Design and Analysis of Array Weighted Wideband Antenna using FRFT

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1 The Interntionl Arb Journl of Informtion Technology, Vol., o. 4, July Design nd Anlysis of Arry Weighted Widebnd Antenn using FRFT Adri Sty Srinivs Ro nd Prudhivi Mllirjun Ro Deprtment of ECE, Adity Institute of Technology nd Mngement, Indi Deprtment of ECE, Andhr niversity College of Engineering, Indi Abstrct: The bemwidth of liner rry depends on number of elements in the rry nd frequency of the input signl. The min requirement of widebnd bemformer is, the min bem pttern should be constnt even there is chnge in input signl frequency. Vrious methods were proposed in literture, one method is clled elementl lowpss filtering designed by using Finite Impulse Response (FIR) digitl filters. In this pper, the elementl lowpss filtering method ws implemented using Frctionl Fourier Trnsform (FRFT) nd performnce nlysis ws crried out with rry weighting.. Keywords: Antenn rry, widebnd ntenn, FRFT, rry weighting. Received December 8, ; ccepted My 4, ; published online August 5,. Introduction Arrys of brodbnd signls hve been pplied in sonr, rdio, rdr, coustic imging etc., these re often difficult to design becuse of highly frequency dependent rry properties. Figure shows the directivity pttern of simple element liner rry. The figure shows, the minlobe width decreses with increse in frequency. This cuses, some signls to be received with distorted spectr, nd lso frequencydependent null loctions impir the bility to cncel brodbnd interference. Response in db elements ntenn X: Y: MHz Figure. Plne wve response of liner rry with input frequency. In the pst, brodbnd bemformers hve been studied extensively due to its pplictions [3, 4, 6]. It ws pprent tht, for uniformly weighted liner rry the lrgest sidelobes re down pproximtely 4 percent from pe vlue. The sidelobe levels cn be further decresed by using non uniform spcing method. The presence of sidelobes mens tht the rry is rditing energy in untended directions. In multipth environment, the sidelobes cn receive the sme signl from multiple ngles. This is the bsis for fding experienced in communictions [5]. The sidelobes cn be suppressed by weighting, shding or windowing the rry elements. Arry element weighting hs numerous pplictions in res such s digitl signl processing, rdio stronomy, rdr, sonr, nd communictions. From Frctionl Fourier Trnsform (FRFT) nd relted concepts [,, 8], it hs been seen tht the properties nd pplictions of Continuous Time Fourier Trnsforms (CTFT) is specil cse of FRFT. We hve explined the usefulness of FRFT in the design of brodbnd bemformers with uniform spcing nd dvntges of it hve been reported [7]. This pper gives the suitbility nd results of certin rry weighting methods in the design of brod-bnd ntenn rry using FRFT. This pper is orgnized s: section presents theory for bemforming, section 3 presents implementtion of lowpss filter with FRFT, section 4 focused on some rry weighting methods, section 5 gives the simultion results nd section 6 represents conclusions respectively.. Bemforming Theory For liner rry, the fr field response for n input frequency ω nd incident ngle θ (mesured reltive to brodside) is given by [6]: P ω sinθ c ( θ, ω) = exp j x D ( x, ω) dx () where c nd θ re the propgtion speed nd ngle of impinging signl nd D(x,ω) the frequency response with respect to the ngulr frequency ω nd loction x. Obviously, in generl p(ω,θ) is function of both ω nd θ, while for frequency invrint bemformer, we require tht the bem pttern p(ω,θ) be independent of ω. For weighted liner rry of + eqully spced

2 374 The Interntionl Arb Journl of Informtion Technology, Vol., o. 4, July 3 unidirectionl elements, the fr field response cn be represented from the bove eqution : ω sin θ P, ex p j x D x, c ( θ ω) = ( ω) n= n = ( ωτ θ) ( ω) = ex p jn sin D x, where, τ =d/c is the interelement spcing divided by the speed of wve nd D(x,ω) is the response of the filter connected to ntenn element t x. The inter-null bemwidth of uniformly excited rry is given by [4]: θ BW = sin π 4π Mωτ Mωτ () (3) where, M=+. This expression clerly indictes tht the bemwidth is inversely proportionl to frequency (or proportionl to normlized frequency). It lso implies tht n increse in either the number of elements or interelement spcing results in decrese in the bemwidth s well. Figure presents the trnsmission signl of signl from ech ntenn element nd is pssed through lowpss filter. The weighted sum of lowpss filters re combined t liner combiner [3]. Here, we hve used FRFT to design lowpss filter. FRFT, which is controlled by single continuous ngulr prmeter α. So, FRFT cn be represented s the rottion of signl in time-frequency plne [8] s shown in Figure 3. Figure 3. Rottion of signl in time frequency plne with n ngle α. The generliztion of the CTFT is obtined if we consider rottion through n rbitrry ngle α in the (t, ω) plne. Thus, the FRFT of signl f(t), cn be expressed s [7]: F f (t ) K t,t f (t )dt [ ] = ( ) ( ) = ϕ π( ϕ ϕ+ ϕ) K t,t K exp j t cot t t cos ec t cot K = exp[ j( π sgn( φ) / 4 φ / ) ] [ sin φ]. 5 (6) φ where φ=π/. The ernel function K (t,t) hs the following spectrl expnsion: (5) K π = ( t, t) ψ ( t ) exp j ψ ( t) (7) = Figure. Filter-nd-sum structure. 3. Implementtion of Low Pss Filter using FRFT The CTFT, which is defined by the following pir [8]: F ( ω) = f ( t ) ex p( jω t) dt π f ( t) = F ( ω ) ex p( jω t) dω π (4) The CTFT reflects to the ssumption tht the signl of interest hs sttionry frequency content. However, signl representtions using intermedite ngulrly coupled xes hold some promise for nlyzing signls with time-frequency coupling, e.g., liner-frequency modultion. Angulr trnsform of the CTFT, which is clled s the Angulr Fourier Trnsforms (AFT) or where ψ (t) denotes th Hermite-Gussin function, nd t denotes the vrible in the th-order Frctionl Fourier Domin. The th order Hermite-Gussin function is defined s (=,,, ): ψ 4 ( t) H ( π t) exp( π t ) = (8)! where H denotes th order Hermite polynomil hving π rel zeros. In the eqution 7, exp represents the th power of the eigenvlues. When =, the FRFT reduces to the ordinry fourier trnsform, where t denotes the frequency-domin vrible. Here, we hve dopted the design low pss filter using FRFT bsed on procedure of tunble Finite Impulse Response (FIR) filters. A FIR digitl filter opertion is liner convolution of the finite durtion impulse response with the input signl sequence x(n). The impulse response h(n), of iser window is given s [9]: j h( n) = h ( n) w( n) (9) d where h d (n) is the desired or idel impulse response, nd w(n) is the iser window sequence. Since multipliction in the time domin corresponds to

3 Design nd Anlysis of Arry Weighted Widebnd Antenn using FRFT 375 convolution in the frequency domin which cn be expressed s complex convolution opertion given s: H π π ( ω) = W ( λ) H d ( ω λ) π dλ where, H(ω) is the frequency response of filter H d (ω), is desired or idel frequency response of lowpss filter, nd W(ω) is frequency response of iser window. From the bove eqution, the trnsition bndwidth of is proportionl to minlobe width of iser window W(ω). Figure 4 shows the vrition of minlobe width with order of FRFT. It is observed tht s the FRFT order is reduced the min lobe width of FRFT iser window shrins. This feture hs been used here to tune the trnsition bndwidth of lowpss filter. - =. =.4 3=.6 () 4.3. Hmming Weights The hmming weights re defined by: w( where =,,, -. + ) = cos(π /( )) () 4.4. Kiser-Bessel Weights The iser-bessel weights re defined by: I πα λ w( ) = I[ πα] where =,,, - nd α> Blcmn-Hrris Weights (3) The blcmn-hrris weights for -9dB sidelobe level re defined by: Response in db π w ( + ) = co s 4 π 6 π co s co s (4) ormlized frequency Figure 4. Vrition in iser window response with order of FRFT. 4. Arry Weighting Methods There re vst number of possible window functions vilble tht cn provide weights for use with liner rrys. Some of the more common window functions re considered for nlysis of widebnd ntenn. 4.. Binomil Weights Binomil weights will crete n rry fctor with no sidelobes, provided tht the element spcing d λ. The binomil weights re chosen from the rows of Pscl s tringle. The first five rows re shown in Tble. Tble. Pscl s tringle. = = = 3 = = Gussin Weights The gussin weights re defined by: w( + ) = e where =,,, nd α. α () 4.6. uttll Weights The uttll weights re defined by: π w ( + ) = cos 4 π 6 π cos..6 cos 5. Simultion Results The design results were evluted using computer simultion, for frequencies, 5.8MHz, nd, for element ntenn rry with uniform spcing. Figures 5, 6, 7, 8, 9 nd show the ntenn pttern nd the frequency chrcteristics of the widebnd ntenn with FIR Low Pss filter nd FRFT Low Pss Filters for Binomil, Gussin, Hmming, Kiser-Bessel, Blcmn-Hrris nd uttll weighting methods elements widebnd ntenn with Binomil weighting 5.8MHz elements widebnd ntenn with Binomil weighting - FRFT 5.8MHz (5) Figure 5. Response of -element widebnd ntenn rry designed with binomil weights.

4 376 The Interntionl Arb Journl of Informtion Technology, Vol., o. 4, July elements widebnd ntenn with Gussin weighting 5.8MHz elements widebnd ntenn with Gussin weighting - FRFT 5.8MHz Figure 6. Response of -element widebnd ntenn rry designed with Gussin weights elements widebnd ntenn with Hmming weighting 5.8MHz elements widebnd ntenn with Hmming weighting - FRFT 5.8MHz Figure 7. Response of -element widebnd ntenn rry designed with Hmming weights nd FRFT Low Pss Filter elements widebnd ntenn with Kiser - Bessel weighting 5.8MHz elements widebnd ntenn with Kiser - Bessel weighting - FRFT 5.8MHz Figure 8. Response of -element widebnd ntenn rry designed with Kiser-Bessel weights elements widebnd ntenn with Blcmn - Hrris weighting 5.8MHz elements widebnd ntenn with Blcmn - Hrris weighting - FRFT 5.8MHz Figure 9. Response of -element widebnd ntenn rry designed with Blcmn-Hrris weights elements widebnd ntenn with uttll weighting 5.8MHz elements widebnd ntenn with uttll weighting - FRFT 5.8MHz Figure. Response of -element widebnd ntenn rry designed with uttll weights. 6. Conclusions In this pper we hve discussed bout ppliction of rry weighting on the design of widebnd ntenns. From the Figures 5, 6, 7, 8, 9 nd, it is evident tht if the low pss filters re implemented by using FRFT, gives better invrint rdition ptterns compred to norml FIR low pss filters. Figures 5, 9 nd shows tht the Binomil weights, Blcmn-Hrris weights nd uttll weights gives lower side lobe levels, more thn 8dB with FIR filters nd more thn 6dB with FRFT filters. From Figure 8 it is evident tht iserbessel weights gives better directionl chrcteristics. References [] Almeid L., The Frctionl Fourier Trnsform nd Time-Frequency Representtion, IEEE Trnsctions on Signl Processing, vol. 4, no., pp , 994. [] Cndn C., Kuty M., nd Ozts H., The Discrete Frctionl Fourier Trnsform, IEEE Trnsctions on Signl Processing, vol. 48, no. 5, pp ,. [3] Chou T., Frequency-Independent Bemformer with Low Response Error, in Proceedings of IEEE Interntionl Conference on Acoustics, Speech, nd Signl Processing, SA, vol. 5, pp , 995. [4] Goodwin M. nd Elo G., Constnt Bemwidth Bemforming, in Proceedings of IEEE Interntionl Conference on Acoustics, Speech, nd Signl Processing, SA, vol., pp. 69-7, 993. [5] Gross F., Smrt Antenns for wireless communictions, McGrw-Hill Boo Compny, ew Yor, 5. [6] Liu W. nd Weiss S., A ew Clss of Brodbnd Arrys with Frequency Invrint Bem Ptterns, in Proceedings of IEEE Interntionl Conference on Acoustics, Speech, nd Signl processing, Cnd, vol., pp , 4. [7] Ro S., Mllirjun P., Murlidhr V., nd y K., Frequency Invrint Bem Ptterns using Frctionl Fourier Trnsform, Interntionl Journl of Multidisciplinry Reserch nd Advnces in Engineerin, vol., no., pp. 3-34,. [8] Snthnm B. nd Mcclelln J., The Discrete Rottionl Fourier Trnsform, IEEE Trnsctions on Signl Processing, vol. 44, no. 4, pp , 996. [9] Shrm., Sxen R., nd Sxen C., Shrpening the Response of n FIR filter using Frctionl Fourier Trnsform, The Journl of the Indin Institute of Science, vol. 86, pp , 6.

Design nd Anlysis of Arry Weighted Widebnd Antenn using FRFT 377 Adri Sty Srinivs Ro received his M Tech. degree from Andhr niversity, Vishptnm in 4 nd he is PhD student in Andhr niversity.

5 Design nd Anlysis of Arry Weighted Widebnd Antenn using FRFT 377 Adri Sty Srinivs Ro received his M Tech. degree from Andhr niversity, Vishptnm in 4 nd he is PhD student in Andhr niversity. He is hving 5 yers of teching experience in vrious engineering colleges. Presently, he is woring s Professor, Deprtment of ECE, Adity Institute of Technology nd Mngement, Teli. His interests include signl processing, dptive ntenn rrys nd communiction systems. Prudhivi Mllirjun Ro received his ME degree from Andhr niversity in 985 nd PhD degree from Andhr niversity in 998. He is now woring s Professor, Deprtment of ECE, ACE, Vishptnm. His interests include EMI/EMC, ntenn rry syntesis nd pplied electromgnetic pplictions. He hs published 5 ppers in vrious journls/proceedings in the field of ntenn rrys nd EMI/EMC.

Design and Analysis of Broadband Beamspace Adaptive Arrays using Fractional Fourier Transform

Design and Analysis of Broadband Beamspace Adaptive Arrays using Fractional Fourier Transform Interntionl Journl of Electronics nd Communiction Engineering. ISSN 9742166 Volume 5, Number 2 (212), pp. 19326 Interntionl Reserch Publiction House http://www.irphouse.com Design nd Anlysis of Brodbnd