Root Locus Properties of Adaptive Beamforming and Capon Estimation for Uniform Linear Arrays

Transcription

1 Root Locus Propertes of Adaptve Beamformng and Capon Estmaton for Unform Lnear Arrays Allan Stenhardt Alphatech phone: emal: Abstract In ths paper we explore propertes of the eroes of the transfer functon (Z transform) of the weght vector arsng n adaptve beamformng and drecton of arrval estmaton (Capon) usng sample matrx nverson. Our analyss sheds nsghts on propertes of dagonal loadng, as well as hgh-resoluton propertes of Capon's estmate. The analyss also provdes hnts at how to extend these propertes to nonunform array manfolds. Specfcally we prove the followng theorem. Root locus theorem for ULAs: Let w be the clarvoyant weght vector of dmenson N for a length N unform lnear array (ULA), gven by w = R - v, where v s the steerng vector to the target, and R s the (ensemble) covarance matrx. Then all N eroes of the Z transform of w le on the unt crcle. (Note, snce the sample matrx yelds an unbased estmator, the root locus for the adaptve beamformer has mean root loc on the unt crcle as well.) We then dscuss three applcatons of ths theorem: (I) Dagonal loadng: We show that the roots of the weght vector follow a trajectory (root locus) from the quescent pattern to the nterference angles as the nterference-to-nose rato grows. Dagonal loadng can then be vewed as a regularaton process that relaxes the root loc along ths trajectory. (II) Capon: The spectrum dynamc range s maxmed when the eroes are all on the unt crcle; therefore, our result provdes an alternatve nsght nto the hgh-resoluton propertes of Capon estmaton. (III) Non-ULA extensons: We fnd n our proof that the root locus behavor results from symmetry propertes of the MVDR objectve functon. Ths suggests gudelnes for successful approaches to generalng Capon estmaton and dagonal loadng to non-ula settngs. xxxx

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3 Root Locus Propertes of Adaptve Beam Formng and Capon Estmaton for Unform Lnear Arrays A. Stenhardt / Alphatech L. Scharf (as of 8:24am,3/7)/CSU

4 Problem and result Let v r be a length N Vandermond steerng vector, r = [, exp( jω ),.., exp( jω ( N ))] v t T ω t t where ωt s the target arrval angle n normaled coordnates It s well known that ths vector has exactly N- nulls,.e., ts Z transform has all unt crcle roots: V ( ) = N = Exp( jω) ( Exp( jω) ) = ( Exp( jω) ) Let R be a Toeplt matrx (sample matrx for nterference), and let us form the SMI MVDR weght vector: r r R v w = r H r v R v Theorem: the weght vector w has all ts roots on the unt crcle N

5 Matlab examples *quescent roots *fully adaptve roots *loaded adaptve roots target manfold error case target No manfold error case nterference nterference Matlab proves the theorem. Matlab shows that nulls drft f array snt lnear: multpath or manfold error predctor Expanded Vew: roots Mgrate to nterference

6 Why we care Yes No Even manbeam nullng never leads to fnte nulls!

7 Proof Lemma: Ths s a surprsng result! Proof of Lemma: Dr Guerc and Dr Zatman thnk so! ALL NULLS ALWAYS INFINITELY DEEP!!!!!! Proof of theorem: MVDR solves mn r v H r w= w H Rw = f Wener Khntchne, objectve f= 2π π π S( ω) N π = w e jω 2 dω S( ω)>0 ω Or f= * ( ω) W ( ) W ( ) dω, Z exp( jω) 2π S = wth W ( ) = π Let J be ant-dentty. The JRJ=R, JRJw=v=Jv, so w=jw Hence roots appear as recprocals. Are they unt modulus? W ( Z) = n = ( ) /( ), W () =

8 Reformulaton of MVDR cost functon: 2π π π S( ω) n ( ) /( ) 2 dω If I replace a root by ts nverse, constrant s preserved, and f root s NOT on the unt crcle I have a dfferent weght vector. But weght vector s unque by convexty. Hence we nvoke reducto ad absurdum QED