A Novel Algorithm to Improve Resolution for Very Few Samples

Transcription

1 Vol 8, No 0, 207 A Novl Algorithm to Improv Rsolution for Vry Fw Sampls Sidi Mohamd adj Irid STIC Laboratory, Faculty of Tchnology Univrsity of Tlmcn, BP9 Chtouan Tlmcn, Algria Samir Kamch STIC Laboratory, Faculty of Tchnology Univrsity of Tlmcn, BP9 Chtouan Tlmcn, Algria Abstract This papr prsnts a nw tchnic to improv rsolution and dirction of arrival (DOA) stimation of two closd sourc, in array procssing, whn only fw sampls of rcivd signal ar availabl In ths conditions, th dtction of sourcs (targts) is mor arduous, and vn braks down To ovrcom ths problms, a nw algorithm is proposd It combins spatial smooth mthod to widn th spatial rsolution, bootstrap tchniqu to stimat incrasd sampl siz, and a high rsolution tchniqu which is Multipl Signal Classification (MUSIC) to stimat DOA Through diffrnt simulations, prformanc and ffctivnss of th proposd approach, rfrrd to as Spatial Smooth and Bootstrappd tchniqu SSBoot, ar dmonstratd Kywords Dirction of arrival (DOA) stimation; Bootstrap; Multipl Signal Classification (MUSIC); rsolution; spatial smoothing; array procssing; Uniform Linar Array (ULA) I INTRODUCTION Whn two sourcs ar vry clos in spac in th ambiguity rang, th radar dtcts thm lik on targt Th spatial rsolution limits for two closly spacd sourcs in th contxt of array procssing is still an activ rsarch []-[3] In fact, thr has bn a trmndous involvmnt in th invstigation of how DOA stimation of many closd sourc (targts) can b stimatd Most of thm, [4], [5], ar basd on high rsolution mthods, g Multipl Signal Classification (MUSIC) or Estimation of Signal Paramtr via Rotational Invarianc Tchniqu (ESPRIT), and dtct sourcs using ignvalus obtaind from covarianc matrix of sampls owvr, th main issu of high rsolution mthod for DOA s stimation is prdtrmination of th modl ordr, sinc ths tchniqus rquirs imprativly numbr of sourcs, as input paramtr within stimation This stimation is basd on information thortic critria lik AIC (AKAIKE) and Rissann s minimum dscription lngth critrion (MDL) algorithms to stimat DOA of sourcs [2], [5]-[7] In othr hand, prformanc of ths tchniqus stays vry poor for low sampls, low SNR, corrlatd sourc signals and prsnc of impulsiv whit nois To improv th rsolution, a spatial smoothing tchniqu is usd This tchniqu divids th array into multipl ovrlapping sub-arrays In ach sub-array, th corrlation matrix is stimatd from bootstrappd data sampls W xploit th ida of th author in [3] and applid MUSIC in ach sub-array; to stimat th numbr of sourcs as numbr of paks [], [8], [9] Unfortunatly, th most xisting mthods ar lss fficint and lost larg prformanc or vn brakdown whn only fw sampls of rcivd signal ar availabl To rduc this hurtful ffct and improv th robustnss of th covarianc stimator, a robust non-paramtric bootstrap mthod stimator was proposd [0]-[3] Basd on tim random sampling of original data, to stimat its sampling distribution without any modl assumption In this work, a nw algorithm is proposd It combins spatial smoothing, a high rsolution mthod (MUSIC) and Bootstrap tchniqu to stimat closly spacd numbr of sourcs and thir DOA s whn only fw sampls of rcivd signal ar availabl First, it s usd bootstrap mthod to stimat th covarianc matrix, thn spatial smoothing curvs up th antnna array into L sub-ntworks In ach sub-ntwork, MUSIC algorithm allows to stimat th numbr of closly spacd sourcs and thir DOA s Numrical simulations ar givn to assss th prformanc of th usd tchniqu Th papr is organizd as follows Data, array modl and MUSIC dscription ar introducd in Sction 2, followd by spatial sampling modl dscription in Sction 3 Thn bootstrap tchniqu is prsntd in Sction 4 Th proposd algorithm SSBoot is dscribd in Sction 5 Simulation rsults ar givn in Sction 6 Finally, discussion and conclusion ar givn in Sction 7 II PROBLEM FORMULATION Just to simplify th notation, w assum a Uniform Linar Array (ULA) composd of M snsors, with quip-spacing d=λ/2 as shown in Fig ; whr λ is th wavlngth of th sourc signal Considr a K narrowband far-fild uncorrlatd sourc impinging on th array with (M > K), such that sourcs hav a dirction of arrival (DOA) θ k, with k= K A Array Signal Modl Th rcivd snapshots at this array, at instanc t ar givn by [], [4] 38 P a g

2 Vol 8, No 0, 207 Fig Localization of two closly spacd sourcs impinging on ULA Th rcivd signal corruptd by additiv whit Gaussian nois is prsntd at instanc t by mathmatical quation [], [5], [6]: Whr () [ ] (2) is th string matrix (MxK) full rank, [ ] (3) and ach column is writtn in function of th rcivd signal as follows: a k ( k j 2 ( d / )sin ) (4) ( t) y ( t) y T M ( t (5) y ) T S ( t) S( t) S K ( t) (6) n ( t) n t n T ( ) M ( t) (7) Suprscript () T prsnts th transpos opration Whr y k (t) dnots th output of k th snsors, s q (t) sourc signal and n k (t) is a stationary nois modl, tmporally whit, zro-man Gaussian random procss indpndnt of th sourc signals Th covarianc of rcivd data is [], [7], [8]: 2 AR A I R yy E YY S (8) Whr R S E S(t)S (t) (9) Th suprscript () stands for th conjugat transposition, σ 2 is varianc and I indicat th idntity matrix Furthrmor, th covarianc matrix is stimatd by [2], [3], [7], [8]: R yy Y Y (0) N Th ignvalus ar givn as follows: 2 k k M 2 whr th first K ignvalus blong to th sourc signal, and th last (M-K) to th nois MUSIC plots th psudo-spctrum [2], [9]: VMusic () a E E a( ) n n Whr En is th (Mx(M-K)) nois subspac composd of th ignvctors associatd with th nois If w assum two closly spacd sourc whr thir DOA ar θ and θ 2 such as: 2 with 5 B Spatial Smoothing In this sction, it s dscribd th us of spatial smoothing in proposd algorithm in ordr to improv rsolution of vry clos spacd sourcs Th ordinary spatial smooth consists of dividing th whol array into L sub-arrays shiftd on anothr by on snsor; th rst of snsors ar ovrlappd as shown in Fig 2 It stimats th corrlation matrix as th avrag of all corrlation matrics from th sub-arrays and can b rprsntd as: (2) Fig 2 An ULA antnna is dividd into L sub array Our mthod is basd on rprsntations of Abd-Mraim t al in [7] who dividd th whol array into intrlaving subarrays In ach sub-arrays th rcivd signal is givn by: 39 P a g

3 Vol 8, No 0, 207 Y m y(0) y2(0) y(( y ( T) N L )) (0) y(( N L )) ( T ) y ( T ) 2 (3) Whr m is th array and varis from to L, and N L = M/L Th sam, th string matrix for th m array is givn by: Am ( ) j sin j 2sin j ( N )sin L j sin j 2sin L q q j ( N )sin q (4) Whr T is a snapshot numbr and q is th numbr of sourc signal rcivd in ach sub-array In this cas, and unlik rsults of Abd-Mraim in [3], [9], w ar sur that numbr of snsors is always gratr than numbr of sourcs, and thrfor it rspcts th assumption to apply MUSIC Thus, that spatial smoothing or spatial sampling considrably improvs th rsolution Indd, according to (3) and (4), th angular part in th matrix output ar multiplid by a factor (NL-) which is gratr than Thrfor, th angular sparation is widnd and th rsolution is improvd C Bootstrap Rplication In this sction, non-paramtric bootstrap rsampling tchniqus ar prsntd, dsignd for indpndnt and idntically distributd data owvr, th assumption of iid data can brak down during opration ithr bcaus data ar not indpndnt or bcaus data ar not idntically distributd [7], [8]Th original data points: (5) with probability for ach sampl A bootstrap sampl X* is obtaind through rplacmnt of original data points by random sampling (n tims) [0]-[2] Som bootstrap sampls can b: with n sampls (6) W assum that th x i s ar indpndnt idntically distributd (iid), ach having distribution F Bootstrap proposs to rsampl from a distribution chosn to b clos to F in som sns This could th mpirical distribution, rsampling from is rfrrd to as non-paramtric bootstrap [0] At th nd w obtain: (7) rin, w crat a numbr B of rsampls Th rsampld boostrapis an unordrd collction of n sampls points drawn randomly from with rplacmnt, so that ach has probability of bing qual to any on of th X j s In othr trms [8], [9], []: [ ] (8) This mans that is likly to contain rpats Th probability that a particular valu x i is lft out is (9) W xploits th rsampl bootstrap algorithm dscribd in [9] to rproduc sampls and us it in our proposd SSBoot algorithm III SPATIAL SMOOT BOOTSTRAPPED SSBOOT ALGORITM Firstly, th proposd mthod is basd on incrasing th numbr of snapshots rcivd on array ntwork using bootstrap tchniqu Scondly, ach of sub-arrays is procssd sparatly and finally th avrag DOA stimation is considrd Dtrmination numbr of sourcs first is ssntial for highrsolution mthod It should us AIC or MDL algorithm to dtrmin th modl ordr But, in this work, w followd th sam spirit givn in [3], [7] W stimatd th sourc numbr using bamforming or Capon mthod applid to th global array output If q paks appar, w r-apply MUSIC algorithm by rstricting our rsarch in intrvals around ach q paks Applying spatial smoothing yilds to divid th array ntwork into L ovrlapping sub-arrays thus, w obtain L diffrnt DOA s stimats Among ths L sts, w kp only th highst numbr of paks in ach intrval Our nw algorithm, w namd SSBoot can b summarizd as follows: Stp : Applying bootstrap tchniqu to gnrat nw sampls by sampling with rplacmnt of original data Stp 2: First stimation numbr of sourcs on global array ntwork using Capon mthod Stp 3: Dfining st of intrvals whr sarch ar rfind Stp 4: Divid th global antnna array into L shiftd ovrlappd sub-arrays Stp 5: On ach sub-array, w apply MUSIC Algorithm Th numbr of MUSIC spctrum paks quals to numbr of sourcs 320 P a g

4 RMSE Dtction rat Dtction rat (IJACSA) Intrnational Journal of Advancd Computr Scinc and Applications, Vol 8, No 0, 207 Stp 6: Th numbr of sourcs is slctd from p intrvals for L sub-arrays that prsnt maximum numbr paks, Stp 7: Computing th final DOA, aftr sorting and calculating th avrag from ach intrval and slcting sub-arrays with maximum paks (20) Whr, l =p diffrnt sub-arrays rprsnts p stimats DOA from IV SIMULATION AND RESULT To illustrat th prformanc of th proposd mthod, som numrical rsults ar prsntd to analys and compar bhaviour stimation of th nw proposd algorithm which is namd SSBoot A Uniform Linar Array (ULA) is constitutd of N=0 snsors spacd of half-lngth wav lngth is mployd Assum that thr ar two closly spacd uncorrlatd narrowband signal sourcs with th sam wavlngth λ, θ = 32 and θ 2 = θ +δθ, whr δθ is a vry small angl diffrnc Simulation rsults wr obtaind basd on 000 Mont Carlo simulation Prformanc of bootstrap for varying snapshots for arrival angls of and 80 rspctivly, ar illustratd in Fig 3 Whn a fw sampls (20 snapshots) ar rcivd th MUSIC spctrum rspons is almost flat and th DOA is difficult to xtract, but whn ths sampls ar bootstrappd at 200, 000 th 2000 sampls, th rsponss incrass and th paks bcom noticabl owvr, it dmonstrats th ffctivnss of th bootstrap mthod to improv th dtction and stimation of DOA Fig 4 prsnts th probability of targt dtction in prcntag for various angular sparations; it illustrats th prformanc achivd by our mthod for fw snapshots with low SNR In fact, for rcivd low sampls, th dtction is wak; it incrass slowly whn SNR incrass But whn ths sampls ar bootstrappd at 000 snapshots th stimation rat improvs and rachs th maximum rat with low SNR owvr, our algorithm SSBoot bootstraps th rcivd sampls and uss th spatial sampling to improv its stimation prformanc for th sam numbr of snapshots Indd th vry clos spacd sourcs ar dtctd for low SNR Fig 5 dpicts th probability of dtction rat in prcntag for various SND in db; it shows that for fw sampls th dtction narly braks down With bootstrap at 000 snapshots, th dtction is slightly achivd bcaus of low SNR valus Th SSBoot proposd mthod ovrcam this limitation by nsuring a highst dtction rat for low SNR and vry clos sparation sourcs Fig 6 illustrats, th DOA s MSE (Man Squar Error) vs SNR for L=2, and angl diffrnc δθ =5, it can b obsrvd that th MSE for Only bootstrappd MUSIC mthod and our tchniqu SSBoot that uss Bootstrap, spatial smooth and MUSIC hav almost th sam stimation accuracy It mans that SSBoot improvs th rsolution with no stimation accuracy nhancmnt Snapshots Probability of targt dtction, L=2, SNR = 5dB Boot000 Snapshots SSBoot 000 Snapshots angular sparation in dgr Fig 4 Angular sparation vs dtction rat Probability of targt dtction, L=2,=2, 20 Snapshots Boot000 Snapshots SSBoot 000 Snapshots SNR vs Dtction rat Fig 5 SNR vs Dtction rat -5 L=2 Boot000 SSBoot SNR in db Fig 3 MUSIC Spctrum for various snapshots Fig 6 RMSE vs SNR 32 P a g

5 Vol 8, No 0, 207 V CONCLUSION In this papr, w hav introducd a nw tchniqu basd on th combination of bootstrap tchniqu, spatial smoothing and MUSIC mthod to improv rsolution and th stimation of closd sourc numbr It was shown that for th cas of small sampl siz, th bootstrap tchniqu is usd to stimat and valuat th rsampl data Th spatial smoothing was also prsntd as spatial sampling mthod, which provids diffrnt sub-arrays and widns th angl sparation of closd sourc whn MUSIC Algorithm is applid Th rsults prsntd in this papr prov that our mthod is attractiv whn fw sampls ar availabl and outprforms th ordinary tchniqu at difficult scnarios spcially for vry clos sourc and low SNR Simulations hav shown that spatial sampling and bootstrap tchniqus outprforms DOA stimation, whn MUSIC mthod is applid for small sampl siz and vry clos sourcs But it s dmonstratd that SSBoot tchniqu can t improv th stimation accuracy REFERENCES [] T Bao, Mohammd El Korso, Ouslimani, Cramér Rao Bound And Statistical Rsolution Limit Invstigation For Nar-Fild Sourc Localization, Elsvir Digital Signal Procssing, (205), pp2 7 [2] S Marcos, Ls méthods à aut Résolution, traitmnt d antnns t analys spctral, (Éditions rmès 998) 2nd Ed [3] S M Irid,S Kamch, A Novl Algorithm To Estimat Closly Spacd Sourc DOA, Intrnational Journal of Elctrical and Computr Enginring (IJECE), Vol 7, No 4, August 207, pp [4] Akaik, A Nw Look At Th Statistical Modl Idntification, IEEE Tran, on aut control AC-9 6, (974) [5] R Ahmd, JN Avaritsiotis, A Study of MCA Larning Algorithm for Incidnt Signal Estimation, (IJACSA) Intrnational Journal of Advancd Computr Scinc and Applications,Vol 5, No 2, 204, pp [6] R Schmidt, Multipl Emittr Location and Signal Paramtr Estimation, IEEE Trans Antnnas Propagation (986) [7] E Grosicki, K Abd-Mraim, Y ua, A Wightd Linar Prdiction Mthod For Nar-Fild Sourc Localization, IEEE Trans Signal Procss 53(0) (2005) , [8] D Grnir, E Boss, Dcorrlation Prformanc of DEESE and Spatial Smoothing Tchniqus for Dirction-of-Arrival Problms, IEEE Transactions On Signal Procssing, Vol 44, No 6, Jun 996, PP [9] Y Albagory, M Nofal, S Alzoghdi, A Novl Location Dtrmination Tchniqu for Traffic Control and Survillanc using Stratosphric Platforms, (IJACSA) Intrnational Journal of Advancd Computr Scinc and Applications, Vol 4, No 0, 203,, pp [0] RF Brcich, AM Zoubir And P Plin, Dtction of sourcs using bootstrap Tchniqus, IEEE Trans signal Procssing, vol 50, no 2, pp206 25, Fb 2002 [] A Zoubir, D R Iskandr Bootstrap Tchniqu for Signal Procssing, Cambridg Univrsity Prss, Nw York 2004 ISSN [2] Z Lu, A Zoubir, Sourc Enumration in Array Procssing Using a Two-Stp Tst, IEEE Transactions On Signal Procssing, VOL 63, NO 0, May 5, 205 [3] Z Lu, A Zoubir, F M aardt, Sourc Enumration Using Th Bootstrap For Vry Fw Sampls,9 th Europan Signal Procssing Confrnc (EUSIPCO 20, Barclona, Spain, 20 PP [4] Jiang, S Pnnock, And P Shphrd, A Novl W-MUSIC Algorithm for GPR Targt Dtction in Noisy and Distortd Signals, IEEE Radar Confrnc (2009) IEEE, pp -6 [5] AS Achour, LPA Bamformr for Tracking NonstationaryAcclratd Nar-Fild Sourcs, (IJACSA) Intrnational Journal of Advancd Computr Scinc and Applications,Vol 5, No 3, 204, pp [6] X Zhang, M El Korso, M Psavnto, Angular rsolution limit for dtrministic corrlatd sourcs, 203 IEEE Intrnational Confrnc on Acoustics, Spch and Signal Procssing, ICASSP, 203, pp [7] P Stoica, RL Moss, Spctral Analysis of Signals, Parson/Prntic all, Uppr Saddl Rivr, NJ, 2005 [8] L Schmidr, D Mllon, M Saquib, Intrfrnc Cancllation and Signal Dirction Finding with Low Complxity, IEEE Transactions on Arospac and Elctronic Systms Vol 46, No 3 July 200, pp [9] P Stoica and Y Sln, Modl-ordr slction: a rviw of information critrion ruls, IEEE Signal Procssing Magazin, vol 2, no 4, pp P a g