Coherent Targets DOA Estimation Using Toeplitz Matrix Method with Time Reversal MIMO Radar. Meng-bo LIU, Shan-lu ZHAO and Guo-ping HU

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1 7 nd Inernaional Conference on Wireless Counicaion and Newor Engineering (WCNE 7 ISBN: Coheren arges DOA Esiaion Using oepliz Marix Mehod wih ie Reversal MIMO Radar Meng-bo LIU, Shan-lu ZAO and Guo-ping U Air and Missile Defense College, Air Force Engineering Universiy, Xi'an 75, China eywords: DOA, MIMO radar, Mixed nor, Mulipah variable, Peas searching ehod. Absrac. In his leer, he direcion of arrival (DOA esiaion for uli-inpu uli-oupu (MIMO radar in ulipah environen is invesigaed. he signal odel is based on he assupion ha he signal is ransied by widely spaced anennas and received by soe unifor linear arrays (ULAs. A novel ehod, which includes opiizing he ransier ulipah variable by r ixed nor and he receiver ulipah variable by peas searching ehod, is proposed based on he received signal odel. he ehod can use he ulipah echo power o ae he esiaion perforance beer, especially wih he increase of he ransi anennas or ULAs. Siulaion resuls verify he usefulness of our ehod. Inroducion Muli-inpu uli-oupu (MIMO radars []-[], which can capure he spaial diversiy of he arge s radar cross secion (RCS, have he poenial advanages over he radiional radars. I has been deonsraed ha MIMO radar shares ore degrees of freedo for ransission bea foring [3], iproved deecion perforance [4] and higher sensiiviy for oving arges. Uilizing MIMO radar for direcion of arrival (DOA esiaion in ulipah environen has drawn a grea deal of aenion in recen years. In [5], he paper explores how ransi diversiy can iprove he direcion finding perforance of a radar uilizing an anenna array a he receiver. In [6], he bea-space axiu lielihood ehod (BSML is developed o reduce he copuaional burden and reain esiaion precision. In [7], ulipah effec, which can change bea-paern pea direcion, increase side-lobe level and eiher produce false peas in bea-paern, is proposed syseically, ec. owever, he sudy for DOA esiaion in ulipah environen described before ofen regards he ulipah echo signals as he noise or cluer, which cerainly wases a large aoun of signal energy. In his leer, based on he assupion ha he signal is ransied by widely spaced anennas and received by soe unifor linear arrays (ULAs, a novel DOA esiaion ehod which can use he ulipah echo power o iprove he esiaion perforance effecively is derived. R MIMO Radar Array Signal Model Consider a MIMO radar syse consising of colocaed receive and ransi arrays boh equipped wih unifor linear array (ULA, having N anennas and M anennas separaed by half a wavelengh, respecively. he arge disance is assued uch larger han he aperure and heighs of he ULAs. he DOA for each arge wih respec o he colocaed ransier and receiver is denoed by (,,,. Assue also far-field arges, which include boh coheren and non-coheren arges, are illuinaed by narrowband orhogonal wavefors. he oupu of he ached filers for he receive sensors can be expressed as jπ f ( = r ( βe ( ( + ( X a a s W ( where β and f are he radar cross secion (RCS fading coefficien and Doppler frequency of he jπ ( M sin jπ ( N sin h arge, respecively. a ( = [,,e ], a ( = [,,e ], and s( = [ s ( l s ( l s ( l], l,, L =, L denoes he nuber of snapshos. W ( is an addiive Gaussian 96 r

2 whie noise vecor of zero ean and covariance arix σ I N. In he R MIMO radar, he oupu vecor X ( in (7 is energy noralized by facor ε, phase conjugaed, ie reversed, and reransied ino he ediu. hen he R ransied signal is ε X (, and he R oupu can be given as j π f ( ε ( βe r ( ( Y = a a X + V ( ( ( ( ( j 4 π f e r r = ε β a a a a s + Ṽ where V and Ṽ are he observaion noise for he R sage and he accuulaed noise fro boh he convenional and R sages, respecively. Siilar o he noises W (, he noises V are also eporally and spaially coplex whie Gaussian processes wih zero-ean and variance σ. As describe in Appendix A, he accuulaed noises can be proved as approxiaely whie noise. hen, Apply he ached filer arix s ( o he received signal in (, and we have ( 4 ε β π ( ( r r ( ( j f Z = e a a a a + Vs ( (3 hus, afer he vecorizaion operaor of he arix Z ( consruced as (, a coposie daa vecor can be Z ( = ε A Dη + v (4 where A =[ a (,, a ( ], D = [ a ( a ( a (,, ( ( ( ] j 4π f j 4π f β e β r r a a a, r r η = [ e,, ], and v = vec[ Vs ( ]. Le us exaine he saisical characerisics of he noise vecor v afer ached filer. According o he basic properies of he ronecher producs, heir covariance arices can be given as ( [ ] ( M E vv = s( I E VV s ( I = σ s( s ( IM = σ I where vec[ Vs ( ] = [ s( IM ] vec ( V M M, and he covariance of he noise Ṽ is σ I ( ( r ( r ( = N ( ( and he oupu in (4 can be changed as M (5. Fro (4, we have a a a a a a (6 ( = ε N ( ( + (7 Z A A η v Proposed DOA Esiaion Mehod In his secion, a new ehod naed as reduced-diension oepliz arix ehod is proposed in his par. A firs, we define a ( a ( = Gb ( (8 jπ ( M sin jπ ( M sin where G = [ G,, G M ], G = [ M ( J M M ( M ], and b( =[e,,,e ]. hen, we ( ( also define = = (,,,,,, M diag M M G G G C. Using he received signal vecor Z ( in (7 - lef uliplied by he arix G G, we can obain ( ( ( (9 Z - - = ε N G G A A η + G G v = B η + v - B =[ b(,, b( ], η = ε Nη, and v = G G v is addiive Gaussian whie noise arix. As he where ( 97

3 arix G is sparse, he reduced-diension ransforaion can add less copuaion load. In order o reove he coherency, we can esablish a oepliz arix via he new received signal vecor in (9, which can be denoed as R z z z z z z z z z ( M ( M = M M M + (,,(,, ( = A ηφη Φ η Φ η q = A ( diag ( η A ( + q jπ sin jπ sin where Z = [ z ( M,,,, zm ], q is he noise arix, and Φ = diag(e,,e. As a resul, in (, since A ( is a full-colun-ran Vanderonde arix and diag ( η is a diagonal arix, he ran of R is no relaed wih he coheren signals. Perfor eigenvalue decoposiion of (, and we can ge R = UsΣsUs + UnΣnU n ( where Σ s is a diagonal arix whose diagonal eleens conains he larges eigenvalues; Σ n sands for a diagonal arix whose diagonal enries conain he salles M- eigenvalues; U s and U n are he signal subspace and he noise subspace, respecively. I is well nown ha he signal subspace is orhogonal o he noise subspace. herefore, a cos funcion can be given as ( Q = U a ( ( n hen, he iniu of Q can be achieved for rue DOAs, and he esiaion process can be expressed as = [,, ] = arg in Q (3 ere, we eploy an ieraive approach o iprove he esiaion perforance, and he approach can be given as Sep : Perfor (3 o ge he iniial value. ( i Sep : Updae he esiaed DOAs based on firs-order approxiae of Q, which can be given as [9] ( i+ ( i dq = + µ d ( i = (4 dq = Re diag ( ( n n diag ( ( d a U U a (5 Re j diag ( a ( UnUn a ( where i is he ieraive ies, and µ is a real nuber beween zero and one. Sep 3: Repea his procedure unil he difference beween wo consecuive ieraions is lile han a hreshold, i.e. ( i+ ( i ν (6 5 where ν is a posiive sall consan (e.g., ν =. 98

4 Siulaion Resuls As described in Figure, in he case of L M, copared wih he ESPRI-lie ehod and he FBSS- MUSIC ehod, he proposed ehod has lower copuaion load. owever, for he low snapshos, he copuaion coplexiy of he proposed ehod is higher han he ehod in [], which is caused by he ieraive approach and peas searching in Secion III-B. Wha s ore, as a price, he proposed ehod can provide beer esiaion perforance. 8 he proposed ehod ESPRI-lie he ehod in[] FBSS- MUSIC 7 Coplexiy Snapshos Figure. Coparison of copuaion coplexiy beween proposed ehod and oher ehods. We now show soe siulaion resuls evaluaing he esiaion perforance of he proposed ehod in ulipah environen. We consider a unifor linear array (ULA of N= anennas. hree non-coheren narrowband signal sources coing fro =, =, =3 3 are considered. he wavelengh is λ =, while he heigh of receiver sensors is h=. All signals are of equal power σ s. he oal nuber of snapshos is chosen o be L=. he esiaion perforance is exained over Mone Carlo rials. Exaple : In his exaple, we exaine he effeciveness of he ehod in opiizing for B ( r=, =. We assue he received ULA is P =, h =. Fig. shows he DOA esiaion perforance wih differen nuber of ransier anennas. he ore he nuber of ransi anennas has, he beer he perforance of esiaion will be goen for MIMO radar, especially for he angle resoluion. In fac, for he virual ransied signal B, he ehod by r ixed nor opiizing can obain he larges ransier ulipah variable and reflecion coefficien, which will iprove he ransied signal power. - ulipah M= ulipah M=3 ulipah M=5 - agniude (db angle (degree Figure. Esiaion perforance wih differen nuber ransier anennas. Exaple : In his exaple, we exaine he effeciveness of he peas searching ehod in p opiizing for G. We assue he ransier anennas is M=5. Fig.3 shows he spaial specru funcion a differen receiver sub-array. In he case of h = 3, he DOA esiaion perforance a = and =4 3 is relaive poor; Liewise, In he case of h = 5, he DOA esiaion perforance a = is very poor. he reason is ha he receiver ulipah variable is very lile for soe angle, which will reduce he echo power largely. As a resul, he new peas searching ehod described in secion can selec he bes peas and side lobe o iprove he esiaion perforance. 99

5 - ulipah p= h=3 ulipah p= h=5 - agniude (db angle (degree Figure 3. Esiaion perforance a differen receiver sub-array. Exaple 3: In his exaple, we exaine he esiaion perforance by he RMSE versus SNR. In Figure 4, we assue he ransier anennas is M=5. In he case of P =, h =, we only consider he ransier ulipah variable and he perforance in ulipah environen is beer han in non-ulipah environen, especially under relaively low SNR, which illusraes ha he ehod in opiizing for B can iprove he DOA esiaion perforance by using he ransier ulipah signals power; In he case of P =, h = 3, since he receiver ulipah variable a soe angle is very lile (Fig., he perforance is poor. owever, In he case of P = 3, h = : : 3, we can selec he os effecive receiver ulipah variable fro differen receiver ULAs by he peas searching ehod and he esiaion perforance is he bes. non-ulipah ulipah P= h= ulipah P= h=3 ulipah P=3 h=::3 RMSE SNR Figure 4. RMSE versus SNR for DOA esiaion perforance. Conclusion In his leer, we have proposed a novel DOA esiaion ehod, which can ae advanage of he ulipah echo power. he ehod can avoid he effecs of differen ulipah variable and arges glin, which resuls in he spaial diversiy for ransier and receiver. We deonsrae our ehod has uch beer perforance in conras o he non-ulipah environen or wih he increase of he ransi anennas or ULA. he arge racing and posiioning in ulipah environen will be considered in near fuure. References [] E. Fisher, A. aiovich, R.S. Blu, D. Chizhi, L. Ciini, and R. Valenzuela, MIMO radar: an idea whose ie has coe, in Proc. of he IEEE Radar Conference, pp.7-78, Apr 4. [] E. Fisher, A. aiovich, R.S. Blu, L. Ciini, D. Chizhi, and R. Valenzuela, Spaial diversiy in radars-odels and deecion perforance, IEEE rans. on Signal Processing, vol. 54, pp , Mar 6. [3] P. Socia, J. Li, and Y. Xie, On probing signal design for MIMO radar, IEEE rans. Signal Process, vol. 55, pp , Aug 7.

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