Properties of the partial Cholesky factorization and application to reduced-rank adaptive beamforming

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Properties of the partial Cholesky factorization and application to reduced-rank adaptive beamforming

Olivier Besson, François Vincent

To cite this version:

Olivier Besson, François Vincent. Properties of the partial Cholesky factorization and application to reduced-rank adaptive beamforming. Signal Processing, Elsevier, 2020, 167, pp.107300-107309.

�10.1016/j.sigpro.2019.107300�. �hal-02572620�

an author's https://oatao.univ-toulouse.fr/25983

https://doi.org/10.1016/j.sigpro.2019.107300

Besson, Olivier and Vincent, François Properties of the partial Cholesky factorization and application to reduced- rank adaptive beamforming. (2020) Signal Processing, 167. 107300-107309. ISSN 0165-1684

Properties of the partial Cholesky factorization and application to re duce d-rank adaptive beamforming

Olivier Besson^∗,François Vincent

ISAE-SUPAERO, 10 Avenue Edouard Belin, Toulouse 31055, France

Keywords:

Adaptive beamforming Cholesky factorization Reduced rank Wishart matrices

a b s t r a c t

Reduced-rankadaptivebeamformingisawellestablishedandefficientmethodology,notablyfordistur- bancecovariancematriceswhicharethesumofastronglow-rankcomponent(interference)andascaled identitymatrix(thermalnoise).Eigenvalue orsingulardecompositionisoftenusedtoachieve rankre- duction.Inthispaper,westudyandanalyzeanalternative,namelyapartialCholeskyfactorization,asa meanstoretrieveinterferencesubspaceandtocomputereduced-rankbeamformers.First,westudythe anglesbetweenthetruesubspace andthat obtainedfrom partialCholeskyfactorizationofthe covari- ancematrix.Then,astatisticalanalysisiscarriedoutinfinitesamples.Usingpropertiesofpartitioned Wishartmatrices, weprovideastochasticrepresentationofthebeamformerbasedonpartialCholesky factorizationandofthecorrespondingsignaltointerferenceandnoiseratioloss.Weshowthatthelatter followsapproximatelyabetadistribution, similarlytothe beamformerbasedoneigenvalue decompo- sition.Finally,numericalsimulationsarepresented whichindicatethatareduced-rankadaptivebeam- formerbased onpartial Choleskyfactorizationincursalmostnoloss, and caneven performbetterin somescenariosthanitseigenvalueorsingularvalue-basedcounterpart.

1. Motivationofthework

Enhancing retrieval of a signal of interest (SoI) buried in noise and interference by means of adaptive filtering is a very widespread problemin many engineering applications, including radar sonar andcommunications [26], aswell asin finance with theselectionofamean-varianceefficientportfolio[23].The most widelyusedapproachconsistsindesigningalinearfilterw,which preserves the SoI through some constraints, and minimizes the output power, hence tends to cancel or at least attenuate inter- ference. In other words, one tries to minimize w^{Hw, under a} unit-gainconstraintw^Hv=1,where^{standsforthep}×pcovari- ancematrixofthemeasurementsandvdenotestheSoIsignature.

The solution is wopt=^−1v/(^vH−1v), which provides the opti- malsignaltointerferenceandnoiseratio(SINR),whichwedenote asSINRopt=SINR(^wopt)^.

In array processing applications,when =R_N contains noise only(i.e.,thermalnoiseandinterference),thisfilterisreferredto as the minimum variance distortionlessresponse (MVDR) beam- former. If the SoI is presentin the measurement, in which case =R_S+N, one speaks ofminimum power distortionlessresponse (MPDR) [26].When ^{is known, there is no difference between}

∗ Corresponding author.

E-mail address: olivier.besson@isae-supaero.fr (O. Besson).

the two filters. However, in practice ^{is unknown and must} be estimated from a set of n independent and identically dis- tributedsamplesx_i,with(supposedlycommon)covariancematrix

=E

x_ix^H_i

.Useofthesamplecovariancematrixˆ ^inlieuofthe trueonemakesabigdifferencebetweentheMVDRandMPDRsce- narios. Indeed,while only 2p−3 samples are neededto achieve anaverageSINRlossequalto−3dBintheMVDRcase,thisfigure risestoabout(^p+1)(¹+SINRopt)^{intheMPDRcase,whichcanbe} muchlarger.Inmostpracticalcases,thenumberofavailablesnap- shotsismuchlower, whichresultsina significantdegradation in theMPDRcontext.

In some cases one might even face a situation where n<p, which makes ˆ ^{singular, and therefore its inverse does not ex-} ist. Inorder to address thissituation andto improve the rateof convergence,twomainapproacheshaveemergedintheliterature.

The firstis based on diagonalloading [1,5,6], a simpletechnique which was shown to achieve, provided that the loading level is properlychosen,a fastconvergencerate, typicallyofthe orderof twicethenumberofinterferingsignals[2,6,7,10].Thesecondmain categoryisthat ofreduced-rank adaptivefilterswheretheweight vectorw isconstrainedto liein alow-dimensional subspace,see e.g.,[11,14,15,17,22].Thisapproachisparticularlysuitablewhenthe interferenceplusnoisecovariancematrixisthesumofalow-rank term and a scaled identity matrix, i.e., when R_N=U^UH+σ^2I

where U is a p×r matrix of the eigenvectors of R_N and ^is

https://doi.org/10.1016/j.sigpro.2019.107300

thediagonalmatrixofeigenvalues. Actually,forlarge interference to noise (INR) ratio wopt is approximately, up to scaling factor, theprojection ofv on thesubspace orthogonal to U, i.e., wopt

α(^I−UU^H)^{v.Inpractice,anestimate ofUismadeavailable from} the eigenvalue decomposition (EVD) of ˆ ^{or the singular value} decomposition(SVD) ofthedatamatrixX=

x₁ x₂ ... xn

. AnalysisoftheSINRlosswasconductedin[21]whereitwasstated thatitapproximatelyfollowsaBetadistribution.In[16],theSINR lossisanalyzed startingfromthe asymptoticpropertiesofeigen- valuesandeigenvectors.Whiletheproofisrigorous,itholdsonly forn→∞,whilereduced-rankadaptivefilteringisespeciallyinter- estinginlowsamplesupport.

While EVD or SVD are the usual tools forretrieval ofprinci- pal subspace, one might investigate alternative, possibly simpler factorizationsthat could yield a similar performance in terms of adaptivebeamforming:thisis theobjective ofthe presentpaper.

Our goal is not to propose a new reduced-rank adaptive beam- former,ratherto suggest anotherimplementationandto validate its performance through a theoretical analysis. Towards this end weproposetouseapartial(truncated)Choleskyfactorization,that iswe suggest to use a conventional (with no pivoting) Cholesky factorizationalgorithm,applied toˆ, andtostop it afterr itera- tions.We are particularlyinterestedin thecasewhere ˆ ^isesti- mated fromn<p snapshots andis thus of rankn. Also, we will focus on matrices =U^UH+σ^{2I (MVDR case) or} =Pvv^H+ U^UH+σ^{2I(MPDR case),whereUformsabasisfortheinterfer-}

encesubspace.

The paperisorganizedasfollows.InSection 2,we willdefine thepartialCholeskyfactorizationandwillshowthatitisarather accuratemethodtoretrievetheprincipalsubspaceof^whenthe latteristhe sumofa low-rankmatrix anda scaled identityma- trix.InSection3wewillintroduceareduced-rankadaptivebeam- formerbased on thepartial Cholesky factorizationandderive its statistical properties. The simulations of Section 4 will compare thisbeamformer toits counterpart usingSVDandfinally conclu- sionswillbedrawninSection5.

2. PartialCholeskyfactorization

In thissection,we introducethe partialCholeskyfactorization andwestudytheanglesbetweenthesubspaceobtainedfromthe partialCholeskyfactorizationof^andtheinterferencesubspace, intheMVDRaswellasintheMPDRcase.

2.1.Definition

Let us start fromthe (full) Choleskyfactorization [12,19] ofa positive definite matrix A which yields a p×p lower triangular matrixL,withpositivediagonalelements, suchthat A=LL^H.The principleofthepartialCholeskyfactorizationissimplytostopthe factorizationafterrsteps(columns),yieldingap×rlowertriangu- larmatrix,whosepropertieswillbestudiedbelow.IfAisfull-rank, andnopivoting is used,then stoppingafter riterationsprovides thesameresultasselectingthefirstrcolumnsofa fullCholesky factorizationofA.IfAhasrankn<p,thenthefullCholeskyfactor- izationdoesnotexist,andstoppingafterr≤nsteps yieldsalow- rankapproximation, which will be used to obtain a basis of the interferencesubspace. Note that, when A has rank n, symmetric pivotingcanbeusedtoproduceap×nlower triangularmatrixL withpositivediagonalelements,andapermutationmatrix^such that of ^A=LL^H, see e.g., [12], Algorithm 4.2.4. Now, with a positivesemi-definitematrixA,Choleskyfactorizationwithpivot- ingcanalsobestoppedafterrsteps,wherer≤rank(A),toproduce alow-rankapproximationofA.Suchatechniquehasbeenusedin machinelearningwhereitservesthepurposeoffindingalow-rank approximationoftheKernelmatrixseee.g.,[3,9],anddifferences

amongmethods concernthestrategy forpivoting. In[18]conver- genceresultswereobtainedwhichprovetheeffectivenessofsuch amethod.

The partial Cholesky factor of ^{, which will be denoted as} pchol(^{, r), is thus the p}×r (with r≤rank(^{)) lower triangular} matrixwithpositivediagonalelementsG=(^G_G⁻¹

−2),whereG₋₁ isa r×r lowertriangular matrixwithpositive diagonalelements andG₋₂isa(^p−r)×rmatrix,definedfrom

=

11 12

21 22

=

11 12

21 21⁻11¹12

+

0 0 0 2.1

=

G₋1

G₋2

G^H₋₁ G^H₋₂ +

0 0 0 2.1

(1)

where 2.1=22−21⁻₁₁¹12, G₋₁G^H₋₁=11 and G₋₂G^H₋₁=21.Fromapracticalpointofview,G=pchol(,r) can be obtained, e.g., by usingonly rsteps of Algorithm4.2.2 of GolubandLoan[12].

2.2. Angleswiththeinterferencesubspace

Letusnow examinethe abilityof thepartial Choleskyfactor- ization to retrieve the principal subspace of ^{when the latter} isoftheform=U^UH+σ^2Iwhere=diag(λ1,...,λ^r)^isthe diagonal matrix of eigenvalues and U is the matrix of eigenvec- tors. Towardsthis end, let usstudythe angles betweenthe sub- spacespannedbypchol(^{,r) andthesubspacespannedbyU.Let} us partition U as U=[U₁

U₂] where U₁ is r×r. From (1), one has 11=U₁^UH1 +σ^2Irand21=U₂^UH1,sothat

range(^G)^=range 11

21

=range

U1^UH1+σ^2Ir

U2^UH1

=range

U1

U₂

^UH1+

σ^2Ir

0

=range

U₁ U2

+

σ²

U^H₁ ⁻¹⁻¹ 0

=range

U1+1

U2

. (2)

ForlargeINR,i.e.,forλkσ^2,1isamatrixwithsmallelements, hencerange(G)should beclosetorange(U).The anglesbetween thetwosubspacesareobtainedfromthesingularvaluesofthefol- lowingmatrix[12]

M=U^H

U1+1

U2

(^U1+1)^H(^U1+1)+U^H₂U2

−H/2

=

Ir+σ^{2−1 I}r+^H1U1+U^H₁1+^H11 −H/2

=

Ir+σ^{2−1 I}r+2σ²⁻¹+σ⁴⁻¹ U^H₁U1

−1⁻¹−H/2

Ir+σ²⁻¹ Ir−σ²⁻¹−1 2σ⁴⁻¹

U^H₁U1 −1

⁻¹+3 2σ⁴⁻² Ir−σ⁴

2⁻¹ U^H₁U1 −1

⁻¹+1 2σ⁴⁻²

=Ir−σ⁴ 2^−1UH2

Is+U2U^H₂ ⁻¹U2⁻¹. (3) Note that one should go up tosecond orderin the expansion.If we let θk,k=1,...,r denote the angles between the subspaces,

Fig. 1. Distance between range pchol

UU ^H+ σ²I , r

and range( U ) using the par- tial Cholesky decomposition. r = 2 interference with varying INR in the field of view of a p = 16 element uniform linear array.

one has Tr

MM^H

=r

k=1cos²θk. Now, the square distance be- tween the two subspaces is d²=r

k=1θk²r

k=1sin²θk and, for largeINR,

r

k=1

sin²θkσ^{4Tr −1UH}2

Is+U2U^H₂ ⁻¹U2⁻¹

. (4)

Although this is not zero, as would be the case with the sin- gular value decomposition whoser principalleft singularvectors sharethesamesubspaceasU,thedistancebetweenrange(G)and range(U)goestozeroasINR⁻¹ whenINRgrowslarge.

This is illustrated in Fig. 1 where we display the distance betweenrange(G) andrange(U) inthecaseofa p=16element uniform linear array with inter-distance half a wavelength. Two narrowband interference are present in the field of view with respective directions of arrival −10^◦ and10^◦,and a varying INR.

As canbe observed, thedistancegoesto zeroasboth INRsgrow large. Note that this figure concern the asymptotic case where ^{is known. Of more practical interest is the case where is} estimated from n snapshots with n possibly smaller than p. In this case, a different analysis should be conducted. In the next section, we provide some results about the statistical properties of pchol_ˆ

,r while numerical results about its application to adaptivebeamformingisthesubjectofSection4.

Let us now study what happens when =Pvv^H+U^UH+

σ^2I, i.e., when the SoI is present in the measurements. With no loss of generality, one can assume that v=ep where ep=

0 0 ... 0 1T

.Then,onlythe(p,p)elementof^ismod- ified, compared tothe MVDRcasewhere =U^UH+σ^2I.Since

thisaffectsonly22 itmeansthatthepartialCholeskyfactorization isleftunchanged,thatis

pchol

Pvv^H+U^UH+σ^2I,r =pchol

U^UH+σ^2I,r (5)

which implies that the angles between

range

pchol

Pvv^H+U^UH+σ^2I,r

and range(U) are still given by the analysis above. Note that this holds true provided that the first r components of v are zero. In contrast, the angles be- tweenrange(U) andthefirstreigenvectorsofU^UH+σ^2Iareno

longer zero.Indeed,it isonlyknownthat thefirstr+1principal eigenvectors of Pvv^H+U^UH+σ^{2I share the same subspace as} U v

. Accordingly, the subspace spanned by the r principal

Fig. 2. Distance between range eig

Pvv ^H+ UU ^H+ σ²I , r

and range( U ). The dis- tance between range

pchol

Pvv ^H+ UU ^H+ σ^{2I , r}

and range( U ) is given by Fig. 1 . r = 2 interference with varying INR in the field of view of a p = 16 element uniform linear array. Signal to noise ratio is 0 dB.

eigenvectors contains a contribution from v and one cannot recover exactly the interference subspace, in contrast with the MVDR case. Therefore, when one wants to retrieve the interfer- encesubspaceinaMPDRcontext,apartialCholeskyfactorization, stopped afterr steps withrthe number ofinterfering signals,is a meaningful approach, because it brings the SoI component in a part of the matrix which is not used in the partial Cholesky factorization.

Fig. 2 displaysthe angles obtainedwith an EVD in the same scenario as before but with the SoI present. Note that the an- glesobtained witha partial Choleskyfactorization are still given by Fig. 1,due to (5). Comparingthe two figures, it appears that thedifference in termsof subspace proximity isnot very impor- tant. Therefore,fromthe point ofview of retrievingthe interfer- encesubspaceinaMPDRcontext,apartialCholeskyfactorization possessesinterestingproperties.

3. Reduced-rankadaptivebeamformingusingthepartial Choleskyfactorization

Inthissection, we consider theuse ofa partial Choleskyfac- torization for reduced-rank adaptive beamforming purposes. Let us assume that n independent and identically distributed com- plexGaussianvectorsx_iwithzeromeanandcovariancematrix areavailableandgatheredinthematrixX=

x₁ x₂ ... xn

. We denote the distribution of X as X=^d CNp,n(⁰,,I_n)^{. We are} especially interestedin the casewhere the numberof snapshots is less than the size of the observation space, i.e., n<p. Let S=XX^H be the samplecovariance matrix andlet G=pchol(^S,r) withr≤min(p,n)denoteitspartialCholeskyfactorization.Assaid above,Gconstitutesausefultooltoretrievetheinterferencesub- space, which leads naturally to consider the following reduced- rankadaptivebeamformer

w=P^⊥_Gv (6)

where P^⊥_G=I−G(^GHG)^{−1GH is the orthogonal projector on the} subspaceorthogonal torange(G). Inthesequel, weprovide asta- tisticalanalysisoftheSINRlossoftheadaptivebeamformerin(6). Priortothat,wealsoprovidesomestatisticalpropertiesofG.The mainresultsare statedbelowwhiletechnicalproofsare deferred