An Improved Music Algorithm for DOA Estimation of Coherent Signals

Transcription

1 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp Senor & Tranducer 04 by IFSA Publhng, S. L. An Improved uc Algorthm for DOA Etmaton of Coherent Sgnal Yan Gao, Wenjng Chang, Zeng Pe, Zhengjang Wu School of Computer Scence and Technology, enan Polytechnc Unverty, o. 00 Century oad, Jaozuo, Chna Center for ado Admntraton & Technology Development, Xhua Unverty, Chengdu, Chna Tel.: E-mal: eceved: Aprl 04 /Accepted: 30 June 04 /Publhed: 3 July 04 Abtract: The etmaton of drecton of arrval (DOA) the man tudy of the drecton fndng and locaton of rado gnal, USIC algorthm the mot clac DOA etmaton. USIC algorthm can etmate ndependent gnal DOA effectvely, but t falure to coherent gnal. Concernng the ue, the modfed USIC algorthm wor when the correlatve gnal et. owever, the etmaton performance of modfed USIC algorthm wll deterorate n the cae that the gnal-noe rate low. Th paper mprove the modfed USIC algorthm by ung matr decompoton. Fnally, the computer mulaton reult prove that th propoed method ha better drecton fndng performance than the modfed USIC algorthm wth low gnal to noe rato. Copyrght 04 IFSA Publhng, S. L. Keyword: Drecton fndng and locaton, USIC algorthm, odfed USIC algorthm, Improved modfed USIC algorthm.. Introducton ado potonng an mportant wor of rado management. It the neceary mean to mantan the ar rado wave order, afeguard the legtmate rght of uer and combat the llegal occupaton of frequency reource. In moble communcaton, wrele locaton technology a method that determne the phycal poton of the rado gnal tranmttng termnal n a certan radu by ung the wrele gnal. The etmaton of drecton of arrval (DOA) of multple narrowband gnal an mportant problem n array gnal proceng ncludng radar, onar, rado atronomy, and moble communcaton []. Drecton of arrval etmaton to confrm multaneouly the patal poton of multple gnal n a certan pace, whch the angle of the multple gnal that reach the reference array. The far-feld gnal angle etmaton manly ung the array gnal proceng technology []. Array gnal proceng alo often referred to a the patal gnal proceng, where the patal pectrum etmaton a new patal gnal proceng technology developed n the lat 30 year. The USIC algorthm the mot clac patal pectrum etmaton algorthm, whch propoed by Schmdt [3]. USIC an acronym whch tand for ultple Sgnal Clafcaton. It bac dea egenvalue decompoton about the covarance matr of the arbtrary array output data, thereby we wll get the gnal ubpace whch correpondng to the gnal component and the noe ubpace whch orthogonal to the noe component, then we could contruct the pectral functon by ung the orthogonalty of the two ubpace, and etmate the gnal ncdence drecton by mean of earchng the 75

2 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp pectrum pea, fnally achevng the gnal hgh reoluton parameter etmaton. USIC algorthm ha a good etmaton performance. It etmaton complety moderate, but the USIC algorthm jut adjuted to the ndependent gnal. It neffectve to coherent gnal [4, 5]. Therefore the drecton fndng of coherent gnal ha been the focu of reearch, and many modfed USIC algorthm have been propoed to mprove the performance of the etmaton of the coherent gnal, that preproceng the data to elmnate or weaen the correlaton between ource. Th paper modfe the tradtonal USIC method utlzng the dea of patal moothng algorthm, and mae further mprovement on the bac of the modfed algorthm, compared wth the modfed method by computer mulatng, lat verfed the mproved algorthm of accuracy and effectvene.. uc Algorthm.. Array Sgnal athematcal odel In order to facltate the analy, Fg. how the deal tate of array gnal mathematcal model [6-8]: σ varance and the noe n each channel uncorrelated. 5) Each of the recevng channel ha the ame properte. Aume the S () t the wavefront gnal whch the th gnal ource ncdent the array and S () t narrowband gnal [9], o S () t can be repreented by the followng form: S () t = ()ep{ t jω t}, () where (t) for =,,, D the comple envelope of S () t, and w the angular frequency of the gnal S () t, we uppoe the D gnal ha the ame center frequency, o: π c ω = ω0 =, () where c the peed of electromagnetc wave, and the common gnal wavelength, t ndcate the tme electromagnetc wave through the antenna array. Then accordng to the narrowband aumpton [0], et the followng appromate: ( t t ) ( t), (3) Then the delayed wavefront gnal : θ θ d Fg.. Array gnal model. ) The detected gnal ha the ame unrelated polarzaton. Generally conder the gnal narrowband, and each gnal ource ha the center frequency w 0, and the number of the detected gnal ource D. ) We conder the unform lnear array that compoed of otropc enor, and > D. d the pacng between adjacent element, and d not more than half wavelength of the hghet frequency. 3) Antenna array the far feld of each gnal ource, namely the gnal receved by the antenna array the plane wave. 4) nm () t the recever noe component aumed to be Gauan dtrbuted wth zero mean and S ( t t) = ( t t)ep[ jω0( t t)] ()ep[ t jω0( t t)], (4) = S ( t)ep( jwt ) So, f the frt array element a reference pont, the comple receved gnal of the m th enor for the th gnal at tme can be epreed a: 0 πd nθ as ()ep[ t jm ( ) ] For m=,,, and =,,, D, (5) where a the mpact that the m th enor for the th gnal. We aume each enor nondrectonal, o defne a =. The DOA of the th ource gnal d nθ θ, and ( m ) the propagaton delay c between the frt enor and the m th enor. Conder the meaurement noe and all gnal ource, the output gnal of the m th enor can be epreed a: D πd nθ m ( t) = S ( t)ep[ j( m ) ] + n = m ( t), (6) 76

3 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp where nm () t the meaurement noe, the volume labeled m mean the volume belong to the m th enor and the volume labeled mean the volume belong to the th gnal ource. We et am( θ ) the repone functon of the m th enor for the th gnal. It can be ummarzed a follow: πd nθ am( θ) = ep[ j( m ) ], (7) Then the receved gnal m () t of the m th enor can be wrtten a: D () t = a ( θ ) S () t + n () t, (8) m m m = where S (t) the gnal trength of the th gnal. Ung vector notaton for the receved gnal of enor, the data model can be preented a: where X ( n) = AS( n) + ( n), (9) X () t = [ (), t (), t, ()] T t, (0) St () = [ (), t (), t, ()] T t, ()... e e... e jφ jφ jφd = [ ( θ), ( θ),... ( θd )] = A a a a j( ) φ j( ) φ j( ) φ D e e... e () π d φ = nθ, (3) t () = [ n(), t n(),..., t n ()] t T, (4) where =,,, D and T upercrpt denote tranpoe of a matr. The DOA problem fndng proper etmaton of θ, θ,, θd from a fnte number ( ) of data ample or naphot of m () taen at tme, =,, []. A a reult, the array gnal can be naturally regarded a the uperpoton of everal pace harmonc wth the noe nterference, thu the DOA etmaton problem wll be combned wth the pectral etmaton... The Egen-decompoton of Array Covarance atr ae the proceng about array output matr, then the covarance matr can be receved a: = E[ XX ], (5) where ndcate the conjugate tranpoe of a matr. Wth the above aumpton, the noe and gnal waveform are uncorrelated, and the noe at each enor a tatonary zero meanng comple whte Gauan proce. So, from (9) and (5), the covarance matr of receved gnal become: where = E[( AS + )( AS + ) ] = AE[ SS ] A + E [ ], (6) = A A + = E[ SS ], = σ I, (7) where the gnal correlaton matr. the noe correlaton matr. σ the noe power at each enor and I an dentty matr. In practce, the data covarance matr not avalable but a mamum lelhood etmate baed on a fnte number ( ) of data ample can be obtaned a: = X() X (), (8) = And the etmaton of drecton of arrval of ource baed on th ample covarance matr [9, ]. Accordng to the theory of matr egendecompoton, the array covarance matr can be decompoed. Frt the noe-free vector of receved gnal gven by: A A =, (9) For the unform lnear array, the matr A n () Vandermonde matr, a long a the followng condton met: θ θ, j (0) j Then, the column of the matr A are ndependent of each other, n addton, f the matr nonngular matr, and > D, then ung the aumpton made n prevou ecton, A A n (9) a ran D matr and t can be hown a: From (5), we obtan: ran( A A ) = D, () =, () 77

4 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp where the ermte matr, and t egenvalue are real number. Bede, potve defnte matr, o A A potve emdefnte havng D potve egenvalue and D zero egenvalue. The gnal-plu-noe model can be wrtten a the followng: AA σ I = +, (3) where σ > 0, the full ran matr, o ha potve real egenvalue (,,, ) correpondng to the egenvector( v, v,... v ). oreover, ermte matr, o t egenvalue are orthogonal to each other: v v = 0, j (4) j The number of egenvalue whch aocated wth the gnal D. It value equal to the σ plu the egenvalue of matr A A, and the D egenvalue σ. That to ay, σ the mnmum egenvalue of the matr. In the net ecton, we can get the drecton of arrval θ by eplotng the above properte of the egendecompoton..3. The USIC Algorthm Prncple [3, 4] After the egendecompoton of the covarance matr, we receved the followng concluon: Compute the egenvalue of that are n decreang order, then t egenvalue have the property: σ (5)... D D+ = = = n We can obtan from (5), beng D larger egenvalue of aocated wth gnal and D maller egenvalue of aocated wth the noe. Smultaneouly, the egenvector of matr belongng to thee egenvalue are alo correpondng to the gnal and the noe. Therefore, thee egenvalue (egenvector) can be dvded nto gnal egenvalue (egenvector) and noe egenvalue (egenvector). We et egenvalue of matr the th, = D+, D+,, egenvector v, then we have: v, and t correpondng = v, (6) If = σ the mnmum egenvalue, then: v = σ v, (7) Equaton (3) and (7) may be combned a: σ v = ( A A + σ I) v, (8) Epand the rght de of (8), then compared wth the left, (8) may be rewrtten a: A A v = 0, (9) For the A A the nonngular matr, and multaneouly, the matr ( A A) and et. Then, both de of (9) are multpled by ( A A) A. We can derve: ( A A) A A A v = 0, (30) and then (30) may be mplfed to A v = 0, = D+, D+,, (3) Equaton (3) tate that noe egenvalue correpondng to the egenvector are orthogonal to each column of matr A that n fact a teerng vector correpondng to the angle of arrval. The USIC algorthm ue th property to etmate drecton of ource. We can then contruct the ( D) noe ubpace E n panned by the noe egenvector uch that: E = [ v, v,..., v ], (3) n D+ D+ We defne the pectrum functon of the USIC algorthm a: P mu ( θ ) = a ( θ) E ( ) nen a θ =, En a( θ ) (33) Equaton (33) ugget that the denomnator the nner product of the teerng vector and the noe ubpace matr. When the teerng vector a( θ ) orthogonal to each column of matr E n, the denomnator zero. But due to the preence of noe, the denomnator a mnmum value, whch reulted n the pectrum defned n (33) havng a harp pea. Therefore, from the analy above, t can be een that we can etmate the angle of arrval by obervng the pea n the cae of the ncdent angleθ vary..4. Implementaton Step of the USIC Algorthm [5] ) Accordng to the receved gnal vector, we obtan the etmaton of the followng covarance matr: 78

5 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp = X() X (), (34) = Then the covarance matr can be epreed a: = + σ, (35) AA I ) By order of the egenvalue of the matr, we can acqure the gnal ubpace whch compoed of the egenvector correpondng to the D egenvalue that equal wth the gnal ource and the noe ubpace whch compoed of the egenvector correpondng to the D egenvalue, then we obtan the noe matr E : E = [ v, v,..., v ], (36) n D+ D+ 3) The DOA of the ncdent gnal ource can be etmated by locatng the larget pea of the functon P ( θ ) = a ( θ) E E a( θ), (37) mu n n Etmate ource drecton from the pectrum obtaned by the uperpoton of narrowband USIC pectra evaluated acro everal bn [6]. 3. Propoed ethod for DOA Etmaton 3.. odfed USIC Algorthm (USIC) The USIC algorthm only for the patal pectrum etmaton of ncoherent gnal. When gnal ource are coherent, the coherent gnal wll unte nto one gnal, then the ndependent gnal ource that receved by the array wll decreae, whch lead to the array covarance matr ran reduce and the number of larger egenvalue le than the ncomng gnal. Spatal pectral curve doe not preent the pea, thu cannot obtan the correct gnal DOA etmaton. ence, f we want to etmate the coherent gnal DOA accurately, we mut remove the correlaton between the gnal [7]. In USIC algorthm, the receved gnal model gven by: X ( n) = AS( n) + ( n) We defne a new matr Y( n ) a:, (38) Yn ( ) = J X( n), (39) n where X ( n ) denote the comple conjugate of X ( n ), I an dentty matr, and J the echange matr. It can be hown a: J = m m, (40) Obvouly, J J = I. The covarance matr y epreed a: y = E Y( n) Y ( n) = J A( A) J + σ I = J J The matr D aumed uch that: j ( ) φ j ( ) φ j ( ) φ D, (4) D = dag[ e, e,..., e ], (4) π d where ϕ = nθ, for =,,, D, and we obtan: J A = AD, (43) Ung (4) and (43), the utlzng of the order of dagonal matr product echangeable, o the covarance matr can be rewrtten a: y = A A + δ I =, (44) y Then the covarance matr can be fnally defned a follow: = + = + J J, (45) ( y )/ ( )/ It can be een from the formula dervaton proce, the eence of the modfed muc algorthm the pecal tuaton of the patal before and after moothng algorthm, whch the length of ubarray equal wth the number of array element [8]. 3.. Improved odfed USIC Algorthm (IUSIC) The patal moothng algorthm ha array element n the unform lnear array. The enor dvded nto P ( P = q+ ) ubarray wth q array element, and the covarance matr of the receved data. Bede, the prncple of matr decompoton algorthm [9] a follow: 79

6 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp From the l + row to l+ p row of the covarance matr, we defne a new matr () l a follow: () l γl+, γl+, γ l+, = γl+ p, γl+ p, γ, l+ p, l = 0,,,( p) Accordng to the matr matr gven by: (0) () ( p) = (46) () l, the modfed, (47) It clearly that the matr P [( P+ ) ] dmenon. Therefore, a long a meetng certan condton [0, ], the matr ran equal wth the gnal ource D. Obvouly, the matr decompoton algorthm wrtten n (47) the matr recontructon under the deal condton. Smlar wth the forward-bacward patal moothng algorthm [], we add revere mooth tem n the modfed matr n order to mprove the performance of coherent oluton. Then the mproved matr repreented a: In the frt eperment, we conder three uncorrelated ource ( p = 3 ) wth drecton of arrval 0, 40, 60. Fg. how the clac USIC and the modfed USIC algorthm wth S=0 db. It can be verfed from Fg. that n all ucceful mulaton of the clac USIC method the USIC algorthm ha alo been ucceful. In the econd eperment, we conder three ource ( p = 3 ) wth drecton of arrval 0,40,60. The frt two ource are coherent whch rrelevant wth the thrd ource. The S of both ource 0 db. We have preformed the clac USIC and the modfed muc algorthm n Fg. 3. Th fgure how the conventonal USIC falure when the ource are coherent. By comparon, the USIC can etmate the gnal DOA accurately. In the thrd eperment, we conder three ource ( p = 3 ) wth drecton of arrval 0,40,60. The frt two ource are coherent, but rrelevant wth the thrd ource. The S of both ource -5 db. We have preformed the modfed USIC and the mproved modfed USIC algorthm n Fg. 4. Th fgure how the etmaton performance of the USIC degrade when the gnal-to-noe low. On the contrary, the mproved modfed USIC pectrum tae the form of harper pea n whch angular reoluton mproved. =, (48) (0) ( p) (0) ( p) J J J J Baed on the above IUSIC algorthm, the q =, P = the pecal crcumtance of the patal moothng algorthm. We mprove the covarance matr n (48), t can be rewrtten a: = J J, (49) [ ] Ung (4), (49) can be mplfed a: = [ ], (50) y Through ngular value decompoton (SVD) on the matr, we wll obtan the gnal ubpace, noe ubpace and ngular value. Thu ung the USIC algorthm, we can etmate the gnal ource DOA. The IUSIC algorthm combned whch the patal moothng algorthm and the matr decompoton technology. Compared to the matr decompoton algorthm, the IUSIC method ha le computatonal load n reolvng the uncorrelated and correlated ource. Fg.. Clac USIC and USIC algorthm. 4. Smulaton eult In all mulaton, a unform lnear array of eght enor ued a hown n Fg.. Spacng between element aumed to be 0.5. Fg. 3. Clac USIC and USIC algorthm n the cae of coherent. 80

7 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp Fg. 4. USIC algorthm and IUSIC algorthm. 5. Concluon In th paper, a new algorthm whch ha le computatonal load than the matr decompoton and ha better reoluton capablty than the patal moothng algorthm preented for the cae of low S. It uperorty over USIC method wa hown through mulaton. Smulaton alo proved the advantage of propoed method n reolvng coherent ource. Smulaton reult have confrmed the theoretcal analy and have howed the effectvene of the propoed method n the cae of low S. Acnowledgement Th wor wa upported by Schuan ey technology reearch and development program (0fz005). eference []. Yufeng Zhang, Zhongfu Ye, Chao Lu, An effcent DOA etmaton method n multpath envronment, Sgnal Proceng, Vol. 90, Iue, 00, pp [].. Krm, B. Otterten, Two decade of array gnal proceng reearch: the parametrc approach, IEEE Sgnal Proce, Vol. 7, Iue 4, 996, pp [3].. O. Schmdt, ultple emtter locaton and gnal parameter etmaton, IEEE Tranacton on Antenna Propagaton, Vol. 36, Iue 4, 988, pp [4]. an Fang-ng, Zhang Shou-ong, Separaton of coherent multpath gnal wth mproved USIC algorthm, Sytem Engneerng and Electronc, Vol. 6, 004, pp [5]. J. Dmochow, J. Benenty, Drecton of arrval etmaton ung the parameterzed patal correlaton matr, IEEE Tranacton on Audo, Speech and Language Proceng, Vol. 5, Iue 4, 007, pp [6].. Wa, T. Kalath, Detecton of gnal by nformaton theoretc crtera, IEEE Tranacton on Acoutc, Speech and Sgnal Proce, Vol. 33, Iue, 985, pp [7]. G. Xu,.. oy, T. Kalath, Detecton of number ource va eplotaton of centro-ymmetry property, IEEE Tranacton on Sgnal Proceng, Vol. 4, Iue, 994, pp. 0-. [8]. Y. Wu, K. W. Tam, On determnaton of the number of gnal, n Proceedng of the Internatonal Conference on Sgnal and Image Proceng, onolulu, awa, Augut 00, pp [9]. Al Olfat, Sad ader-efahan, A new gnal ubpace proceng for DOA etmaton, Sgnal Proceng, Vol. 84, Iue 4, 004, pp [0]. X. L, X. a, S. Yan, C. ou, Super-reoluton tme delay etmaton for narrowband gnal, IET adar, Sonar & avgaton, Vol. 6, Iue 8, 0, pp []. P. Palanamy,. Kalyanaundaram, A. aghunandan, A new DOA etmaton algorthm for wdeband gnal n the preence of unnown patally correlated noe, Sgnal Proceng, Vol. 89, Iue 0, 009, pp []. Anne Ferreol, Pacal Larzabal, at Vberg, Stattcal analy of the USIC algorthm n the preence of modelng error, tang nto account the reoluton probablty, IEEE Tranacton on Sgnal Proceng, Vol. 58, Iue 4, 00, pp [3].. T. Wllam, S. Praad, An mproved patal moothng technque for bearng etmaton n a multpath envronment, IEEE Tranacton on Acoutc, Speech and Sgnal Proceng, Vol. 36, Iue 4, 988, pp [4]. ao Ye, D. De Groat, amum lelhood DOA etmaton and aymptotc Cramer_ao bound for addtve unnown colored noe, IEEE Tranacton on Sgnal Proceng, Vol. 43, Iue 4, 995, pp [5]. Kang Chunme, Yuan Yehu, The applcaton of muc method to detet the coherent ource on the bacground of ocean clutter, Acta Electronca Snca, Vol. 3, Iue 3, 004, pp [6]. Vnod V. eddy, B. P. g, Yng Zhang, Andy W.. Khong, DOA etmaton of wdeband ource wthout etmatng the number of ource, Sgnal Proceng, Vol. 98, Iue 4, 0, pp [7]. A. rata, T. ormoto and Z. Kawaa, DOA etmaton of ultra-wdeband E wave wth USIC and nterferometry, Antenna and Wrele Propagaton Letter, Vol., Iue, 003, pp [8]. D. Kundu, odfed USIC algorthm for etmatng DOA of gnal, Sgnal Proceng, Vol. 48, 996, pp [9]. X. Q. u, Y. Xu,. S. Chen, Fat teratve oluton of ntegral equaton wth matr decompoton algorthm and multlevel mple pare method, IET crowave, Antenna & Propagaton, Vol. 5, Iue 3, 0, pp [0]. A. D, ultple ource locaton-a matr decompoton approach, IEEE Tranacton on Acoutc Speech Sgnal Proceng, Vol. 33, Iue 4, 985, pp []. A. D, L. Tan, atr decompoton and multple ource locaton, n Proceedng of the IEEE Internatonal Conference on Acoutc Speech Sgnal Proceng ICASSP, Vol. 33, 984, pp. -4. []. Grant Thompon, Performance of patal moothng algorthm for correlated ource, IEEE Tranacton on Sgnal Proceng, Vol. 44, Iue 4,996, pp

8 Senor & Tranducer, Vol. 75, Iue 7, July 04, pp Copyrght, Internatonal Frequency Senor Aocaton (IFSA) Publhng, S. L. All rght reerved. ( 8