Beamforming Based on Suppression Transforming Error for Array Interpolation

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1 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, 18 Beamformig Based o Suppressio Trasformig Error for Array Iterpolatio Shexiag Ma, ei Pa, Xi Meg Abstract Aimig at the iheretly problems that iterpolated trasformatio techique ca ot work effectively over a large trasformatio area, ad whe the burst iterferece occurs to the out of trasformatio area, it ca ot be suppressed. I this paper, we propose a improved algorithm by desigig the weights of iterpolated array to reduce the ifluece of trasformatio errors. The key feature of this algorithm is that miimizes the error of output sigal power which is caused by trasformig errors ad guaratees the weight is orthogoal to the trasformig errors of the desired sigal directio. Numerical simulatios cofirm the validity of the proposed algorithm. I comparig with the existig algorithms, the proposed iterpolated array beamformig algorithm ca suppress iterferece sigal accurately, prevet the mai lobe from shiftig effectively ad make the ull deeper i the large trasformatio area. Besides, it also icreases the output sigal to iterferece plus oise ratio (SINR), ad has lower complexity. Keywords beamformig, iterpolated array, ull depth, trasformig error I. INTRODUCTION ith the study of array atea, dealig with more sigals with Wfewer elemets becomes the mai goal. To achievig this, i 199s, the iterpolatio techology was proposed by B. riedlader firstly [1]. The freedom of array is icreased ad the irregular array is trasformed ito uiform liear array (ULA). Sice the, the iterpolated array has bee used i directio of arrival (DOA) estimatio [-6]. I recet years, it has become a focus i applyig iterpolated array to beamformig. I 1997, a arbitrary real array is trasformed ito several idetical cofiguratio subarrays usig iterpolatio techology was discussed by Ta-Sug Lee, so as to coheret iterfereces ca be suppressed by usig spatial smoothig [7]. I 7, a coheret beamformig method usig iterpolatio techique based o uiform circular array is proposed by ZhagYag [8]. I 1, a beamformig method based o iterpolated array which is applied to coformal array was proposed by P. Yag [9]. I that paper, it makes the adaptive beamformig ca be directly applied to coformal This work was supported i part by the Natioal Natural Sciece oudatio of Chia uder Grat No , No Shexiag Ma is with the School of Electrical ad Electroic Egieerig, Tiaji Uiversity of Techology, Tiaji 3384, Chia (phoe: ; masx_tjut@16.com). ei Pa is with School of Electrical ad Electroic Egieerig, Tiaji Uiversity of Techology, Tiaji 3384, Chia ( @stud.tjuedu.c). Xi Meg is with the Maritime College, Tiaji Uiversity of Techology, Tiaji 3384, Chia ( megxi@tjuedu.c). array. owever, the study o the relatio betwee trasform errors ad area is show that: the iterferece ca ot be suppressed whe it falls outside the trasform area; besides, whe the trasform area is too large, it occurs to zero drift ad the mailobe offse I other words, the agle-sesitive exists i iterpolated array [1]. I order to solve this, a lot of improved algorithms are proposed. I 199, multi regios iterpolated array is proposed by B. riedlader [11]. I that paper, the trasformatio area is divided ito several regios, but the itegrated error will be itroduced ad the suppressio performace will be degraded. The method of robust iterpolated array is proposed by M. Pesaveto i [1]. The mai idea is to impose costraits o the trasform iterval. owever, it is difficult to determie the threshold outside the trasformatio area. I the same year, the optimal trasformatio matrix was desiged by P. yberg [13, 14]. Yet, the optimal coversio matrix is more complicated ad higher computed. I 16, a ew approach was proposed by u Datig to reduce the effect of the error [15]. The key operatio is that the covariace matrix of iterpolated array is corrected by the error covariace matrix. But the ull depth formed at the iterferece is shallow. I view of this situatio, we proposed the iterpolated steerig vector ad covariace matrix are costraied by the trasformatio error ad its covariace matrix respectively. It ca ot oly achieve beamformig i the large agle usig iterpolated array, but also get better performace, deeper ull ad greater output sigal to iterferece plus oise ratio (SINR) whe dealig with iterferece sigal. II. CONVENTIONAL INTERPOLATED ARRAY A. Sigal Model A geeral real ULA with M isotropic elemets ad d spacig is cosidered. Mutual couplig is igored. If the N radiatig arrowbad sourcces are observed by this array, the received sigal vector ca the be writte as: x ( t) = As( t) + ( t) (1) where s( t) = [ s ( t) s ( t) s ( T 1,, N t deotes the vector of radiated sigals; ( t) = [ 1 ( t), ( t),, M ( t) ] deotes the = is defied vector of oise; A [ a( θ 1 ), a( θ ),, a( θ N as M N array maifold, a( θ ) represets the steerig vector ISSN:

2 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, 18 of the array i directio θ ad it is give by a d siθ T [ ], k = π λ. jkd siθ jk ( M 1) ( θ ) = 1, e,, e The covariace matrix is give by E[ x( t) x( t) ] where E[ ] R = () traspose. deotes expectatios, deotes cojugatio B. The Method of Iterpolated Array The study of array sigal processig foud that the more elemets, the better resolutio whe the elemet spacig is costa There are several methods to icreasig resolutio effectively. or example: cumulat [16], liear predictio [17] ad array iterpolatio [1]. Iterpolated array ca covert every array ito ULA, ad the trasformatio matrix ca be processed offlie, so iterpolatio techique is used. Assumig that the sigal falls withi the regio Θ ad it is evely divided as: Θ = [ θ l, θl + θ,, θ r ] (3) where, θl ad θ r represet the left ad right boudaries of Θ respectively, θ deotes the iterpolatio step. The i this regio, the maifold matrix of real ULA is expressed as: A = [ a( θ l ), a( θl + θ ),, a( θ r (4) So the iterpolated maifold matrix i the same area ca be described as: A = [ a( θl ), a( θl + θ ), a( θ r (5) Obviously, if we wat to implemet the array expasio, there is a fixed relatio betwee the real ad iterpolated array, such that: mi BA A (6) B where deotes the robeius orm. Whe the umber of poits to be iterpolated are larger tha the elemets of real array ad A is full rak, the trasformatio matrix ca be solved as: B = AA ( AA ) 1 (7) The covariace matrix of iterpolatio array ca be writte as: ~ R = BRB = BARs A B + BR where Rs deotes sigal autocorrelatio matrix, RN = σ I represets oise autocorrelatio matrix. The equatio (8) ca be simplified as: ~ R = BARs A B + σ BB (9) The oise power is estimated by sapshots i fact, so it ca be expressed as: M N 1 = M N i= 1 ˆ σ σ (1) i N B (8) ere, ( i = 1, M N ) σ are the M N i small eigevalues of the matrix R ~, M deotes the umber of virtual elemets. It ca be see from the equatio (9) that the Gaussia white oise is cotamiated after iterpolatio. It is ecessary to white the colour oise. So the whiteed covariace matrix ca be described as: ~ R = R σ ˆ BB + ˆ σ I (11) Accordig to the actual eeds, there are several adaptive beamformig algorithms, such as Least Mea Square (LMS) [18], Sample Matrix Iversio (SMI) [19], Miimum Variace Distortioless Respose (MVDR) [18, ]. Sice the MVDR oly eeds to kow the directio of the desired sigal, it is used i this paper. The desired sigal is icidet from θ, the whiteed covariace matrix is used to beamformimg accordig to the MVDR algorithm ad ca be expressed as: mi w Rw w (1) s. wa( θ ) = 1 III. BEAMORMING O INTERPOLATION ARRAY BASED ON SUPPRESSION TRANSORMING ERROR A. Aalysis of Error o Iterpolatio Trasform Array Sice the iterpolated poits are limited, the trasformatio error is defied as: erroe BA A B = mi (13) A where E = BA A deotes trasformatio error matrix i the Θ. It ca be see from equatio (13) that trasformatio error is always exists ad determied by iterpolated step ad agle rage. or istace, 4 elemets λ spacig real ULA was trasformed ito 8 elemets λ spacig iterpolated ULA. Whe the trasformatio area is selected from to18 ad step is.1, the relatio betwee trasformatio error ad agle rage is show i ig. 1. ISSN:

3 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, 18 Mappig Error Agle Rage(degrees) ig. 1 Mappig error for differet agle rage ig. 1 is show that trasformatio error is icreased as trasformatio agle rage icreasig. Whe trasformatio agle is less tha 3, the trasformatio error is cotrolled to 1-3 or less. As the agle regio icreasig, the error is rapidly icreased. Whe the agle is 18, the error is greater tha.5. B. Beamformig of Iterpolatio Array Based o Suppressio Trasformig Error The above aalysis is show that whe trasformatio agle is too large, the ifluece of trasformatio error ca ot be eglected. I order to solve this problem, the beamformig based o suppressio trasformig error was proposed. Assumig that the sigal falls withi the regio Θ, the trasformatio error matrix E is calculated from the formula (13), ad the error of output sigal ca be writte as: ye ( t) = w E (14) The correspodig output power is give by P = E [ ye( t) ] E[ w Rew ] e = (15) where Re = E E is called error covariace matrix i this paper. I order to suppress the impact of trasformatio error, we miimize the error of output sigal power whe the weight is orthogoal to the trasformatio error i the desired directio. It ca be described as: mi w Rew s. w E( θ ) = w (16) Cosiderig (1) ad (16), the proposed algorithm i this paper is deoted as: mi w {( 1 α ) w Rw + αw Rew} s. w a( θ ) = 1 s. w E( θ ) = R = 1 α R + αr Let ( ) e as: (17), the above formula ca be simplified mi w Rw w s. w[ a( θ ) + E( θ = 1 The optimal weight is calculated as: w = 1 R [ a( θ ) + E( θ 1 [ a( θ ) + E( θ R [ a( θ ) + E( θ (18) (19) α is defied as a costat belogig to [,1). The largerα, the better orthogoal betwee weight ad error, but over largeα will reduce the suppressio performace. ig. shows a curve which is the eigevalues of R with differetα. Whe the weight ad error are orthogoal,α ca't affect the small eigevalue of R greatly. Soα is chose as.3. Eigevalue(dB) α=. α=. α=.3 α=.4 α=.6 α= Idex ig. Eigevalues of R with differetα IV. SIMULATION AND RESULT ANALYSIS I this sectio, the proposed algorithm is compared with the covetioal iterpolated virtual array (covetioal IVA) [1], multi regio iterpolatio virtual array (multi regio IVA) [11] ad costrait weight iterpolated virtual array (costrait weight IVA) [15]. The 4-elemet spacig λ ULA is exteded to 8-elemet spacig λ ULA. A. The Situatio of No Burst Iterferece 1) Small trasformatio agle Simulatio 1: aalysis the beamformig performace of iterpolated array. The desired sigal icidets from directio, add oe idepedet iterferece, arrivig agle is -. SNR=dB, INR=4dB. The trasformatio areas are defied as [-3, ], iterpolated step is selected as.1, ad sapshots are. ISSN:

4 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, Multi Regio IVA ig.3 Aalysis beamformig uder small trasformatio agle ig. 3 shows that beamformig is accurately implemeted by covetioal IVA [1], costrait weight IVA [15] ad the proposed method i this situatio. Simulatio : aalysis output SINR for differet iput SNR. Settig iput SNR rage from db to 5dB. Other coditios such as simulatio 1. Output SINR(dB) SNR(dB) ig.4 Aalysis output SINR uder small trasformatio agle ig. 4 shows that the covetioal IVA [1], costrait weight IVA [15], ad proposed algorithm have the same performace whe the SNR is small. But the proposed algorithm becomes greater tha the others whe SNR is icreased. ) Small trasformatio agle Simulatio 3: aalysis the beamformig performace of iterpolated array. The desired sigal icidets from directio, add oe idepedet iterferece, arrivig agle is -. SNR=dB, INR=4dB. [-9, 9 ] is divided evely ito 6 sectios as the trasformatio area of Multi Regio IVA [11]. The trasformatio areas of the others are defied as [-9, 9 ], step size is selected as.1, sapshots are ig.5 Aalysis beamformig uder large trasformatio agle As show i ig.5, i the case of large trasformatio agle, the ifluece of trasformatio error ca ot be igored. The iterferece comig from - ca ot be suppressed whe covetioal IVA [1] is adopted. The costrait weight IVA [15] ad Multi Regio IVA [11] ca form a shallow ull to suppress the iterferece. I the same situatio, a deep ull (about -4 db) i the - is formed by the proposed algorithm. That is to say, the beamformig performace is the best tha the others. Simulatio 4: aalysis output SINR for differet iput SNR. Settig iput SNR rage from db to 5dB. Other coditios such as simulatio 3. Output SINR(dB) Multi Regio IVA SNR(dB) ig.6 Aalysis output SINR uder large trasformatio agle It ca be kow from the ig. 6 that the output SINR of covetioal IVA [1] is the worst, sice iterferece ca ot be suppressed. Because the impact of trasformatio error is cosidered by costrait weight IVA [15] ad proposed algorithm, the output SINR are better tha covetioal IVA [1]. Yet, with the icrease of the iput SNR, the proposed algorithm has obvious advatage. B. The Situatio of Burst Iterferece Simulatio 5: aalysis the beamformig performace of ISSN:

5 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, 18 iterpolated array. O the basis of simulatio 1, add aother burst iterferece icidet from 4, INR=4dB. [-9, 9 ] is divided evely ito 6 sectios as the trasformatio area of Multi Regio IVA [11]. The trasformatio area of covetioal IVA [1] is selected as [-3, ], the costrait weight IVA [15] ad proposed algorithm are defied as [-9, 9 ]. -1 C. aalysis of beamformig performace with differet iterpolatio step Simulatio 7: aalysis beamformig accuracy uder differet iterpolatio step. The desired sigal icidets from directio, add oe idepedet iterferece, arrivig agle is -. SNR=dB, INR=4dB. The trasformatio area of proposed algorithm is [-9, 9 ], sapshots are, step are set to.1 ad 1 respectively Multi Regio IVA ig.7 Aalysis beamformig uder burst iterferece ig. 7 shows that burst iterferece ca ot be suppressed by covetioal IVA [1]. Costrait weight IVA [15] ad proposed algorithm trasform area are selected as [-9, 9 ], which ca suppress ay iterferece ad the proposed algorithm has better suppressio performace. Simulatio 6: aalysis output SINR for differet iput SNR. Settig iput SNR rage from db to 5dB. Other coditios such as simulatio _step is.1 _step is ig.9 Aalysis accuracy with differet iterpolatio step ig. 9 shows that the iterferece also ca be restraied by the proposed algorithm, but it is obvious that the depth of the ulls is shallow whe the iterpolatio step is 1. Simulatio 8: aalysis beamformig performace uder large iterpolatio step. O the basis of simulatio 7, settig trasformatio area of costrait weight IVA [15] is [-9, 9 ], step size is Output SINR(dB) Multi Regio IVA SNR(dB) ig.8 Aalysis output SINR uder burst iterferece ig. 8 shows that burst iterferece ca ot be suppressed by covetioal IVA [1], so the output SINR is the worst (the maximum is about 3dB). The output SINR of costrait weight IVA [15] ad the proposed algorithm are high, ad the proposed algorithm is close to the 8 elemets ULA (the maximum is about 31dB) _step is 1 _step is ig.1 Aalysis beamformig with large iterpolatio step ig. 1 shows that while step is 1, the costrait weight IVA [15] aligs the desired sigal with the mailobe, but the iterferece ca ot be suppressed. Therefore, the proposed algorithm has better performace i beamformig. V. CONCLUSION I this paper we proposed a ew method that miimizes the ISSN:

6 INTERNATIONAL JOURNAL O CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 1, 18 error of output power i the case of orthogoal betwee weights ad trasformatio error, so that the error power cotributio is miimize whe the iterpolated array is beamformig. Compared with the existig algorithms, it solves agle-sesitive. The iterferece ca be suppressed better; the ull ad the output SINR became deeper ad larger. Shexiag Ma (196-) is curretly a professor ad a doctoral tutor, the mai research areas are sigal processig, atea techology, etc. ei Pa (1991-) is a master's graduate studet ad the mai research areas are array sigal processig, adaptive beamformig, etc. Xi Meg (1981-) is a associate professor ad master tutor, ad the mai research areas are sigal processig, mobile commuicatios, etc. REERENCES [1] riedlader B, Directio fidig usig a iterpolated array, IEEE Iteratioal Coferece o Acoustics, Speech, ad Sigal Processig, vol. 5, pp , 199. [] riedlader B, Weiss A J. Directio fidig for wide-bad sigals usig a iterpolated array, IEEE trasactios o sigal processig, vol.41, o. 4, pp ,1993. [3] Weiss A J, Gavish M, Directio fidig usig ESPRIT with iterpolated arrays, IEEE trasactios o sigal processig, vol. 39, o. 6, pp , [4] riedlader B, The root-music algorithm for directio fidig with iterpolated arrays, Sigal processig, vol. 3, o. 1, pp. 15-9, [5] Weiss A J, riedlader B, Stoica P, Directio of arrival estimatio usig MODE with iterpolated arrays, IEEE trasactios o sigal processig, vol. 43, o. 1, pp. 96-3, [6] Lau B K, Cook G, Leug Y, A improved array iterpolatio approach to DOA estimatio i correlated sigal eviromets, 4 IEEE Iteratioal Coferece o Acoustics, Speech, ad Sigal Processig, vol., o., pp [7] Lee T S, Li T T, Adaptive beamformig with iterpolated arrays for multiple coheret iterferers, Sigal Processig, vol. 57, o., pp , [8] Yag Zhag, Zhou Zou, Zeju Lv, Beamformig of Coheret Sigals Based o Uiform Circular Array, Joural of Electroic Sciece ad Techology, vol. 36, o. 1, pp. -3, 7. [9] Yag P, Yag, NIE Z P, Robust adaptive beamformer usig iterpolated arrays, Progress i electromagetics research B, vol. 3, pp. 15-8, 1. [1] Wexig Li, Yipeg Li, Adaptive beamformig method for arc legth based virtual atea array, IEEE Coferece Publicatios, pp , 11. [11] B. riedlader, A. J. Weiss, Directio fidig usig spatial smoothig with iterpolated arrays, IEEE Trasactios o Aerospace ad Electroic Systems, vol. 8, o., pp , 199. [1] Pesaveto M, Gershma A B, Luo Z Q, Robust array iterpolatio usig secod-order coe programmig, Sigal Processig Letters, vol. 9, o. 1, pp. 8-11,. [13] P. yberg, M. Jasso, B. Otterste, Array mappig: optimal trasformatio matrix desig, IEEE Iteratioal Coferece o Acoustics Speech, ad Sigal Processig, vol. 3, pp [14] P. yberg, M. Jasso, B. Otterste, Array iterpolatio ad bias reductio, IEEE Trasactios o Sigal Processig, vol. 5,o. 1, pp , 4. [15] Datig u, Jupig Geg, Vitual Array Beamformig Based o the Costrait of Iterpolatio Mappig Errors Usig Orthogoal Weights, Chiese Joural of Radio Sciece,vol. 3, o. 1, pp. 9-15, 17. [16] Doga M C, Medel J M, Applicaio of cumulats to array processig-part I: aperture extesio ad array calibratio, IEEE Trasactios o Sigal Processig, vol. 43, o. 5, pp , [17] Peg u, Study o Beamofrmig Algorithm of Array With Virtual Elemets, M.S. thesis, Northwester Polytechical Uiversity, Chia, 6. [18] P. M Maikar, Guja N. Jagtap, G. N. Mulay, Aalysis of miimum variace distortioless respose ad least mea square beamformig algorithm for smart atea, 16 Iteratioal Coferece o Iteret of Thigs ad Applicatios (IOTA), pp [19] Larry L. orowitz, oward Blatt, Cotrollig adaptive atea arrays with the sample matrix iversio algorithm, IEEE Trasactios o Aerospace ad Electroic Systems, AES-15, o. 6, pp , [] J. Capo, igh-resolutio frequecy-waveumber spectrum aalysis, 1969 Proceedigs of the IEEE, vol. 57, o. 8, pp ISSN: