2-D Parameter Estimation Using Arrays With Multidimensional Invariance Structure A. SWINUI.EIIUI1ST AND T. KAIIATII 1. 1NTRODUCTlON HEN SIGNALS ARE COLLECTED by an array W of spatially distributed sensors, a variety of pa- ranielers (e.g., amplitude, phase, wavelength, azimuth and elevation angles of arrival, range, etc.) may he esti- iiinLcd per signal source. Although most of the so-cnllcd high-resolufion algorithnrs (e.g., hlUSIC 111, autoregres- siw modeling techniques, eic.) have I)een presented in llie ronl.cxt of cst,iiiiat,iiig a single parnnirt.cr prr source, giicrnlizntioiis to the iiiultiple paranirtrr case nrs rcln- 1.ively shiglitforward (e.g., for MUSIC, see 121). 'lhe inain dificulty associated with these niet~hods is l,liat, hoth computational and storage costs Lend to increase rapidly with the dimension of the parameter vector. The increased costs are usually prohibitive even for the two- dimensional case, and the result is that in practice, sys- f,eins typically employ iron-paraaretric techniques (i.e., l~eaiirforniiiig, 2D FFTs) to solve wliat are in rea1it.y pnra- weiric problems. Though lliese classical techniques are more e,asily implemented, their perforilialice is known to lie poor. 'The receiitly iiit.roduced Weighted Sul)space Fil,t.iiig (\\'SF) algoritliin [a, 4, 51 has been sliown lo provide esl.iiiiates which asymptotically achieve the CraniCr-Rao 'This work was supported in part by the SDIO/IST Progmn~ tnnnnged by the Oflice of Naval Research under Conlract NOOU14- 85.1<-0550. by the Joint Services Program at Slsnford Universily (US Anny. US Navy. US Air Force) under Conlrct UAAL03-88-C- 001 I, and by grilnls from Rorkwell Inlernsl,iond and llir General Rleclric Company. bo~iiid for paraiiiekr rst,iiiiatioii prohleiiis iii iinrroml,niid sensor array processing [G, 71. The tnaiii drawback of {,lie WSF approach is that the resulting parameter search must in general be initialized with estimat,es of reason- alh qiinlit.y. For imilt.iplc parninct.cr prohlrins, tlic in- l,inl csl,iii)at,cs tsius1. also 1~: 1,rolwrly nrsociahl; i.r.. cs- tiiiia1,t-s frotii dilferrtit. pnraiiirt,cr iliiimisions IIIIISI. IK no- signed to R particular source. Additioimlly, siws t.l>c WSF algorit.liiii ulliiiial.ely reqriircs a iiiiilt.idiiiieiisioiial parameter search, oiic ivould hope that. t,lm iiiitial es1.i- mates could be obtained as eficielitly as possible. In llie siiigle paran~c1.cr per source tax. if tlw nrrny is coi~~poscd d I,WO i,lciil,iral biil. dirplarrd sitlmrr.iys, f.lv I?SI?RI'I' nlgoril,liiii [RI caii IIC ciiiploycd lo liud 1.11~ iiii- tial es1,iiiiates iii a coiiil"it.nt,ioiinIly ellicieiit. maiiiicr. 'lb use ESl'RlT for the miiltiple parameter case. ~II arm! with displaceinelit invarialices iii more t.liaii one diiiirii- sion would be required. Ilowever, performing ESPIII'I' separately for each dimeiisioii of the parameter vecl.or aoiild not take advantage of the full iiivariaiice slrurlure of such an array. In nddition, such an approach pro- vides no mechairisin for associating tlir est,iiiratrs froill dilTeereiil diincrisiowa wil.li a parl.ictilar sourre. A siiiiilnr problem was addresscrl in In] iii the coiit.e.ut ofstale space IIIO~CIS for Imiklrin sninplril 2-11 tiiw srrirs. I IIP hry to iisiug 1.11~ gwiBwf.ric idws of li:Sl'lll'l~ Lir I.lie iiiiill.il)le pnrniiict,cr proldriii is recogliiaiq I,l~nl. 1.l~ ESPRIT algorilhm is a qwcial raw of \\'SF for R spe- c.ific array parainct,erizat.ioii [5, IO]. Oiice t.liis coiinrct.ioii is made, it is seen that. ext,endiiig the algorit,hlll to ar- rays with inult.iditiiensioiial iiivariaiices simply amouiik to a re-paramet,erisatioii of 1.h \VSF problem. \Vliile llie ESPRIT/WSF ininimizat.ioii is accomplished very em- ciently in the siiigle parainet,er case, a search lecliiiiqiie is reqiiired for the miiltiple invariance parameterization. Coi~sequently, iii t,Iiis paper we develop a sabopiirrtnl, hut. computationally eficiriit. algorillim for finding parailleter rstiiiiat,es in t,lw iiiiill,iplc invnriatire casc. If the opt.inlal \VSF solritioii is hired. 1~1~~s~ vsl,iiiinf,cs n~ny I,e ~isrd as initial coiidil.ioits for the rcsult.iiig grntliriit. search. 111 t.he next sdioti, a Iuirf descript.ion of the narrow- hand dab model is given for arrays wi1.h iilult.iple iden- tical subarrays. The H'SF algorit.liiii for such arrays is considered iii Sect,ion 3, and 1,he corresponding suhopti- mal approach is descrilwd in Srcl.ioii 4. Some rrpreseiita- live siiiiulat.ioii rrsiills arc givrn in Scrlioii 5. Alll~oiigli ,. 950 23ACSSC-1218910950 $1.00 (31989 MAPLE PRESS Authorized licensed use limited to: IEEE Editors in Chief. Downloaded on August 17, 2009 at 19:55 from IEEE Xplore. Restrictions apply.; .. ;:; IOCIIR on I.IE t~vo-~lii,i,~iisi~~iinl msc, 1,lic coiicept,s preseiited are easily extcnded to Iiigltcr diiim- sioiial parameter vectors. 11. DA7A Mons1, {\'e will assume ail M-element array of sensors, d aar- rowband far-field emitters, and we will define a(0) E C" to be tlie array response for a iiarrowlmiid ciiiitlcr al. 1)OA 0. The ou1,put z E CA* of the array at time 1 is giveii by m(1) = Gs(1) + n(1) , 5v1icrc s(t) E C* is tlie ainpIit.iideniid p~insc or I.J~ signals nt time 1, n(t) is additive noise, and where def G = [a(@1) ... a(@d)]. Assiiiiiiiig tlie noise is spatially white' and iiiicorrc- Iahl with the signals, we have Itzz = &{z(t)z*(/)) = GR,,G' + nZI , wliere & ~lenotes erpectatioii, R.,, is tlic covariance nia- 1.ris of tlie ciiiittcr signals, and u2 is the noise variance at, cach sensor. The covariance R,, is assumed to be full rank d (no riiiit,y correlated sigi~als)~ and the coliiiuiili 01 G are assiiiiied to be liiiearly iiidependeiil.. 'The cigriide- coiiiposition of It,, Iias t,he,following form: Af R.~~ = Chieief = E,A,E: + u2~,,~;, (I) ... 2 Ad > Ad+l = ... = hnj = d. The spaii of the ;=I where E, = (el ... ed], E, = [ed+l ... qc], and A1 2 d eigenvectors E, defines the signal subspace, and the orthogonal complement spanned by E,, defiiies the uoise subspace. This terminology is a conseqiience of the fact i,linL span(E,) =span((=) I span(E,,). 'J'liis inipIics1,liat :I (I x d iiialhix T exisl,s snlkfyiiig E, = GT. 'l'v csniiiiiie the specinl striictiire that, rcsu11~~ for arrays !ril,li iiinltiple invariances, assume the array is coinposed of 11 identical subarrays of 111 sensors, each displaced in liine or space from tlie reference