Research on Modified Root-MUSIC Algorithm of DOA Estimation Based on Covariance Matrix Reconstruction

Transcription

1 Sensors & ransducers, Vol. 78, Issue 9, September 04, pp. 4-8 Sensors & ransducers 04 by IFSA Publshng, S. L. Research on Modfed Root-MUSIC Algorthm of DOA Estmaton Based on Covarance Matrx Reconstructon Changgan SU,, Yumn LIU, Zhongyuan YU, We WANG, Ywe PENG, Wen ZANG and esheng WU State Key Laboratory of Informaton Photoncs and Optcal Communcatons, Bejng Unversty of Posts and elecommuncatons, Bejng 00876, Chna Insttute of Semconductors, Chnese Academy of Scences, Bejng 00083, Chna el.: E-mal: Receved: 7 June 04 /Accepted: 9 August 04 /Publshed: 30 September 04 Abstract: In the standard root multple sgnal classfcaton algorthm, the performance of drecton of arrval estmaton wll reduce and even lose effect n crcumstances that a low sgnal nose rato and a small sgnals nterval. By reconstructng and weghtng the covarance matrx of receved sgnal, the modfed algorthm can provde more accurate estmaton results. he computer smulaton and performance analyss are gven next, whch show that under the condton of lower sgnal nose rato and stronger correlaton between sgnals, the proposed modfed algorthm could provde preferable azmuth estmatng performance than the standard method. Copyrght 04 IFSA Publshng, S. L. Keywords: DOA estmaton, Root-MUSIC, Covarance matrx reconstructon.. Introducton Root-Multple sgnal classfcaton (Root- MUSIC) s a classc method of the azmuth estmaton algorthms. In ths algorthm, the orthogonalty of sgnal subspace and nose subspace s used to calculate the drecton of arrval (DOA) [, ]. hs approach results n good performance when the sgnals n space are ndependent of each other or they have a small correlaton. Nevertheless, there exst some coherent sgnals generated by the multpath transmsson and chromatc dsperson n a real envronment. hese coherent sgnals reduce the rank of the covarance matrx, leadng to a stuaton that the algorthm cannot dvde the sgnal subspace and nose subspace correctly. In ths scenaro, some steerng vectors of coherent sources and nose subspace wll be not orthogonal to each other, thus causes the omsson of spatal spectrum estmaton [3, 4]. Modfed Root-MUSIC algorthm s proposed n ths paper. By reconstructng and weghtng the covarance matrx of receved data, a new covarance matrx wll be used for to calculate the DOA of sgnal. When the correlaton exst n sgnals and n crcumstances that a low sgnal nose rato (SNR) and a small sgnals nterval, the modfed Root-MUSIC algorthm can estmate the sgnals more accurately than the standard algorthm.. Research Method.. Sgnal Model Consder a unform lnear array (ULA) whch conssts of M array elements, the spacng s d, λ s 4

2 Sensors & ransducers, Vol. 78, Issue 9, September 04, pp. 4-8 the carrer wavelength. ypothesze there are N narrowband sgnals ncdent on each array element wth an angle, as Fg.. {() R x t x ()} t ARSA σ IM =Ε = +, (6) where A denotes the array manfold matrx, R S s defned as R =Ε { S( t) S ( t)}, (7) S Fg.. Unform lnear array. Assumng that there s a Gaussan whte nose n the envronment, wth a zero mean and the varance σ, whch has no correlaton wth sgnals. he data n receved by array elements can be expressed as the followng model, where A s defned as X () t = AS() t + N() t, () A αθ αθ αθ N = [ ( ), ( ),, ( )], () denotes the array manfold matrx, αθ ( ) s defned as jπsn( θ ) / λ j π( M )sn( θ ) / λ αθ ( ) = [, e,, e ], (3) denotes the steerng vector. St () s defned as St () = [ s(), t s(), t, s ()] t, (4) denotes the vector of source waveforms. Nt () s defned as Nt () = [ n(), t n(), t, n ()] t, (5) denotes the vector of nose receved by array elements []... Standard MUSIC algorthm handles wth the covarance matrx of the receved data R, dvde the characterstc space nto two subspaces by egenvalue decomposton (EVD), sgnal subspace and nose subspace, usng the orthogonalty of the two subspaces to estmate the DOA of the sgnals. Defne R as follows N M denotes the covarance matrx of sgnals, σ denotes the nose power, I M s the M M unt matrx. Utlzng the EVD for R, two matrx can be get, S and U. S denotes the sgnal subspace, consstng of N large egenvalues. U denotes the nose subspace, consstng of M N small egenvalues. It can be proved that S and U are orthogonal [], therefore, MUSIC spatal spectrum can be constructed as follows [5] ( θ ) = a ( θ) UU a( θ), (8) P MUSIC Root-MUSIC algorthm [6-9] s a polynomal form of standard MUSIC, fndng of the roots of polynomals nstead of searchng the space spectrum peak. Defne the polynomal as where f ( z) = e B( z), = N +, N +,, M, (9) e denotes M N feature vectors whch correspondng to the small egenvalues of the covarance matrx. Bz ( ) s defned as M Bz () = [,, zz,, z ], (0) denotes the steerng vector of array sgnals. On account of Bz ( ) and nose subspace s orthogonal, the polynomal can be changed as f ( z) = B UU B, () Under the condton that the covarance matrx and the sgnal number are known, the DOA of sgnals can be calculated by solvng the roots of polynomal z as follow λ θ = arcsn( arg( z)) =,,, N, () π d.3. Modfed Root-MUSIC he modfed Root-MUSIC algorthm s put forward n ths secton. As s well known that, for the deal ndependent sgnal sources, the covarance matrx R has oepltz nature, whereas n realty, the oepltz performance s destroyed because of lmted snapshot, system 5

3 Sensors & ransducers, Vol. 78, Issue 9, September 04, pp. 4-8 error and coherent sources. hs knd of stuaton reduces the performance of DOA estmaton whch base on the covarance matrx decomposton. herefore, the covarance matrx R can be changed nto oepltz matrx [0] as follow R R IR I * = ( + )/, (3) where the symbol * denotes conjugate operaton, I * s a M M reverse unt matrx, R denotes the conjugate matrx of R. Do EVD wth R, S and U wll be get. In order to reduce the effects of mprecse source number estmaton, a consderable method of weghtng wth U s as follow Success, f ˆ θ ˆ ˆ ˆ θ + θ θ < θ θ, (8) fal, other hen the probablty of success (POS) estmaton η wll be Nsuccess η = 00%, (9) N all where N success denotes the number of success estmaton, N all denotes the number of estmaton n total. he smulaton results are shown as Fg., Fg. 3 and Fg. 4. =[ u, u,, u,], (4) β β β U λ λ λ M N M N 0.9 Modfed Root-MUSIC where λ ( =,,, M N) s egenvalues of the covarance matrx R, β [0,] denotes the weghtng coeffcent. By weghtng the egenvectors, dfferent egenvectors can make dfferent functons n MUSIC spectrum that the algorthm could mantan a preferable estmaton performance under the underestmaton of source number condton. Replace U wth the newu, the formula f( z ) s changed as follow f ( z) = B U U B, (5) Consderng the easness of root, the formula f( z ) could be rewrote n a further form M ( ) ( ) ( ) f z = z B z U U B z, (6) Under the same condton above, the DOA of sgnals can stll be calculated by formula, λ θ = arcsn( arg( z)) =,,, N, (7) π d 3. Results and Dscusson Some smulaton results wll be presented n ths secton n order to prove the correctness and effectveness of the modfed algorthm whch has be put forward earler n the artcle. Consderng a unform lnear array (ULA) whch conssts of 0 array elements. he nterelement spacng d s 0.5λ. he number of narrowband sgnals N s. wo narrow-band coherent sources arrve at the array n ncdence angles θ, θ, the estmated values of whch are ˆ θ and ˆ θ. Defne an estmaton result as RMSE( ) RMSE( ) Probablty of Success SNR(dB) Fg.. DOA estmaton RMSE aganst SNR. Modfed Root-MUSIC Δθ( ) Fg. 3. DOA estmaton RMSE aganst Δ θ. Modfed Root-MUSIC SNR(dB) Fg. 4. POS aganst SNR. 6

4 Sensors & ransducers, Vol. 78, Issue 9, September 04, pp. 4-8 Fg. shows the DOA estmaton root mean square errors (RMSE) of two methods, the standard Root-MUSIC, the modfed Root-MUSIC respectvely, aganst the SNR of sgnals, under the condton of θ = π /6, θ = π /3.Usng 00 Monte Carlo smulatons to calculate the RMSE, t could be defned as 00 ( ( ) ( ) ) 00 j=, j j=, j RMSE = ˆ ˆ 00 θ θ + θ θ, (0) As we can see from Fg., under the condton of hgher SNR, 0 db, e.g., the two methods have good performance of DOA estmaton, wth the [RMSE (standard), RMSE (modfed)]=[0.08, 0.0 ]. Wth the decrease of the SNR, the RMSE of standard Root-MUSIC algorthm ncreases gradually, nevertheless, modfed Root-MUSIC algorthm stll has a recevable performance, 0dB, e.g., wth the [RMSE(standard), RMSE(modfed)]=[0.83, 0.38 ]. he curves n Fg. ndcate that the modfed algorthm has a lower RMSE than the standard whch means more precse azmuth estmaton under the same SNR. Fg. 3 shows the DOA estmaton RMSE of two methods, the standard Root-MUSIC, the modfed Root-MUSIC respectvely, aganst the angle dfference between two sgnals Δθ= θ θ, under the condton of SNR = 0dB. As shown n Fg. 3, the x axs s the angle dfference Δ θ, and the y axs s the DOA estmaton RMSE, usng a logarthmc coordnates axes because of the large value. When Δ θ s relatvely small, e.g., the correlaton between the two sgnals s strong, also the mutual nterference between them should not be neglected, wth the [RMSE (standard), RMSE (modfed)]=[33.7, 4. ], unable to estmate the DOA of sgnals correctly both. When Δ θ s an approprate value, 8, e.g., wth the [RMSE(standard), RMSE(modfed)]=[.55, 0.43 ]. he curves n Fg. 3 ndcate that the modfed algorthm has a lower RMSE than the standard. It s known that the correlaton and mutual nterference between sgnals wll be stronger wth the decrease of the Angle dfference. By reconstructng and weghtng the covarance matrx of receved data, the modfed algorthm reduces the coherence of sgnal, whch provdes more precse azmuth estmaton. Fg. 4 shows the probablty of success (POS) estmaton of two methods aganst the SNR of sgnals, under the condton of θ = π /6, θ = π /3. As shown n ths fgure, the x axs s the SNR, and the y axs s the POS estmaton, range [0,]. Wth the ncrease of SNR, the POS of the two methods are both gradually ncreased. In the same SNR, the modfed Root-MUSIC has a hgher success probablty than the standard one. 0 db, e.g., wth [POS(standard), POS(modfed)]=[0.3, 0.53]. Another advantage s that to reach the condton of POS=, the modfed method need a smaller SNR than the standard method, wth [SNR(standard), SNR (modfed)]=[4 db, 0 db]. he curves n Fg. 4 ndcate that the modfed algorthm has a hgher POS than the standard whch means more effcent DOA estmaton under the same SNR. 4. Concluson Smulaton results show that, under the condton of lower SNR and stronger correlaton between sgnals, the modfed Root-MUSIC algorthm could provde preferable DOA estmatng performance than the standard method by reconstructng and weghtng the covarance matrx of receved sgnal wthout ncreasng computatonal complexty, whch has good reference value. Acknowledgements he author acknowledges the fundng of followng scence foundaton: Natonal Natural Scence Foundaton of Chna (Grant Nos , , , and 6750). References []. Schmdt R., Multple emtter locaton and sgnal parameter estmaton, IEEE ransacton on Antennas and Propagaton, Vol. 34, Issue 3, 986, pp []. R. Jeffers, K. L. Bell,. L. Van rees. Broadband passve range estmaton usng MUSIC, IEEE Internatonal Conference on Acoustcs, Speech, and Sgnal Processng (ICASSP'0), Orlando, USA, 3-7 May 00, pp [3]. Plla S. U., Kwon B.. Forward/backward spatal smoothng technques for coherent sgnal dentfcaton, IEEE ransactons on Acoustcs, Speech and Sgnal Processng, Vol. 37, Issue, 989, pp [4]. Gardner W. A., Napoltano A., Paura L. Cyclostatonarty: alf a century of research, Sgnal Processng, Vol. 86, Issue 4, 006, pp [5]. Zhang Cong, u Mou Fa, Lu uan Zhang, Vrtual array based spatal smoothng method for drecton fndng of coherent sgnals, Acta Electronca Snca, Vol. 38, Issue 4, 00, pp [6]. Rao B. D, ar K. V. S., Performance analyss of Root-MUSIC, IEEE ransactons on Acoustcs, Speech, and Sgnal Processng, Vol. 37, Issue, 989, pp [7]. Chevaler Pascal, Ferreol Anne, Albera Laurent. gh-resoluton drecton fndng from hgher order statstcs: he q-music algorthm, IEEE ransactons on Sgnal Processng, Vol. 54, Issue 8, 006, pp [8]. Zoltowsk M. D., Kautz G. M., Slversten S. D. Beamspace Root-MUSIC, IEEE ransactons on Sgnal Processng, Vol. 4, Issue, 993, pp [9]. Zoltowsk M. D, Slversten S. D, Mathews C. P., Beamspace Root-MUSIC for mnmum redundancy 7

5 Sensors & ransducers, Vol. 78, Issue 9, September 04, pp. 4-8 lnear arrays, IEEE ransactons on Sgnal Processng, Vol. 4, Issue 7, 993, pp [0]. Jen Der Ln, Wen sen Fang, Yung Y Wang, et al., FSF MUSIC for jont DOA and frequency estmaton and ts performance analyss, IEEE ransactons on Sgnal Processng, Vol. 54, Issue, 006, pp Copyrght, Internatonal Frequency Sensor Assocaton (IFSA) Publshng, S. L. All rghts reserved. ( 8