A DOA Estimation Algorithm without Source Number Estimation for Nonplanar Array with Arbitrary Geometry

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1 Internatonal Journal of Coputer heory and Engneerng, Vol., o. 4, August, A DOA Estaton Algorth wthout ource uber Estaton for onplanar Array wth Arbtrary Geoetry e Ke, Zhang Xaon, an Peng, Yu Yang and Yan h Abstract he perforances of ost of the hgh resoluton ethods always depend on the estaton of the source nuber. In real applcaton, when the estated nuber of sgnals s not correct, the orthogonalty between sgnal subspace and nose subspace can not be antaned any ore. And the perforance of DOA estaton algorth wll deterorate severely. In ths paper, a hgh resoluton algorth called - MVM wthout source nuber estaton and egendecoposton for Drecton-Of-Arrval (DOA) estaton s proposed, whch s an proveent of the Mnu Varance Method (MVM). Furtherore, t s sutable to nonplanar arrays of arbtrary geoetres. oe representatve coputer sulatons are presented to llustrate the perforance coparson between dfferent algorths and dfferent arrays. Index ers nonplanar array, DOA, MVM I. IRODUCIO In recent decades, the proble of estatng sgnal paraeters fro data collected by an array of sensors has receved uch attenton. And a great varety of hghresoluton algorths for source localzaton [,] have been reported n the lterature. Although they have superor accuracy and resoluton perforance over conventonal beaforng technques, ost hgh-resoluton algorths depend on the source nuber estaton. Indeed, for those ethods, we ust truncate the covarance atrx n sgnal and nose subspaces. hat truncaton depends on the estaton of the source nuber. In real applcaton, when the nuber of sgnals s underestated or overestated, the orthogonalty between sgnal subspace and nose subspace can not be antaned any ore. And the perforance of DOA estaton algorth wll deterorate severely. In [3,4], the perforance of MUIC algorth under errors of source nuber s analyzed, and the proposed algorths based on the egendecoposton of saple covarance atrx can estate DOA properly when the source nuber s unknown. Although the hgh-resoluton algorths reported n the lterature have deonstrated varyng degrees of success, ther applcatons are lted. In any stuatons, planar array can t eet the requreents of any actual detecton systes. Whle nonplanar array ounted on the planes, bobs, shps, nes, and underwater vehcles wll brng e Ke, Zhang Xaon, an Peng, Yu Yang are wth College Of Marne, orthwestern Polytechncal Unversty, X an, Chna (eal:hk@nwpu.edu.cn). Yan h s wth the Delegaton of avy at Kunng,Kunng, Chna (eal: @39.co). 569 any benefts. owever, few algorths have been developed for ths proble. And t s very portant for the desgn and evaluaton of nonplanar array to develop hghresoluton algorth for nonplanar array. In ths paper, a nonplanar array wth arbtrary geoetry s odeled; we present a hgh-resoluton algorth based on the Mnu Varance Method (MVM) wthout source nuber estaton and egendecoposton. Coputer sulaton for the perforance coparson between the proposed algorth and the MUIC algorth s deonstrated. And soe conclusons of the proposed algorth for the nonplanar quaternon array and planar array are reached. II. ARRAY DAA MODEL In ths secton, a odel of arbtrary geoetry s descrbed for the nonplanar array. he array s assued to be - eleent arbtrary nonplanar geoetry, and n narrowband sgnals wth known centre frequency fro far feld, as shown n Fg. Fgure. he nonplanar array wth arbtrary geoetry. Wth the sensor at the orgn of the coordnate as reference, the array output can be wrtten as X ( t) = A ( t) + ( t) () Where X ( t) = [ x( x( x ( t)] represents a data vector, t) = [ s ( s (, s ( t)] the source vector, and n ( )] ( t) = [ n ( n( n ( t the nose vector. he th colun of the array anfold atrx A, whch s represented by a(θ,φ )(=,, n), s the steerng vector for the th source sgnal, and s gven by πf τ πf τ πf τ a (, ) = [ e, e e ] () θ ϕ And τ (=,, ) s the dfferental te delay of the th source sgnal between the th sensor and the coordnate reference

2 Internatonal Journal of Coputer heory and Engneerng, Vol., o. 4, August, u p τ = c (3) = ( x snθ cosϕ + y snθ snϕ + z cosθ ) c Where f s the centre frequency of the th source sgnal and c s the speed of propagaton n the edu of nterest; θ and φ are the elevaton angle and azuth angle for the th source sgnal and u = [snθ cosϕ,snθ snϕ,cosθ ] s the drecton unt vector; p =[x,y,z ](=,, ) s the vector ade up by eleent of the coordnate. Wth the assupton that sgnal sources are ncoherent and noses are ndependent wth zero ean and σ n varance, the spatal covarance atrx of X(t) s R = E[ XX ] = AE[ ] A + E[ ] (4) = AR A +σ I Decopose the spatal covarance atrx R, the egenvectors correspondng to the largest egenvalues λ and the rest are naed as sgnal space U = e, e, e ] and nose space [ [ e +, e +, em U = ] respectvely. Accordng to the egenstructure ethod, we obtan R M = λee + σ e e = = + = U Σ U + σ U In practce, ths atrx s estated usng L snapshots of the actual array output, as shown n equ.6 L Rˆ = XX (6) L = III. ALGORIM A. MUIC Algorth Wth the orthogonalty of the sgnal and nose subspaces, he MUIC spectru estate s gven by U (5) P ( θ, = MUIC α ( θ, U U α ( θ, (7) he estaton of θ and φ s correspondng to the peaks of the MUIC spectru: ( ˆ, θ ˆ) ϕ = arg ax{ P (, )} ˆ, MUIC θ ϕ ( θ ˆ) ϕ (8) = arg n{ α ( U U α( } owever, the MUIC Algorth s senstve to dfferent nose felds and the perforance s degraded n short snapshots. B. Mnu Varance Method Based on the nu varance ethod (MVM), the MVM spectru estate s gven by PMVM ( θ, = (9) α ( θ, R α ( θ, he estaton of θ and φ s correspondng to the peaks of the MVM spectru: ( ˆ, θ ˆ) ϕ = arg ax{ P ( } = arg n{ α ( R α( } MVM (0) he MVM spectru s the axu lkelhood estate of θ and φ, and s proportonal to the power spectral of the sgnal. Coparng wth the MUIC ethod, t shows the advantage that MVM does not requre source nuber estaton and egendecoposton of saple covarance atrx. owever, due to the array aperture ltaton, MVM s usually at lower resoluton. C. -MVM Algorth Cobnng the advantages of MUIC and MVM, we can get better resoluton wthout source nuber estaton and egendecoposton. It s called -MVM Algorth, whch s an proveent on the tradtonal MVM. ransfor the equ.5: σ R = U dag U + λ σ U U () Where s an nteger bgger than ; Apparently, σ /λ s less than, so equ. approaches to nose space, as the approaches to nfnty. lσ R = U U () As a atter of fact, the experent shows that the perforances of the algorth are proved, when s bgger than. o, the -MVM algorth can be expressed as: ( ˆ, θ ˆ) ϕ = arg ax ( ˆ, θ ˆ) ϕ (, ) ˆ = α θ ϕ R α( = arg ax ( ˆ, θ ˆ) ϕ (, ) ˆ α θ ϕ R arg n{ α ( Rˆ = ( ˆ, θ ˆ) ϕ α( α( } (3) For nonplanar array wth arbtrary geoetry, the -MVM algorth can be pleented n steps as followng: ) et the paraeters of the nonplanar array and the arrval angle, and set up the steerng vector fro equaton () and (3), the array anfold atrx A can be derved. ) Create the source sgnal data and rando nose data usng atheatcal odelng, and calculate the data vector X(t) fro equaton (). 3) Calculate the correlaton atrx R of the output of array fro equaton (6). 4) Calculate the -MVM spectru usng the steerng vector and the correlaton atrx wth the followng functon. P MVM ( θ, = (4) α ( θ, R α ( θ, he physcal eanng of the -MVM algorth s a connecton of MVM estators. When s equal to, the -MVM algorth s equal to MVM estator. When approaches to nfnty, the -MVM algorth approaches to 570

3 Internatonal Journal of Coputer heory and Engneerng, Vol., o. 4, August, MUIC algorth. IV. PERFORMACE IMULAIO In ths secton, the perforance of the proposed algorth s evaluated by soe representatve coputer sulatons. uercal exaples are carred out usng two knd of array of 4 eleents, as shown n fgure. Fgure 3. he MUIC spectru wth correct source nuber a. unfor cross array Fgure 4. he MUIC spectru wth wrong source nuber b. nonplanar quaternon array Fgure. he geoetry of array he assued array s that of the unfor cross array and the nonplanar quaternon array wth an ntersensor spacng of λ/, where λ s the wavelength of the source sgnals. he nuber of snapshots s fxed at 5 and two sources wth the sgnal to nose rato(r) of 0dB arrvng at (45, -60) and (30, 30), whch s the elevaton angle and azuth angle of the sources. he results of sulaton are shown fro Fgure 3 to Fgure 7. Fgure 3 s the DOA estaton result usng MUIC algorth wth correct source nuber estaton, whle Fgure 4 s the MUIC spectru wth wrong source nuber estaton. otce that the MUIC algorth can not estate the DOA wth wrong source nuber estaton. Fgure 5 and Fgure 6 s the DOA estaton result usng -MVM algorth wthout source nuber estaton for the nonplanar array and the unfor cross array. otce that the perforance of the -MVM spectru s better than the MUIC spectru, and the perforance of the nonplanar array s better than the unfor cross array. Fgure 5. -MVM spectru for two sources wth the unfor cross array 57

4 Internatonal Journal of Coputer heory and Engneerng, Vol., o. 4, August, MVM algorth by Monte Carlo sulaton ethod of 000 tes, copared to the MVM and MUIC algorth for the unfor cross array and the nonplanar quaternon array, he nuber of snapshots s fxed at 56, and the R range fro 5dB to 0dB. otce that the varance becoes saller along wth the ncrease of R for the nonplane quaternon array and unfor cross array. he varance of the -MVM algorth s saller than the MVM algorth, and the varance of the MUIC algorth s the least. Fgure 6. -MVM spectru for two sources wth the nonplanar array Fgure 7 and Fgure 8 show the devaton and varance of the -MVM algorth by Monte Carlo sulaton ethod of 500 tes, copared to the MVM and MUIC algorth for the nonplanar quaternon array, he R s fxed at 5, and the nuber of snapshots range fro 50 to 500. otce that the devaton and varance becoe saller along wth the ncrease of the nuber of snapshots for the nonplanar quaternon array. otce that the perforance of the MUIC algorth s poor, when the nuber of snapshots s less than 500. In contrast, the perforance of the MVM algorth and -MVM algorth are better for the nuber of snapshots ore than 00. Fgure 9. he effect of R on varance for three algorths wth nonplanar array Fgure 7. he effect of the nuber of snapshots on Devaton for the three algorths Fgure 0. he effect of R on varance for three algorths wth unfor cross array Fgure shows the effect of R on varance usng the -MVM algorth wth unfor cross array by Monte Carlo sulaton ethod of 000 tes, copared to the nonplanar quaternon array, he nuber of snapshots s fxed at 56,R [ 5dB,0dB]. otce that the varance of the nonplanar quaternon array s saller than planar array. he varance becoes saller along wth the ncrease of R for the nonplanar quaternon array and the unfor cross array. Fgure 8. he effect of the nuber of snapshots on varance for the three algorths Fgure 9 and Fgure 0 shows the varance of the - 57

5 Internatonal Journal of Coputer heory and Engneerng, Vol., o. 4, August, ACKOWLEDGME he authors wsh to thank Prof. Xang and the anonyous revewers for ther suggestons and coents whch contrbuted to the fnal for of the paper. Fgure. he effect of R on varance for unfor cross array and nonplanar quaternon array V. COCLUIO In sensor array processng, ost hgh-resoluton algorths depend on the source nuber estaton, the perforance of drecton-fndng decreases sgnfcantly wth underestaton or overestaton of the source nuber. In ths paper, an -MVM algorth wthout source nuber estaton and egendecoposton s proposed for nonplanar array wth arbtrary geoetry. uercal exaples deonstrate the valdty of the proposed algorth, and soe conclusons of the proposed algorth for the nonplanar quaternon array and planar array are reached. REFERECE [] Pascal Chevaler, Anne Ferréol, and Laurent Albera. gh-resoluton Drecton Fndng Fro gher Order tatstcs: he q-muic Algorth, IEEE ransactons on sgnal processng, vol. 54, no. 8, August 006 [] Zhu Weqng, u Juan, Lu Xaodong, Lu Zhyu and Zhu Mn. ource nuber estaton usng egenspace n drecton of arrval (DOA) estate OCEA 009-EUROPE, 009. OCEA ' May 009 Page(s):-6 [3] Lu Pn and Zhang Zhun. A Perforance Analys s of the Modfed MUIC Algorth Wthout Consderng the ource uber Control & Autoaton, Vol. 3, o. 30, 007 [4] Zhang Zh-un, ou Zh-bn and Zhang ao A Modfed MUIC Algorth wthout Consderng the ource uber elecouncaton Engneerng, Vol. 47, o. 5, Oct. 007 [5] Du Jngng, Zhang Yang and ang Bn. Estaton of blnd source nuber n councaton reconnassance gnal Processng, 008. ICP th Internatonal Conference on 6-9 Oct. 008 Page(s):85 87 [6] Qngchun Zhang and Zhenghe Feng. A ovel ource uber Estator Wthout Egendecoposton Antennas, Propagaton & EM heory, 006. IAPE '06. 7th Internatonal yposu on 6-9 Oct. 006 Page(s): 3 [7] Radch, B.M.and Buckley, K.M. he effect of source nuber underestaton on MUIC locaton estates gnal Processng, IEEE ransactons on [see also Acoustcs, peech, and gnal Processng, IEEE ransactons on Volue 4, Issue, Jan. 994 Page(s):33-36 [8] Qao-we Yuan, Qang Chen, Meber and Kuno awaya. Accurate DOA Estaton Usng Array Antenna Wth Arbtrary Geoetry IEEE ransactons on antennas and propagaton, vol. 53, O. 4, aprl