Sparsity Enhanced Beamforming in The Presence of Coherent Signals

Transcription

1 Iteratioal Coferece o Advaced Iformatio ad Commuicatio Techology for Educatio (ICAICTE 23) Sparsity Ehaced Beamformig i The Presece of Coheret Sigals Bao Ge Xu, Yi e Wa, Qu Wa 2, Si Log Tag, Xue Ke Dig, Li Zou 2 Joit Lab of Array Sigal Processig, TogFag Electroic Sciece ad Techology Co Ltd, Jiujiag, Chia, Departmet of Electroic Egieerig, Uiversity of Electroic Sciece ad Techology of Chia, Chegdu, Chia, 673 Abstract I order to fid the directios of coheret sigals, a sparsity ehaced beamformig method is proposed The miimum variace i the proposed method correspods to the orthogoal relatioship betee the oise subspace ad the sparse represetatio of the received sigal vector, hereas the distortless respose correspods to the o-orthogoal relatioship betee the sigal subspace ad the sparse represetatio of the received sigal vector The proposed sparsity ehaced method is carried out by the iterative reeighted Lp-orm costrait miimizatio for directio fidig of coheret sigals Simulatio results are provided to sho that it has better performace tha the existig algorithms i presece of coheret sigals Keyords: distortless respose of sigal subspace; miimum variace of oise subspace; sparse represetatio; directio fidig; coheret sigals Itroductio Source localizatio has bee of iterest i the past fe decades ad played a fudametal role i may applicatios ivolvig electromagetic, acoustic, biomedical, seismic sesig, etc A importat goal for source localizatio methods is to be able to locate coheret sigals i the presece of multipath propagatio [] ay advaced methods for the localizatio of icoheret sigals attai super-resolutio by exploitig the separability of a small umber of sigals The most ell-o existig methods iclude Capo's method [2], beam-formig [3] ad its relevat algorithms [4], ad subspace based methods such as USIC [5] oever, most of them could ot deal ith the coheret sigals Recetly, the usage of sparse feature of sigals has evolved very rapidly, fidig applicatios i several ids of sigal processig problems There has also bee some emergig research of these ideas i the cotext of spatial spectrum estimatio, beam-formig, ad directio fidig by atea array [3,4] Sacchi et al too advatage of Cauchy-prior to itroduce sparsity i spectrum estimatio ad solved the oliear optimizatio problem by iterative approaches [6] Jeffs made use of a Lp-orm pealty ith p to eforce sparse feature deduced from several applicatios, icludig sparse atea array desig [7] Goroditsy ad Rao used a recursive eighted miimum-orm algorithm called focal uder-determied system solver (FOCUSS) to mae use of sparsity of spatial spectrum i the problem of DOA estimatio [8] The or of Fuchs [9] as also ivolved i sparse sigal localizatio uder the assumptio that the umber of sapshots is abudat I [3], 23 The authors - Published by Atlatis Press 568

2 I order to improve the beam-patter, a total variatio miimisatio of the hole beam patter is icorporated to ecourage large array gais accumulated i the mailobe ad small trivial array gais gathered i the side-lobes, hile revisig the sparse costrait oly o the sidelobe All these methods are based o the sparse represetatio of the received vector of the array as a sparse liear combiatio of directio vectors The L pealty for sparsity ad the L2 pealty for oise are ofte ustilized to recover the sparse sigal represetatio To mitigate the effect of measuremet oise ad reduce the calculatio, a ovel DOA estimatio method, L-SVD [], as proposed, hich sparsely represeted the sigal subspace by a overcomplete basis ad assumed that DOAs of icomig sigals are usually very sparse relative to the hole spatial domai It is carried out by L-orm costrait miimizatio due to it is a covex problem oever, L-orm costrait miimizatio has a drabac that larger coefficiets of sigal are puished more heavily tha smaller coefficiets, ulie the more impartial puishmet of the L-orm costrait miimizatio [] This causes the degradatio of sparse sigal recovery performace based o regular L-orm costrait miimizatio I this paper, e focus o the problem of directio fidig for coheret sigals The methodology of the iterative reeighted Lp-orm costrait miimizatio is expaded from the array data to sigal ad oise subspace for directio fidig of coheret sigals aig use of the orthogoality betee oise subspace ad sparse represetatio of received sigal vector, the objective of Lporm costraied miimizatio variace distortless respose (VDR) ca be achieved 2 Problem formulatio First use the equatio editor to create the equatio The select the Equatio marup style Cosider a uiform circular array (UCA) that cosists of atea The radius of the circle is r Assume that the sources s () t come from azimuth i the far field of the array,,2,, K, K is the umber of sources The received sigal vector of UCA ca be expressed as: K x( t) a( ) s ( t) v ( t) () here x () t is the received sigal vector, t is samplig momets, v () t is the receiver oise vector, a ( ) is directio vector correspodig to azimuth The m-th compoet of a ( ) is 2 j2 fcr cos( ( m) ) m( ) e a, m,2,, Whe there are coheret sigals, ie, s ( t) isi ( t) for costat i ad i K, e oly have L differet ad icoheret sigals here L K The sample autocorrelatio matrix of the received vector x ( t, ) is: T R x( t) x ( t) (2) T t here T represets the umber of received sigal vectors of UCA, ad complex cojugate traspositio The sigular value decompositio of the sample autocorrelatio matrix is R UΛU (3) here Λ is a diagoal matrix hose diagoal elemets correspod to the sigular value of R, U is matrix hose colum vectors are sigular vectors of R, 569

3 u, u2, u3,, u, correspodig to the sigular values, 2 3 Accordig to the subspace decompositio approach, the oise subspace of the sample autocorrelatio matrix is: Q u u u (4) L L here L is the umber of icoheret sigals The problem is to estimate the azimuth,,2,, K, for all the sigals hatever they are coheret or icoheret sigals 3 Sparsity ehaced VDR Accordig to the criterio of miimum variace ad distortless respose (VDR), e have the spatial spectrum: f ( ) mi mvdr R (5) st a ( ) here the eightig vector C ad is the searchig grid of azimuth, eg,,,,359 degree,,2,, N ad N 36 Because the above problem is a quadratic optimizatio ith liear costrait, it is easy to calculate the spatial spectrum i the closed-form as give by fmvdr ( ) (6) a ( ) R a( ) oever, it is usually ot sparse eough i the spatial domai To tae the advatage of sparse property of the spatial spectrum, e ehace the criterio of miimum variace ad distortless respose (VDR) by sparsity costrait The problem of sparsity ehaced VDR ca be described as: f arg mi QA (7) F st u A here is the umber of o-zero compoets of the eightig vector N C, A a( ) a( ) a ( ) (8) 2 N Ufortuately, the above problem of oliear optimizatio is a NP-hard problem A remedy is to use the Lp-orm costrait miimizatio istead ad solve it ith iterative adaptive algorithm f arg mi QA G (9) F F here st u A p/2 G diag () It should be orth otig that G To avoid the drabac F p that larger coefficiets of the eightig vector are puished more heavily tha smaller coefficiets, ulie the more impartial puishmet of costrait miimizatio, e ofte choose p 5 The optimizatio problem (9) is ocovex ad ca be reritte as 2 f argmi A Q QA G () st u A oever, if e assume that G is idepedet ith the eightig vector, it is easy to solve the above problem i closed-form as here BA u u ABA u (2) 2 B ( A QQA G ) (3) Therefore, e ca summarize the iterative adaptive algorithm to solve the problem () as the folloig () Iitializatio: f ( ) mvdr ; p/2 (2) Update: diag 2 B ( A QQA G ) ; G ad (3) Upate: BA u ; u ABA u 57

4 (4) Repeat (Step 2) ad (Step 3), util the differece betee obtaied by the adjacet steps is small eough Fially, the spatial spectrum of the sparsity ehaced VDR is give by f (4) Obviously, there are may differeces betee VDR ad SEVDR First, the dimesio of the eightig vector of VDR is the same as the umber of atea of UCA, hereas eightig vector of SEVDR is the same as the umber of the searchig grid of azimuth Secod, the spatial spectrum of VDR is a quadratic fuctio of the eightig vector, hereas the spatial spectrum of SEVDR is directly the eightig vector Fially, the spatial spectrum of SEVDR is explicitly sparse, hereas VDR taes o advatage of sparsity of the spatial spectrum Because the above SEVDR algorithm oly fid the sparse solutio, there may be some sigals that could ot be foud by the pea positio of f Whe the directios of sigals are estimated from the pea positio of f, e should modify the sigal subspace ad oise subspace as u P u ad Q P Q here P is the orthogoal projectio matrix of matrix hose colum vectors are the directio vectors correspodig to the the directios of sigals estimated from the pea positio of f The, e should repeat the above SEVDR algorithm util o sigificat pea is foud i f 4 Simulatio Result We cosider a uiform circular array of = 9 atea ith radius r = 4 meters The avelegth of the arrobad sigals is 5 meters Three zero-mea arrobad sigals i the far-field impige upo this array from distict directios of arrival (DOA), ie, 53, 785 ad 24 degree The first ad third sigal are coheret The total umber of sapshots is T = 64, the sigal to oise 6 ratio (SNR) is 9 db, ad I Figs ad 2, e compare the spatial spectrum obtaied usig our proposed method ith those of VDR ad USIC methods I Fig, e ca see that VDR ad USIC oly detect the secod sigal hose directio vector is orthogoal to the oise subspace, ad are uable to detect the coheret sigals hose directio vectors are ot orthogoal to the oise subspace O the cotrary, i Fig2, e ca see that VDR ad USIC oly detect the secod sigal hose directio vector is orthogoal to the oise subspace, ad are uable to detect the coheret sigals hose directio vectors are ot orthogoal to the oise subspace 5 Coclusio The ovelty ad advatage of our techique is that it is able to fid the directios of coheret sigals Though the proposed sparsity ehaced method is carried out by the iterative reeighted Lp-orm costrait miimizatio, the umber of iteratio is usually ot more tha 2 Simulatio results sho that it has better performace tha the existig algorithms i the presece of coheret sigals 6 Acoledgmet This or as supported by the Natioal Natural Sciece Foudatio of Chia uder grat 6724 ad Excellet Teachig Team Support Project (Graduate studet part) " of Uiversity of Electroic Sciece ad Techology of 57

5 Chia (No A ) ad joit laboratory project of UESTC-73 shortave & ultra-short-ave array sigal processig - - VDR USIC Fig Spatial spectrum of VDR ad USIc SEVDR SEVDR Fig 2 Spatial spectrum of to-step's sparsity ehaced VDR (SEVDR- ad SEVDR-2) 7 Refereces [] Q Wa, B G Xu, J Yi, F Fag, Y Wa, S L Tag, "A approach to maifold estimatio for atea array i situatio of iterfereces," IET Iteratioal Radar Coferece 23, Xi A, Chia, Apr 4-6, 23 [2] J Capo, "igh resolutio frequecy-aveumber spectrum aalysis," Proc IEEE, vol 57, o 8, pp48-48, Aug 969 [3] YP Liu, Q Wa, " Robust beamformer based o total variatio miimisatio ad sparse costrait," Electroic Letters, vol46, o25, pp , 2 [4] Y Zhag, BP Ng ad Q Wa, Sidelobe suppressio for adaptive beamformig ith sparse costrait o beam patter, Electroics Letters, vol44, o, pp65-66, 29 [5] R O Schmidt, "ultiple emitter locatio ad sigal parameter estimatio," IEEE Tras Ateas Propag, vol 34, o 3, pp276-28, ar 986 [6] D Sacchi, T J Ulrych, ad C J Waler, "Iterpolatio ad extrapolatio usig a high-resolutio discrete fourier trasform," IEEE Tras Sigal Process, vol 46, o, pp3-38, Ja 998 [7] B D Jeffs, "Sparse iverse solutio methods for sigal ad image processig applicatios," Proc IEEE It Cof Acoust, Speech, Sigal Process, vol 3, pp , 998 [8] I F Goroditsy, ad B D Rao, "Sparse sigal recostructio from limited data usig FOCUSS: A reeighted miimum orm algorithm," IEEE Tras Sigal Process, vol 45, o 3, pp6-66, ar 997 [9] J J Fuchs, "O the applicatio of the global matched filter to DOA estimatio ith uiform circular arrays," IEEE Tras Sigal Process, Vol 49, No 4, pp72-79, Apr 2 [] D alioutov, Ceti, ad A S Willsy, "A sparse sigal recostruc tio perspective for source localiza tio ith sesor arrays," IEEE Tras Sigal Process, vol 53, o 8, pp3-322, Aug 25 []E J Cads, B Wai, ad S P Boyd, "Ehacig sparsity by re eighted L miimizatio," Joural of Fourier Aalysis ad Applicatios, vol 4, o 5, pp877-95, Dec