visualized

5 E. AKSOY et al./ ISITES207 Baku - Azerbaija 53 i Fig. 2 ad i Fig. 3, mai radiatio ad fudametal harmoic radiatio patters are beig show. Fig. 4 also describes maximum sidebad radiatio levels for first te harmoic level. Figure 2. Optimized f rat Values (Normalized) Figure 3. Normalized radiatio patters for mai ad fudametal harmoic radiatio related to Fig. 2 (Max. SLL=-20dB ad Max. SR=-22.55dB)

6 E. AKSOY et al./ ISITES207 Baku - Azerbaija 54 Oce the results are examied, it is clearly see that mai radiatio is directed to broadside with low SLL (< 20dB) ad uwated sidebad radiatio is reduced lower tha -20dB oly optimizig the frequecy ratio. After 50 idepedet experimets, it is also see that iteratio cout may be decreased till 40 iteratios to have early same results with decreased operatio time. 75 iteratios take 56. secods ad for 40 iteratios elapsed time is secods which is early half of former. Istead of havig fixed D offset differece,, the study is beig repeated with radom values usig same iteratio couts ad the results are early idetical with acceptable SLL ad SR level. Usig variable values allow compesatig real time error which may be occurred o switchig atteuator or amplifier. Figure 4. Maximum levels of first te harmoic radiatios oclusios I this study, a modified variable pulse amplitude excitatio strategy which may be amed as Siusoidal VPA is itroduced ad it is show that this ew approach eables a easily optimizable parameter. I less tha fifty iteratios ad i very short time rage, radiatio patters may be optimized with DE ad results are quite satisfactory. I additio, errors of amplifiers ad atteuators used i switchig devices may be tolerated usig differet offset differece scheme.

7 E. AKSOY et al./ ISITES207 Baku - Azerbaija 55 Appedix A I this appedix, i order to maitai paper s readability, steps for calculatig complex Fourier coefficiets of Sie-VPA is beig show. omplex Fourier coefficiets are calculated with: m, = U T (t)e jmωpt dt. p (A. ) Usig (5), for m = 0: t m, = [ (K T + cos(ω t))dt + (K 2 + cos(ω t))dt], ω = 2πf. (A. 2) p 0 t m, = [(K T t + si(ω p ω t)) t 0 + (K 2 t + si(ω ω t)) ]. (A. 3) t m, = K τ + K 2 K 2 τ + For m > 0: si(ω ω ) = τ + K 2 + si(ω ω ). (A. 4) t m, = [ (K T + cos(ω t))e jmωpt dt + (K 2 + cos(ω t))e jmωpt dt]. (A. 5) p 0 t t K e jmω pt dt + 0 m, = t. (A. 6) K 2 e jmωpt dt + cos(ω t) e jmωpt dt + cos(ω t) e jmωpt dt t 0 t [ I ] m, = [ K e jmωpt t jmω p 0 ± K 2 e jmωpt jmω p t + I + I 2 ]. (A. 7) m, = K jm2π ( e jmω pt ) K 2 jm2π ( e jmω pt ) + I + I 2. (A. 8) I 2

8 E. AKSOY et al./ ISITES207 Baku - Azerbaija 56 After I ad I 2 calculated usig Euclidea equivalet of cosie fuctio, with some basic mathematical operatios coefficiets may be expressed as follows: m, = mπ si(mπτ ) e jmπτ + 2π[(f rat ) 2 m 2 ] {jm + f rat si(2πf rat ) jmcos(2πf rat )}. (A. 9) Refereces [] Bickmore, RW, Hase, R. Time Versus Space i Atea Theory. New York:Academic Press; 966 [2] Yag S, Ga YB, Qig A. Sidebad suppressio i time modulated liear arrays by the differetial evolutio algorithm. IEEE Ateas Wirel Propag Lett 2002;; [3] Fodevila J, Bregais J, Ares F, Moreo E. Optimizig uiformly excited liear arrays through time modulatio, IEEE Ateas Wirel Propag Lett 2004; 3; [4] Aksoy E, Afaca E. Thied ouiform amplitude time modulated liear arrays. IEEE Ateas Wirel Propag Lett 200;9; [5] Poli L, Rocca P, Maica L, Massa A. Hadlig sidebad radiatios i time-modulated arrays through particle swarm optimizatio. IEEE Tras Ateas Propag 200;58(4); [6] Aksoy E, Afaca E. Geeralized represetatio of sidebad radiatio power calculatio i arbitrarily distributed time-modulated plaar ad liear arrays. PIERS Proceedigs 20; [7] Bregais J, Fodevila J, Fraceschetti G, Ares F. Sigal radiatio ad power losses of timemodulated arrays. IEEE Tras Ateas Propag 2008;56(6); [8] Aksoy E, Afaca E. alculatio of sidebad power radiatio i time-modulated arrays with asymmetrically positioed pulses. IEEE Ateas Wirel Propag Lett 202;; [9] Bekele ET, Poli L, Rocca P, D Urso M, Massa A. Pulse-Shapig Strategy for Time Modulated Arrays Aalysis ad Desig. IEEE Tras Ateas Propag 203;6(7); [0] Aksoy E, Afaca E. A iequality for acceleratig the calculatio of relative maximum sidebad level i time-modulated liear ad plaar arrays. IEEE Tras Ateas Propag 204; 62(6); [] Aksoy E. alculatio of Sidebad Radiatios i Time-Modulated Volumetric Arrays ad Geeralizatio of the Power Equatio. IEEE Tras. Ateas Propag 204;62(9); [2] Teat A, hambers B. A two-elemet time-modulated array with directio-fidig properties. IEEE Ateas Wirel Propag Lett 2007;6; [3] Zhu Q, Yag S, Yao R, Nie Z. Directioal modulatio based o 4-d atea arrays. IEEE Tras Ateas Propag 204;62(2); [4] Aksoy E. Time modulatio through variable pulse amplitude i 4D arrays. TELE INFO 204;2 20. [5] Aksoy E. Harmoic beam steerig i 4D liear arrays through pulse differece. TELFOR 204;22;

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