The DOA Estimation of Multiple Signals based on Weighting MUSIC Algorithm

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1 , pp The DOA Estimatio of ultiple Sigals based o Weightig USIC Algorithm Chagga Shu a, Yumi Liu State Key Laboratory of IPOC, Beijig Uiversity of Posts ad Telecommuicatios, Beijig , Chia a bupt_scg@163.com Abstract. The classical USIC algorithm does some preprocessig of the sigal covariace matrix by eigevalue decompositio, ad the two orthogoal subspaces called sigal subspace ad oise subspace are divided. Thus the effect of evirometal oise is limited to a certai extet. With the decrease of the sigal agle iterval ad the sigal-oise ratio, the classical USIC algorithm has certai limitatios i multiple sigals estimatio such as loss ad cofusio, which meas the method of estimatio is uable to distiguish those sigals we eed. To this problem, a modified USIC algorithm is proposed. By itroducig a weightig fuctio i the spatial spectrum, some weighted operatio are give to the steerig vectors i USIC spatial spectrum, makig the most of subspaces ad there eigevalues. The subsequet simulatios are take to verify that uder the coditio of a small sigal agle iterval ad a low sigal-oise ratio, the improved algorithm could provide a better sigal resolutio ad offer more accurate estimated results tha the classical. Keywords: USIC algorithm, weightig fuctio, eigevalue decompositio. 1 Itroductio ultiple sigal classificatio algorithm [1] (USIC) is oe of the methods about array sigal processig [-4]. This algorithm has a good performace of sigal estimatio, for istace, a sigificat estimatio variace which is close to the cramer-rao boud ad a moderate computatioal work. Takig eigevalue decompositio [5] (EVD) with the covariace matrix which is made up of array sigals, the two subspaces called sigal subspace ad oise subspace are divided ad the orthogoality betwee them is utilized to build the spatial spectrum which iclude the sigal parameters, such as Directio of Arrival [6,7] (DOA) ad locatio. The classical USIC algorithm has certai limitatios i DOA estimatio especially i multiple sigals estimatio. Whe the several icidet agles of sigals are close, the divisio of subspaces appears blurred which causes a certai loss ad cofusio of sigal parameters. To get more accurate iformatio about target azimuth, some improvemets are put forward. A ovel weightig fuctio W is itroduced to the USIC spatial spectrum. By some weighted operatio to the steerig vectors i spatial ISSN: ASTL Copyright 016 SERSC

2 spectrum, the improved algorithm could provide a better oise suppressio ad more accurate estimated results. Research ethod Cosider a uiform liear array (ULA) for the coveiece purpose. Some array parameters are give. The umber of array elemets is. The spacig betwee two adjacet elemets is d 0.5, where is the carrier wavelegth. Suppose that there are N ( N ) far-field arrowbad sigals arrive at the ULA with a agle, as show i Fig.1. Fig.1. Uiform liear array The evirometal parameters are also give. Suppose that there is a zero mea ad the variace Gaussia white oise (GWN) i the eviromet, havig o correlatio with the sigals moitored. The data received by array elemets ca be expressed as, X ( t) AS( t) N( t) (1) Where A [ ( 1), ( ),, ( N )] deotes the array maifold matrix, deotes the steerig vector. S( t) [ s 1( t), s( t),, s ( )] T N t j si( ) / j ( 1)si( ) / ( ) [1, e i,, e i ] i deotes the vector of source waveforms, N( t) [ 1( t), ( t),, ( )] T t deotes the vector of oise received by array elemets. Some coditios should be satisfied before we use USIC algorithm to estimate the sigal parameters. First, the umber of array elemets should be more tha the sigal umber. The, the steerig vectors made up by sigals with differet icidet agles are idepedet. Also, a o-sigular matrix for covariace matrix is ecessary. The last, the evirometal oise should satisfy the coditio as follow, Copyright 016 SERSC 103

3 E{ i ( t)} 0 E{ i( t) i ( t)} T E{ i( t) i ( t)} 0 Cosider a covariace matrix of array sigals R X, take eigevalue decompositio ad arrage the eigevalues from large to small, The we get 1 N N 1 (3) R U U U U (4) X s s s Where Us [ u1, u un] deotes the sigal subspace, correspodig to the N large eigevalues of s diag{ 1,, N}. U [ un 1, un, u ] deotes the oise subspace, correspodig to the N small eigevalues of diag{,, }. Whe k N 1,,, N1 N To come to this case, we get, Further, RX k kuk uk R AR A I u X k ( S ) k AR A u S k 1 Auk () (5) 0 (6) 0 (7) N1 AU N( - N ) 0 (8) From the formula above we kow that the array maifold ad oise subspace have orthogoality. It is worth otig that the sigal subspace is spaed by the steerig vector of array maifold. We ca cosider that the sigal subspace ad oise subspace satisfy the orthogoal coditio, marked spau s spau. The the spatial spectrum of USIC is costructed as follow, P ( ) 1 ( ) U U a( ) (9) USIC a The scaig the spatial spectrum withi the searchig scope, the N icidet agles will be foud. The steps of algorithm are as follows, 1. Accordig to the data received by array, calculate the covariace matrix R.. Do eigevalue decompositio with R, get the eigevalues ad correspodig eigevectors 104 Copyright 016 SERSC

4 3. Determie the sigal umbers ad divide the sigal subspace U s ad ocie subspace U. 4. Costruct the spatial spectrum of USIC algorithm P USIC, search the spectral peaks ad fid the icidet agles of sigals. To get more accurate iformatio about target azimuth, some improvemets are put forward. I classical USIC algorithm, if the umber of sigals we estimated is error, the oise subspace divided is ot i coformity with the actual, leadig to a big error eve a failed DOA estimatio, especially i the case of a small SNR. A advisable method is that all the characteristic vectors of the sigal covariace matrix are cosidered ad a ovel weightig should be take to the characteristic vectors. After this, the proper weight coefficiet will reduce the effect of sigal eigevectors ad the oise eigevectors are etirely used. The ovel weightig fuctio W is itroduced to the USIC spatial spectrum, the the spatial spectrum chages as, P q q W 0 0 q 0 0 (10) 1 a ( ) Uˆ Uˆ WUˆ Uˆ a( ) (11) USIC N N N N I the formula above, i, i 1,, is the eigevalues of the array covariace matrix. q 0 is a customized parameter. Differet W causes differet estimatio performace. Whe W is a uit matrix, that meas a uiform weightig is selected to the eigevectors, which evolve ito a classical W e e, where 1 [1,0,,0] T e L belogs to a vector of algorithm. Whe 1 1 oise subspaces, the miimum ier product algorithm correspod. 3 Coclusio This article proposes the cyclic cross correlatio USIC algorithm to estimate multiple sigals with less array elemets. Simulatio results show that the cyclic cross correlatio USIC algorithm ca provide a better estimatio performace especially whe the sigals umber is more tha the array elemets. This algorithm has a better oise suppressio ad a larger array aperture tha the stadard, havig a certai referece value. Copyright 016 SERSC 105

5 Ackowledgemets. The author ackowledges the fudig of followig sciece foudatio: Natioal Natural Sciece Foudatio of Chia (Grat Nos , , , ad ) Refereces 1. Schmidt, R.: ultiple emitter locatio ad sigal parameter estimatio [J]. IEEE trasactios o Ateas ad Propagatio, (1986), 34(3): Kim, J.., Lee, O.K., Ye, J.C.: Compressive USIC: Revisitig the Lik betwee Compressive Sesig ad Array Sigal Processig [J]. IEEE trasactios o iformatio theory, (01) vol.58, o.1, pp: Tweg, R., Porat, B.: Numerical optimizatio method for array sigal processig [J]. IEEE trasactios o aerospace ad electroic systems, (1999) vol.35, o., pp: Otterste, B., Stoica, P., Roy, R.: Covariace matchig estimatio techiques for array sigal processig applicatios [J]. Digital sigal processig, (1998) vol.8, o.3, pp: Jeffers, R., Bell, K.L., Va,.L.: Trees. Broadbad passive rage estimatio usig USIC [C]. IEEE Iteratioal Coferece o Acoustics, Speech ad Sigal Processig, (00) vol.3, pp: Becheikh,. L., Wag, Y.: Joit DOD-DOA estimatio usig combied ESPRIT- USIC approach i IO radar [J]. Electroics Letters, (010) vol.46, o. 15, pp: Li, J.-D., Fag, W.-., Wag, Y.Y.: FSF USIC for joit DOA ad frequecy estimatio ad its performace aalysis [J]. IEEE Trasactios o sigal processig, (006) vol.54, o. 1, pp: Copyright 016 SERSC