Parameter Estimation Using Arrays With Multidimensional Invariance Structure

2 DParameter Estimation Using Arrays With Multidimensional Invariance Structure A SWINUI EIIUI1ST AND 1 1NTRODUCTlON W HEN SIGNALS A R E COLLECTED b y an array o f spatially distributed sensors a variety of paranielers e g amplitude phase wavelength azimuth and elevation angles of arrival range etc m a y he estiiiinLcd per signal source Although most of the so cnllcd high resolufion algorithnrs e g hlUSIC 111 autoregress i w 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 p a r a n i r t r r case nrs rcln1 ively s h i g l i t f o r w a r d e g for MUSIC see 121 l h e i n a i n d i f i c u l t y associated w i t h these niet hodsis l liat hoth computational and storage costs Lend to increase rapidly w i t h the dimension of the parameter vector T h e increased costs are usually prohibitive even for the twodimensional case and the result is that in practice sysf eins typically employ iron paraaretric techniques i e l eaiirforniiiig 2D FFTs to solve wliat are in rea1it y pnraw e i r i c problems Though lliese classical techniques are more e asily implemented their perforilialice is known t o lie poor The receiitly iiit roduced Weighted Sul space Fil t iiig SF algoritliin a 4 51 has been sliown l o 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 NOOU1485 1 0550 by the Joint Services Program at Slsnford Universily US Anny US Navy US Air Force under C o n l r c t UAAL03 88 C001I and by grilnls from Rorkwell Inlernsl iond and llir General Rleclric Company T KAIIATII bo iiidfor p a r a i i i e k r rst iiiiatioii prohleiiis iii iinrroml niid sensor array processing G 71 T h e tnaiii drawback of lie WSF approach is t h a t the resulting parameter search must in general be initialized w i t h estimat es of reasona l h qiinlit y For imilt iplc parninct cr prohlrins tlic i n l i n l csl iii at cs tsius1 also 1 1 rolwrly n r s o c i a h l i r cstiiiia1 t s frotii dilferrtit pnraiiirt cr i l i i i m i s i o n s IIIIISI IKnosigned to R particular source Additioimlly s i w s t l c WSF algorit liiii ulliiiial ely reqriircs a iiiiilt idiiiieiisioiial parameter search oiic ivould hope that t lm i i i i t i a l es1 imates could be obtained as e f i c i e l i t l y as possible In l l i e siiigle paran c1 crp e r source tax if t l w nrrny is c o i p o s cd d I WO i lciil iral biil d i r p l a r r d sitlmrr iys f lv I SI RI I nlgoril liiii RI caii IIC ciiiploycd l o liud 1 11 iiiit i a l es1 iiiiates iii a coiiil it nt ioiinIly ellicieiit maiiiicr lb use ESl RlT for the m i i l t i p l e parameter case I arm I w i t h displaceinelit invarialices iii more t liaii one d i i i i r i i 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 s l r u r l u r e of such a n array In nddition such an approach provides no mechairisin for associating tlir est iiiratrs froill dilTeereiil diincrisiowa wil li a parl ictilar sourre A siiiiilnr problem was addresscrl i n In iii the coiit e ut ofstale space I I I O C I S for I m i k l r i n sninplril 2 11 t i i w srrirs I IIP hry to iisiug 1 1 1 gwiBwf ric i d w s of li Sl lll l Lir I lie iiiiill il le pnrniiict cr proldriii is r e c o g l i i a i q I l nl 1 l ESPRIT algorilhm is a q w c i a l r a w of SF for R spec ific array parainct erizat ioii 5 IO Oiice t liis coiinrct ioii is made it is seen that ext endiiig the algorit hlll to arrays w i t h inult iditiiensioiial iiivariaiices simply a m o u i i k to a re paramet erisatioii of 1 h VSF problem Vliile l l i e ESPRIT WSF ininimizat ioii is accomplished very emciently in the siiigle parainet er case a search lecliiiiqiie is reqiiired for the m i i l t i p l e invariance parameterization Coi sequently iii t Iiis paper we develop a sabopiirrtnl hut computationally eficiriit a l g o r i l l i m for finding parailleter rstiiiiat es in t lw iiiiill iplc invnriatire casc If the opt inlal VSF solritioii is h i r e d 1 1 vsl iiiinf cs s n n yI e i s r das i n i t i a l coiidil ioits for the rcsult iiig grntliriit search 111 t he next s d i o t i a Iuirf descript ion of the narrowhand d a b model i s given for arrays wi1 h iilult iple identical subarrays T h e H SF algorit liiii for such arrays is considered iii Sect ion 3 and 1 he corresponding suhoptimal approach is descrilwd i n Srcl ioii 4 Some rrpreseiital i v e siiiiulat ioii rrsiills arc g i v r n i n Scrlioii 5 A l l l o i i g l i 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 o n I IE t v o l i i i i i s i msc i i n l 1 lic coiicept s preseiited are easily extcnded to Iiigltcr d i i i m sioiial parameter vectors wltcrc I i s n i l n x A s lvcl ioii i i I r B I B B is diagoiial i i i a t r i x of IIRSC dclays wlticlt dcsci v s 1 1le displacement of the it subarray froin tlie refcreiiL 11 DA7A Mons1 T l i e paraiiiet erioal ioii of 2 is very general an 6 j l be iiiade to apply l o a wide v a r i d y of problcius A s a exaiiiple consider t l i e 5 x 5 rectangular a r r a y …