DOA Estimation Method Based on Differential Covariance Matrix

Transcription

1 178 JOURNAL OF COMPUERS, VOL. 8, NO. 4, APRIL 213 DOA Etmaton Method Baed on Dfferental Covarance Matr Je Zhou, Guangjun L and Zh Zheng Unverty of Electronc Scence and echnology of Chna/School of Communcaton and Informaton Engneerng, Chengdu, Chna Emal: @163.com Abtract In mot cae, we etmate the drecton-ofarrval (DOA) of gnal by ung ubpace method, whch obtan DOA etmaton through the egenvalue decompoton of the ample covarance matr. But when the gnal are correlatve, the ample covarance matr not full rank, and o the ubpace method are not applcable. Ung a unform lnear array (ULA), a new method for DOA etmaton propoed n th paper, whch can etmate the DOA of multple gnal n patally nontatonary uncorrelated noe envronment. he method frtly contruct a new covarance matr by ung forward/backward moothng technque, and then the matr tranformed and contruct the dfferental covarance matr. Smulaton reult how that the propoed method can reduce the nfluence of the noe and mprove the performance of DOA etmaton. In addton, t alo applcable to the correlatve gnal. Inde erm DOA etmaton, ubpace method, ample covarance matr, dfferental covarance matr, correlatve gnal I. INRODUCION In array gnal proceng, drecton-of-arrval (DOA) etmaton ung an array of patally dtrbuted enor an eental and mportant problem [1], [2]. It ha typcally played an mportant role n a varety of applcaton uch a wrele communcaton [3], audo proceng [4], radar and onar [], [6]. he problem ha been tuded for decade, and many hgh reoluton method have been propoed. he well-known etmator the mamum-lkelhood (ML) etmator [7]. owever, the ML etmator computatonally qute epenve due to the requred multvarate nonlnear mamzaton. herefore, the reearcher have derved the o-called egentructure or gnal ubpace method uch a MUSIC [8] and ESPRI [9] whch repreent more computatonally attractve method for DOA etmaton [1]. he ubpace method are baed on the orthogonalty between the gnal ubpace and the noe ubpace, whch requre that the ample covarance matr full Manucrpt receved October 1, 212; reved November 2, 212; accepted November 2, 212. Correpondng author: Je Zhou. rank. But for correlatve gnal, the gnal ubpace dverge nto the noe ubpace, and the ample covarance matr not full rank. hu, the ubpace method are not applcable n th cae. Moreover, the etng method [11-1] for DOA etmaton of correlatve gnal are degned for the known whte noe, and can t be appled to other noe cae. In th paper, we have propoed a new DOA etmaton algorthm, whch can be ued n the patal nontatonary uncorrelated noe envronment. o deal wth the correlatve gnal, the forward/backward moothng technque frtly ued to keep full rank of the ample covarance matr. And then the dfferental covarance matr contructed. heoretcal analy and mulaton reult ndcate that the algorthm can reduce the nfluence of the noe and mprove etmaton performance. II. SIGNAL MODEL Aume that K narrowband far-feld gnal ource mpngng on a ULA of M enor element, eparated by L wavelength, wth the phae reference at the center of the array. It uppoed that the DOA of the th gnal θ (=1, 2,, K). he receved gnal of the ULA can be epreed a () t = AS()+ t n () t (1) where S(t)=[ 1 (t) 2 (t) K (t)] a K 1 gnal vector; n(t) the M 1 addtve noe vector; A=[a 1 a 2 a K ] -j2 πl( m)n θ / λ the M K teerng matr and a = [1 Le L e ] - j2 πl( M)n θ / λ (Let z - j2π Ln θ / λ = e, the M 1 teerng vector of the ULA a m M = [1 Lz Lz ] ).Where λ the wavelength of the mpngng gnal. When a gnal DSSS gnal, va carrer demodulaton, t can be wrtten a j () t = Pd () t c () t e ϕ (2) where d (t) and c (t) are, repectvely, the data eral and the PN code; E the gnal power and φ the ntal phae. When the PN code are dfferent, the gnal are uncorrelated. III. DOA ESIMAION ALGORIM 213 ACADEMY PUBLISER do:1.434/jcp

2 JOURNAL OF COMPUERS, VOL. 8, NO. 4, APRIL Multpath effect an mportant reaon whch lead to DSSS gnal beng correlatve. Wthout lo of generalty, t uppoed that the th gnal generate one multpath duplcate. α the comple fadng factor. he covarance matr of the receved data R = ARA + R n (3) where R and R n are, repectvely, the gnal covarance matr and the noe covarance matr. Becaue there are multpath duplcate, R a follow: R = E [ ( t ) ( t) ] j P1 L α1p1 L M O M M O M L PK L α KP K = 2 α1p1 L α1p1 L M O M M O M 2 αkpk αkp L L K In the condton of whte noe, there R n =σ 2 I, σ 2 the noe varance, I the M M dentty matr. Under the crcumtance, the teerng vector alo nclude multpath duplcate, o A=[a 1 a K a 1 a K ], a =[1 z m-1 z M-1 ] and z =ep(-j2πlnφ /λ), where Φ the DOA of the th gnal multpath duplcate gnal. From Eq(4), t can be known that R not full rank when ome gnal are correlatve, and the gnal ubpace are no longer orthogonal to the noe ubpace. hu, the conventonal MUSIC n t applcable. When the noe patal nontatonary uncorrelated, R n the followng dagonal matr whoe dagonal element are dfferent from each other M (4) R = dag( σ σ Lσ ) () n he covarance matr R generally etmated by the followng ample covarance matr M M = 1 ˆ R = 1/ = XX (6) where a n 1 column vector whch cont of n naphot of the th enor element; X a n M matr. In order to compenate the rank lo of R whch caued by the correlaton between gnal, we contruct the forward/backward moothng covarance matr a follow [16]: ˆ ˆ* FB = + R 1/2( R ΩR Ω ) (7) where * note the conjugaton. Ω the revere order matr a follow: L 1 Ω= M N M (8) 1 L he matr R FB need to be tranformed n order to etmate the DOA of gnal, and the tranformaton hould not detroy the orthogonalty between the gnal ubpace and the noe ubpace. Defne the followng convertng matr M 1 = dag(1 ρlρ ) (9) where < ρ < 1. In order to reduce noe, we contruct the dfferental covarance matr a follow: D = j( FB FB ) R R R (1) where j = 1. Net, we wll prove that the condton whch the egenvector of R D are orthogonal to the noe ubpace M >2K (K nclude the number of multpath duplcate). hat, the number of enor element more than twce the number of gnal. Moreover, when M >2K, the gnal ubpace orthogonal to the noe ubpace. When Eq.(6) ubttuted by Eq.(3), we can obtan R = j ( AR A + R + ΩA R A Ω+ ΩR Ω) * * * D n n j AR A + R + A R A + R * * * ( n Ω Ω Ω nω) (11) Obvouly, R n =ΩR * nω and R n = -1 R n. Eq(11) can be further mplfed a: R = j ( AR A + ΩA R A Ω) * * * D j AR A + A R A * * * ( Ω Ω) Drecton vector a meet the followng relatonhp: M 1 * j2 L( M 1)n / = (12) z a e π θ λ a (13) Subttute Eq.(13) nto Eq.(12), there : where Η R = j A( R + GR G ) A * D * Η ja( R + GR G ) A M M M 1 2 (14) G = dag( z z Lz K ) (1) Baed on the above dervaton, R D han t ncluded the noe covarance matr, and thu the effect from noe ha been elmnated. R D can be further converted a the followng matr: * Η -j( R + GRG ) RD = [ A A] * Η j( R + GRG ) (16) [ AA] A a M K matr. When M >K, A column full rank and rank(a)=k. So [A -1 A] and [A A] are all the M 2K matr and column full rank, the rank 2K. he mddle matr of Eq.(16) full rank, t rank alo 2K. Becaue R D a M M matr, o only when M > 2K and rank(r D )=2K, the DOA nformaton of gnal wll not be lot. R D have 2K nonzero egenvalue and M-2K zero egenvalue. he egenvector correpondng to the zero egenvalue are orthogonal to the matr [A -1 A]. hat, the noe ubpace orthogonal to the gnal ubpace. 213 ACADEMY PUBLISER

3 18 JOURNAL OF COMPUERS, VOL. 8, NO. 4, APRIL 213 Becaue a and -1 a are orthogonal to the zero egenvector of R D, we can etmate the DOA of gnal by contructng the followng patal pectrum: M -1 P( θ) = a ( θ) uu a( θ) + a ( θ) uu a ( θ) = 2K+ 1 ( π < θ < π) (17) where u the zero egenvector. he DOA value correpondng to the pectrum-peak the DOA etmaton. IV. SIMULAION RESULS AND ANALYSIS In th ecton, we nvetgate the performance of the propoed algorthm through everal mulaton eample. In our mulaton, the ULA ha 8 enor element, the dtance between two adjacent enor L=λ/2, and ρ=.8. Aume that all gnal are DSSS gnal. It uppoed the array noe the patal nontatonary uncorrelated whte noe, the noe power of the econd, thrd and ffth element are, repectvely, 2, 4 and, and that of other element are all 1. hat, R n =dag( ). he gnal-to-noe rato (SNR) of the th gnal defned a t gnal power to the noe power of the frt enor element. Baed on the correlatve degree between gnal ource, three dfferent cae are condered. Cae 1: All gnal are ncoherent In the frt eample, we llutrate the performance of the propoed algorthm for two equpower ncoherent gnal a well a a comparon wth the conventonal MUISC. Aume that the SNR of two gnal are all - db, and ther DOA are, repectvely, 48 and 13. Fg. 1 how the earchng pectrum of the propoed algorthm, wherea Fg. 2 dplay the conventonal MUSIC pectrum. From Fg.1, we can clearly oberve that the patal pectrum wth the cot functon (17) ehbt two promnent peak cloe to 48 and 13, whch ndcate that the propoed algorthm can etmate effectvely the DOA of two ncoherent gnal. Smlarly, Fg.2 ha alo two peak cloe to 48 and 13, but the peak are not more promnent than thoe of Fg.1. h ndcate that our algorthm can provde a better performance than the conventonal MUISC. he man reaon that the propoed algorthm reduce the array noe. power pectrum (db) Azmuth(degree) SNR=-dB Fgure 1. he patal pectrum of the propoed algorthm. (two ncoherent gnal) power pectrum (db) Azmuth(degree) SNR=-dB Fgure 2. he patal pectrum of the MUSIC algorthm. (two ncoherent gnal) Cae 2: All gnal are coherent In the econd eample, we nvetgate the performance of the propoed algorthm for two coherent gnal. It uppoed that the DOA of gnal 1 74, gnal 2 a multpath duplcate of gnal 1 and t DOA 112. Compared wth gnal 1, gnal 2 arrve at the array after two chp delay. So gnal 2 coherent wth gnal 1. Aume that the SNR of gnal 1 and 2 are, repectvely, -db and -8dB. he earchng pectrum of the propoed algorthm llutrated n Fg. 3. It can be een that the patal pectrum ha two promnent peak approachng to 74 and 112. herefore, our approach can be ued to etmate multple coherent gnal. 213 ACADEMY PUBLISER

4 JOURNAL OF COMPUERS, VOL. 8, NO. 4, APRIL power pectrum (db) Azmuth(degree) SNR=-dB Fgure 3. he patal pectrum of the propoed algorthm. (two coherent gnal) Cae 3: he partal gnal are coherent In the thrd eample, we conder the cae of three gnal. Aume that gnal 2 the multpath duplcate of gnal 1, and gnal 3 rrelevant to gnal 1. he DOA of gnal 1, 2 and 3 are, repectvely,, 74 and 136. her SNR are, repectvely, -db, -8dB and -6dB. he patal pectrum of the propoed algorthm demontrated n Fg.4. A can be een from th fgure, there are three promnent peak approachng to, 74 and 136. herefore, our approach alo applcable when partal gnal are coherent. power pectrum (db) Azmuth(degree) SNR=-dB Fgure 4. he patal pectrum of the propoed algorthm. (three gnal, two are coherent) V. CONCLUSIONS In th paper, the DOA etmaton problem of correlatve gnal ha been dcued n detal, and an algorthm baed on dfferental covarance matr propoed. Smulaton reult how that the propoed algorthm can reduce evdently the nfluence of the noe. Moreover, compared to the conventonal MUSIC, the propoed algorthm can provde a better etmaton performance n the cae of patal nontatonary uncorrelated noe. Furthermore, the algorthm can alo deal wth coherent gnal and partal coherent gnal. ACKNOWLEDGMEN he author would lke to thank the anonymou revewer for ther contructve and helpful uggeton and comment. h work wa upported by a grant from the Natonal Natural Scence Foundaton of Chna (No ). REFERENCES [1] S. U. Plla, Array Sgnal Proceng, Sprnger-Verlag, [2] M. Brandten and D. Ward, Ed., Mcrophone Array, Sprnger, 21. [3] A. Klouche-Djedd and M. Fujta, "Adaptve array enor proceng applcaton for moble telephone communcaton," IEEE ran. Vehcular echnology, vol. 4, pp , [4] D. Khaykn and B. Rafaely, "Coherent gnal drecton-ofarrval etmaton ung a phercal mcrophone array: Frequency moothng approach," IEEE Workhop on ASPAA, pp , 29. [] J. Mn,. Janguo,. We, and C. Fuzhao, "Reearch on target DOA etmaton method ung MIMO onar," n Indutral Electronc and Applcaton, 29. ICIEA 29. 4th IEEE Conference on, pp , 29. [6]. Jang, Y. Zhang,.J. Lu, et al, Jont polarzaton and angle etmaton for robut mult-target localzaton n Btatc MIMO radar, Journal of Software, vol. 7, no. 12, pp , 212. [7] P. Stoca and A. Nehora, Performance tudy of condtonal and uncondtonal drecton-of-arrval etmaton, IEEE ran. Acout., Speech, Sgnal Proce., vol. 38, pp , October 199. [8] R.O. Schmdt, Multple emtter locaton and gnal parameter etmaton, IEEE ran. Antenna and Propagaton, vol. 34, no. 3, pp , March [9] R. Roy,. Kalath, and A. B. Gerhman, ESPRI, etmaton of gnal parameter va rotatonal nvarance technque, IEEE ran. Acout., Speech, Sgnal Proce., vol. 37, pp , July [1] S. B. aen, F. Belll, A. Samet and S. Affe, DOA etmaton of temporally and patally correlated narrowband noncrcular ource n patally correlated whte noe, IEEE ran. Sgnal Proce., vol. 9, no. 9, pp , September 211. [11].J. Shan and M. Wa, On patal moothng for drecton-of-arrval etmaton of coherent gnal, IEEE ran. on ASSP, vol. 33, no. 4, pp , Augut 198. [12] S.U. Plla and B.. Kwon, Forward/backward patal moothng technque for coherent gnal dentfcaton, IEEE ran. on ASSP, vol. 37, no. 1, pp. 8-1, June [13] B. D. Rao and K.V.S. ar, Effect of patal moothng on the Performance of MUSIC and the mnmum-norn method, IEE Proc. -F, vol. 137, no. 6, pp , June 199. [14] D. Kundu, Modfed MUSIC algorthm for etmaton DOA of gnal, Sgnal Proce., vol. 48, pp. 8-91, [1] Y.. an and J.K. Wang, he central DOA etmaton algorthm baed on upport vector regreon for coherently dtrbuted ource, Journal of Software, vol. 7, no. 12, pp , 212. [16] D.A. Lnebarger, R.D. DeGroat and E.M. Dowlng, Effcent drecton-fndng method employng forwardbackward averagng, IEEE ran. Sgnal Proce., vol. 42, no. 8, pp , Augut ACADEMY PUBLISER

5 182 JOURNAL OF COMPUERS, VOL. 8, NO. 4, APRIL 213 Je Zhou wa born n enan, Chna, n e receved the M.S. degree n communcaton and nformaton ytem from Graduate School of Chna Academy of Engneerng Phyc, Manyang, Chna, n 21 and currently workng toward the Ph.D. degree n Communcaton and Informaton ytem, at the Unverty of Electronc Scence and echnology of Chna. reearch nteret are focued on array gnal proceng, mart antenna and wrele communcaton. Guangjun L wa born n ebe, Chna, n 19. e currently a full profeor n School of Communcaton and Informaton Engneerng, Unverty of Electronc Scence and echnology of Chna. reearch nteret manly nclude communcaton gnal proceng and wrele peronal communcaton. Zh Zheng wa born n Schuan, Chna, on Aprl, 198. e receved the M.S. and Ph.D. degree n electronc engneerng (EE) from the Unverty of Electronc Scence and echnology of Chna, Chengdu, Chna, n 27 and 211, repectvely. reearch nteret are focued on array gnal proceng, mart antenna and wrele communcaton. 213 ACADEMY PUBLISER