A bias-compensated MUSIC for small number of samples

(1)

an author's

https://oatao.univ-toulouse.fr/26811

https://doi.org/10.1016/j.sigpro.2017.03.015

Vincent, François and Pascal, Frédéric and Besson, Olivier A bias-compensated MUSIC for small number of samples.

(2017) Signal Processing, 138. 117-120. ISSN 0165-1684

(2)

Short

communication

A

bias-compensated

MUSIC

for

small

number

of

samples

François Vincent

a ,∗

_{, Frédéric Pascal}

b

_{, Olivier Besson}

a

a ISAE-Supaéro, Department Electronics Optronics Signal, 10 Avenue Edouard Belin, 31055 Toulouse, France b L2S, CentraleSupélec, 91192 Gif-sur-Yvette, France

Keywords:

DOA estimation MUSIC G-MUSIC

ﬁnite sample analysis

a

b

s

t

r

a

c

t

The multiplesignal classification (MUSIC)methodis known tobe asymptotically efficient,yet witha smallnumberofsnapshotsitsperformancedegradesduetobiasinMUSIClocalizationfunction.Inthis communication,startingfromG-MUSICwhichimprovesoverMUSICinlowsamplesupport,ahighsignal tonoiseratioapproximationoftheG-MUSIClocalizationfunctionisderived.Thisapproximationresults inclosed-formexpressionsoftheweightsappliedtoeacheigenvectorofthesamplecovariancematrix. Anew methodwhichconsistsinminimizing thissimplifiedG-MUSIC localization functionis thus in-troduced,andreferredtoassG-MUSIC.Interestinglyenough,thissG-MUSICcriterioncanbeinterpreted asabiascorrectionoftheconventionalMUSIClocalizationfunction.Numericalsimulationsindicatethat sG-MUSICincuronlyamarginallossintermsofmeansquareerrorofthedirectionofarrivalestimates, ascomparedtoG-MUSIC,andperformsbetterthanMUSIC.

1. Introductionandproblemstatement

Estimating the directions of arrival (DoA) of multiple sources impinging on an array ofM sensorsis a primordialtaskin most sonar or radar systems [1] . A reference approach to tackle this

problem is by the maximum likelihood estimator (MLE) [2–5] ,

whoseperformance isatbestmatchedasymptotically,butis usu-allymostaccurate intheso-calledthresholdareawheremost

es-timators beginto depart fromthe Cramér-Rao bound (CRB). The

MLEentailsa globalsearch forthemaximumofa K-dimensional

likelihood function, where K stands for the number of sources

and canthus be prohibitive froma computational point ofview. In theeighties,the paradigmofsubspace-basedmethods was in-troduced, relying heavily onthe low-rank structure ofthe noise-free covariancematrix.Exploiting thepartitioningofthespaceas a subspace containing the signals of interest and its orthogonal

complement, the K-dimensional problem was reduced to a

one-dimensional problemwhereeitherK maxima,K eigenvalues orK

roots of a polynomial were to be searched,see e.g. MUSIC [6,7] , ESPRIT[8] orMODE[9] respectively.

MUSIC[6,7] ,whichisoneoftheﬁrstsubspace-basedtechnique introduced andisapplicable toanyarray geometry,hasbeen ex-tensively studied. The MUSIC DoA estimates are obtainedas the K deepest minima of the localization function LˆMUSIC

(

θ

)

deﬁned

∗ _{Corresponding author.}

E-mail addresses: [email protected] , [email protected] (F. Vincent), [email protected] (F. Pascal), [email protected]

(O. Besson).

hereafter.Inthe largesamplecase,itwasdemonstratedthat itis asymptotically unbiased and eﬃcient [10–12] ,i.e. it achieves the CRB either as the number of snapshots T or the signal to noise ratio(SNR)growlarge.Nonetheless,itsperformanceinﬁnite sam-pledegrades.Thisisdetrimentalinpracticalsituationswhere dy-namicallychangingenvironmentsrequirecarryingoutDoA estima-tionwithapossiblysmallnumberofsnapshots.In[13] ,Kavehand BarabellprovidedadetailedstudyofMUSIClocalizationfunction

)

where a(

θ

) stands for the array steering vector and Uˆn=

ˆ

u1 · · · ˆuM−K

where uˆm are the eigenvectors of the

sam-ple covariance matrix with the convention that the correspond-ing eigenvalues ˆ

λ

m are sorted in ascending order. They proved

that, when evaluated at a true DoA

θ

_k, LˆMUSIC

(

θ

)

has a ﬁnite

sample bias, which is generally larger than the corresponding

standard deviation, and is thus the main factor for the loss of resolution and accuracy. In [14] , rigorous expressions for the ﬁ-nite sample bias of MUSIC DoA estimates were derived. In fact,

resorting to random matrix theory (RMT), i.e. considering the

asymptotic regime where M, T → ∞ with M/T → c (denoted

asRMT-regime), itwas proven in[15] that thelocalization func-tion of MUSIC is not consistent. As a corollary, it was demon-stratedthat MUSICcannotconsistentlyresolve sourceswithin the

main beam width. In order to cope with this problem, the

G-MUSIC methodwas introduced which provides a consistent

esti-mate of aH

₍

_θ

₎

_U

nUHna

(

θ

)

in the RMT sense. G-MUSIC estimates

the noise projection matrix as PˆG-MUSIC=mM=1wmuˆmuˆHm where

wm are weightsdeﬁnedhereafter.The differencewiththe MUSIC http://dx.doi.org/10.1016/j.sigpro.2017.03.015

(3)

projector PˆMUSIC=Mm−K=1uˆmuˆHm is twofold: MUSIC uses only

“noise” eigenvectorswhile PˆG-MUSIC makesuseofalleigenvectors,

andMUSIC doesnot attributea differentweighting tothe eigen-vectors.G-MUSICwasshowntoimproveoverMUSICand,although itreliesonanasymptoticassumption,G-MUSICprovedtobe effec-tiveinsmallsamplesupport[15,16] .

Thissaid,theweightsofG-MUSICarenoteasytoobtain: com-putingthemrequiresfindingtherootsofaMthdegreepolynomial orfindingtheeigenvaluesofaM× Mmatrix,seebelowfordetails. Additionally,itisdifficultto havea simpleandintuitive interpre-tationoftheseweights. Inthiscommunication, we start from G-MUSICwhichperformswellforsmallT,andtrytosimplify calcu-lationofitsweightsandtoobtainmoreinsightfulexpressions.Our

approachis based ona high SNR approximation ofthe G-MUSIC

weightsandresults ina simple,closed-formexpression. Interest-inglyenough, theso-approximated weights canbe interpreted as a correction of the biasin MUSIC localization function. The new schemeisthussimplerthanG-MUSICwithoutsacriﬁcingaccuracy, aswillbeshowninthenumericalsimulations.

2. DerivationofsG-MUSIC

In thissection, we derive an approximated andsimpliﬁed ex-pressionofG-MUSICprojectionestimate

ˆ PG-MUSIC= M m=1 wmuˆmuˆHm (1)

andrelatetheso-obtainedestimatetoabiascompensationof MU-SIC.

2.1.Backgroundandapproach

TheweightswmofG-MUSICaregivenby[15]

wm=

⎧

⎨

⎩

1+k>M−K

_ˆ λk ˆ λm−ˆλk− ˆ μk ˆ λm− ˆμk

m≤ M− K −k≤M−K

_ˆ λk ˆ λm−ˆλk − ˆ μk ˆ λm− ˆμk

m>M− K (2)

where

λ

ˆkaretheeigenvaluesofthesamplecovariancematrixand

ˆ

μ

k,k=1,...,M denotetherootsof

f

(

μ

)

= M m=1 ˆ

λ

m ˆ

μ

ˆm,whichshouldlie

around thewhite noise power (WNP),and the latteris assumed

to be small.Moreover, observefrom (2) that the M_{− K smallest} ˆ

μ

m will impact theweights ofthe signal eigenvectors whilethe

weightsofthenoiseeigenvectorsdependontheKlargest

μ

ˆmonly.

Our approximation relies on ﬁnding the roots of (3) by con-sideringthetwoclustersofsolutionsindependently.Rewritingthe functionin(3) as f

(

f

(

μ

)

= M−K m=1 fm

(

μ

)

+ M m=M−K+1 fm

(

μ

)

=fn

(

μ

)

+fs

(

μ

)

.

First,we use the fact that, when searching forthe M− K small-estvaluesof

μ

,fs(

μ

)isapproximatelyconstant,whichleadstoan

approximationof

μ

ˆmform≤ M− K andhenceofthesignal

eigen-vectorsweights.Asforthewm,m≤ M− K,wewillprovideahigh

SNRapproximationofthemdirectly.

2.2. Approximatingthesignaleigenvectorsweights

Proposition1. At high signalto noiseratio, theweightswm ofEq.

(2)appliedtothesignaleigenvectorscanbeapproximatedas

wm≈ −ˆ

λ

−1m

(

T− K

)

−1

M−K k=1 ˆ

λk

m>M− K. (4)

Proof. Firstnotewmform>M− K isrelatedtotheM− K

small-estsolutions of(3) .Thelatterwill betypically ofthesame mag-nitudeastheWNP(dueto theinterlacingproperty

λ

ˆm−1<

μ

ˆm<

ˆ

λ

m) and hence negligible compared to

λ

ˆM−K+1,· · · ,

λ

ˆM. Hence,

they belong tosome interval In where ˆ

λ

m/

(

λ

ˆm−

μ

)

≈ 1form>

M− K whichresultsin fs(

μ

) ≈ Kwhen

μ

∈In.Consequently,the

M_{− K smallest}valuesof

μ

areobtainedbysolving

fn

(

μ

)

+K=T⇔1− 1 T− K M−K m=1 ˆ

λ

m ˆ

λ

m−

n=0 ⇔det

ˆ

n− 1 T− K

ˆ

λ

n

ˆ

λ

n T −

μ

I

_and_where_the lastequivalenceisobtainedbymultiplyingbydet

ˆn−

μ

I

.It fol-lowsthat

μ

ˆmform=1,· · · ,M− K areapproximatelythe

eigenval-uesof

ˆn−

(

T− K

)

−1

ˆ

λ

n

ˆ

λ

n T .

ˆ−1m 1. Itthen ensuesthat,for

m>M− K wm=− M−K k=1

_ˆ

λk

ˆ

λm

− ˆ

λk

− ˆ

μk

ˆ

λm

− ˆ

μk

=− M−K k=1 ˆ

λ

m

k ˆ

λ

2 m

(

1− ˆ

λk

ˆ

λ

−1m

)(

1− ˆ

μk

λ

ˆ−1m

)

≈ −ˆ

λ

−1 m M−K k=1

k=−ˆ

λ

−1m

M−K k=1 ˆ

λk

− M−K k=1 ˆ

μk

≈ −ˆ

λ

−1 m

M−K k=1 ˆ

λk

− Tr

{

ˆ_n₋

₍

_T_{− K}

₎

−1

ˆ

λ

n

ˆ

λ

n T

}

=−ˆ

λ

−1 m

(

T− K

)

−1

_M −K k=1 ˆ

λk

. (6)

2.3. Approximatingthenoiseeigenvectorsweights

Proposition2. At high signalto noiseratio, theweightsapplied to thenoiseeigenvectorscanbeapproximatedas

wm≈ 1 m≤ M− K. (7)

Proof. Letuswrite,form≤ M− K

wm=1+ M k=M−K+1

(4)

=1+ M k=M−K+1 ˆ

λm

(

_λk

ˆ _{− ˆ}

_μk

₎

(

_λ

ˆ_m_{− ˆ}

_λk

₎₍

_λ

ˆ_m_{− ˆ}

_μk

₎

=1+ M k=M−K+1

(

_λm

ˆ

_λ

ˆ−1 k

)(

λk

ˆ

μ

ˆ−1k − 1

)

ˆ

λ

k−1,ˆ

λ

k

ˆk_min lies between the largest noise eigenvalue and

thesmallestsignaleigenvalue.Wenowprovethat

μ

ˆk_miniscloseto

ˆ

λ

μk

min>

λk

ˆ min

1−K T

. (11)

Theprevious equation showsthat

μ

ˆk_min isratherclosetothe

−1_k min

1−K T

−1

1.Coming backto (8) it followsthat,athigh

SNR,wm≈ 1form≤ M− K.

2.4. sG-MUSICanditsrelationtoMUSICbiascompensation

Combining(4) and(7) ,itfollowsthatPˆG-MUSICcanbe

approxi-matedby ˆ PsG-MUSIC=UˆnUˆHn −

M−K k=1 ˆ

λk

T− K Uˆs

ˆ −1 s UˆHs =_Pˆ_MUSIC₋

M−K k=1

λk

ˆ

T− K Uˆs

ˆ −1 s UˆHs (12)

where Uˆs=[uˆM−K+1· · · ˆuM] and

ˆs=diag

(

λ

ˆs

)

with

λ

ˆs=

_ˆ

λ

M−K+1 · · · ˆ

λ

M

T

. The projector in (12) provides an

ap-proximation of PˆG-MUSIC which relies only on the eigenvalues

and eigenvectorsand thus avoids theneed to solve (3) .One can

observe that the noise eigenvectors are attributed a common

weight equal to one as in MUSIC, while the signal eigenvectors

areweightedby ˆ

λ

−1_m

(

T− K

)

−1

M−K

k=1

λ

ˆk

,whichtendstozeroas Tincreasesand/orthesignaltonoiseratioincreases,whichseems logical.

Interestinglyenough, PˆsG-MUSIC can be viewed as a correction

ofthebiasofMUSIClocalizationfunction.Moreprecisely, wewill show that the correctiveterm in the second line of (12) can be interpretedasacompensationofMUSICbiasduetoﬁnitesample support.Asshownin[13] ,seealso[1, Chapter 9] ,onehas

E

PˆMUSIC

− UnUHn =− M i=M−K+1 M j=1, j=i

λi

λ

j T

(

λi

−

λ

j

)

2

ujuHj − uiuHi

. (13)

Wecanrewritethepreviousequationas

E

_Pˆ_MUSIC

_{− U}_n_UH n =− M i=M−K+1 M−K j=1

λ

i

λ

j T

(

λi

−

λ

j

)

2

ujuHj − uiuHi

=− M i=M−K+1 M−K j=1

λ

2 T

(

λi

−

σ

2

)

2

UnUHn+

(

M− K

)

σ

2 T M i=M−K+1

λi

uiuHi

(

λi

−

σ

2

)

2 (14)

where

σ

2 _stands_for_the_WNP._Therefore,_when_evaluated_at_a_true

DoA

θ

k,the averagevalueofMUSIC localizationfunction isgiven

by E

_Lˆ_MUSIC

₍

_θk

₎

₌

(

M− K

)

σ

2 T M i=M−K+1

λ

i

|

aH

a H

₍

_θk

₎

_u i

|

2 =aH

(

θk

)

(

M− K

)

σ

2 T Us

−1 s UHs

a

(

θk

)

. (15)

Comparing (12) to (15) , one can interpret the correction

ˆ

PsG-MUSIC− ˆPMUSICasan O

(

T−1

)

estimate ofthe biasofMUSIC.In

other words, PˆsG-MUSIC can be viewed as a modiﬁcation of

MU-SIC by attempting to remove bias. It is very interesting to note thatthetheorywhichledtoG-MUSICiscompletelydifferentfrom thetheoryfromwhich(13) originates.Withthisrespect,thenew ˆ

PsG-MUSICenablestosortofestablishabridgebetweenthetwo

ap-proaches.Itcaneitherbeviewedasanapproximationand simpli-ﬁcationofG-MUSICand/oracorrectionofMUSIC.

3. Numericalsimulations

In this section, we compare the mean-square error (MSE) of

the DoA estimates obtained by MUSIC, G-MUSIC and sG-MUSIC.

We consider the same scenario as in [15] i.e., a uniform lin-eararray ofM₌20sensors,spacedahalf-wavelengthapart.Two

equi-powered sources are assumed to be present in the ﬁeld of

view ofthe array,withpowerP andDoA 35° and37°.The mea-surementsarecorrupted by whiteGaussiannoise withpower

σ

2

and we deﬁne the signal to noise ratio as SNR= P

σ2. We

con-sider asﬁgure ofmerit the meansquare errordeﬁned asMSE=

P p=1E

(

_θ

ˆ_p₋

_θ

_p

₎

2

_.₅₀₀₀_Monte-Carlo_simulations _are_run_to

es-timate MSE. In Figs. 1–3 we plot MSE as a function of SNR for variousvaluesofT,namelyT=15,T=25andT=75.Ascan be

(5)

Fig. 1. MSE of MUSIC, G-MUSIC and sG-MUSIC DoA estimates versus SNR. T = 15 .

thresholdarea (especiallyforsmallT) andmuch betterthan MU-SIC. As SNR increases the difference between the three methods vanishes. Therefore,sG-MUSIC offers a very goodcompromise: it issomewhatsimplerthanG-MUSICandundergoesamarginalloss inthethresholdarea,atleastwhena limitednumberofsamples isavailable.Ontheotherhand,sG-MUSIChasacomplexitysimilar tothatofMUSICbutprovidesmoreaccurateestimates.

4. Conclusions

Inthiscommunication, startingfromthe G-MUSIClocalization function,wehavepresentedan approximationthatcanbeviewed asabiascompensatedversion ofMUSIC.Indeed,thenewmethod correspondstoamodiﬁcationofMUSIClocalizationfunctionwhich somehowremovesthebiasinthelatter.Moreover,theweightsto beappliedtotheeigenvectorsofthesamplecovariancematrixare obtainedinclosed-form,similarlytoMUSIC,butdonot requireto ﬁndtheMrootsofa non-linearequation asin G-MUSIC. Numer-icalsimulations indicatethat thenewschemeperforms nearlyas wellasG-MUSIC,especiallyinlowsamplesupport,andbetterthan MUSIC.

References

[1] H.L. Van Trees , Optimum Array Processing, John Wiley, New York, 2002 .

[2] Y. Bresler , A. Macovski , Exact maximum likelihood parameter estimation of su- perimposed exponential signals in noise, IEEE Trans. Acoust. Speech Signal Pro- cess. 34 (5) (1986) 1081–1089 .

[3] J.F. Böhme , Estimation of spectral parameters of correlated signals in wave- ﬁelds, Signal Process. 11 (4) (1986) 329–337 .

[4] U. Sandkühler , J. Böhme , Accuracy of maximum-likelihood estimates for array processing, in: Proceesings ICASSP, 12, 1987, pp. 2015–2018 .

[5] A.G. Jaffer , Maximum likelihood direction ﬁnding of stochastic sources: a sep- arable solution, in: Proceedings ICASSP, New York, NY, 5, 1988, pp. 2893–2896 .

[6] R.O. Schmidt , Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag. 34 (3) (1986) 276–280 .

[7] R.O. Schmidt , R.E. Franks , Multiple source DF signal processing: an experimen- tal system, IEEE Trans. Antennas Propag. 34 (3) (1986) 281–290 .

[8] R. Roy , T. Kailath , ESPRIT- estimation of signal parameters via rotational in- variance techniques, IEEE Trans. Acoust. Speech Signal Process. 37 (7) (1989) 984–995 .

[9] P. Stoica , K. Sharman , Novel eigenanalysis method for direction estimation, Proc. IEE-Radar Sonar Navig. 137 (1) (1990) 19–26 .

[10] P. Stoica , A. Nehorai , MUSIC, maximum likelihood and Cramér–Rao bound, IEEE Trans. Acoust. Speech Signal Process. 37 (5) (1989) 720–741 .

[11] P. Stoica , A. Nehorai , Performance study of conditional and unconditional direction of arrival estimation, IEEE Trans. Acoust. Speech Signal Process. 38 (10) (1990a) 1793–1795 .

[12] P. Stoica , A. Nehorai , MUSIC, maximum likelihood and Cramér–Rao bound: fur- ther results and comparisons, IEEE Trans. Acoustics Speech Signal Process. 38 (12) (1990b) 2140–2150 .

[13] M. Kaveh , A. Barabell ,The statistical performance of the MUSIC and the min- imum-norm algorithms in resolving plane waves in noise, IEEE Trans. Acoust. Speech Signal Process. 34 (2) (1986) 331–341 .

[14] X.L. Xu , K.M. Buckley , Bias analysis of the MUSIC location estimator, IEEE Trans. Signal Process. 40 (10) (1992) 2559–2569 .

[15] X. Mestre , M.A. Lagunas , Modiﬁed subspace algorithms for DoA estimation with large arrays, IEEE Trans. Signal Process. 56 (2) (2008) 598–614 .

[16] B. Johnson , Y. Abramovich , X. Mestre , MUSIC, G-MUSIC and maximum likelihood performance breakdown, IEEE Trans. Signal Process. 56 (8) (2008) 3944–3958 .

[17] X. Mestre , Improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates, IEEE Trans. Inf. Theor. 54 (11) (2008) 5113–5129 .