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
Short
communication
A
bias-compensated
MUSIC
for
small
number
of
samples
François Vincent
a ,∗, Frédéric Pascal
b, Olivier Besson
aa 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: francois.vincent@isae-supaero.fr , Francois.Vincent@isae.fr (F. Vincent), frederic.pascal@centralesupelec.f (F. Pascal), olivier.besson@isae-supaero.fr
(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
ˆ
LMUSIC
(
θ
)
=aH(
θ
)
UˆnUˆHna(
θ
)
where a(
θ
) stands for the array steering vector and Uˆn=ˆ
u1 · · · ˆuM−K
where uˆm are the eigenvectors of thesam-ple covariance matrix with the convention that the correspond-ing eigenvalues ˆ
λ
m are sorted in ascending order. They provedthat, when evaluated at a true DoA
θ
k, LˆMUSIC(
θ
)
has a finitesample 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
)
UnUHna
(
θ
)
in the RMT sense. G-MUSIC estimatesthe 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
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−Kˆ 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
ˆkaretheeigenvaluesofthesamplecovariancematrixandˆ
μ
k,k=1,...,M denotetherootsoff
(
μ
)
= M m=1 ˆλ
m ˆλ
m−μ
=M c =T (3)sorted in ascending order. Note that, when c < 1, we have the interlacingproperty that ˆ
λ
m−1<μ
ˆm<λ
ˆm [17] .It followsthat, athigh signal to noise ratio where there is a clear separation be-tweensignal andnoise eigenvalues, thelast K values
μ
ˆm willbewellabovetheclusteroftheM− K smallest
μ
ˆm,whichshouldliearound 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 whiletheweightsofthenoiseeigenvectorsdependontheKlargest
μ
ˆmonly.Our approximation relies on finding the roots of (3) by con-sideringthetwoclustersofsolutionsindependently.Rewritingthe functionin(3) as f
(
μ
)
=Mm=1fm
(
μ
)
, where fm(
μ
)
= ˆλˆm λm−μ one canthusmakethefollowingpartitioningf
(
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,whichleadstoanapproximationof
μ
ˆmform≤ M− K andhenceofthesignaleigen-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 ˆλk
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 toλ
ˆM−K+1,· · · ,λ
ˆM. Hence,they belong tosome interval In where ˆ
λ
m/(
λ
ˆm−μ
)
≈ 1form>M− K whichresultsin fs(
μ
) ≈ Kwhenμ
∈In.Consequently,theM− K smallestvaluesof
μ
areobtainedbysolvingfn
(
μ
)
+K=T⇔1− 1 T− K M−K m=1 ˆλ
m ˆλ
m−μ
=0 ⇔1−T 1 − Kˆ
λ
n Tˆ
n−
μ
I −1ˆ
λ
n=0 ⇔det ˆn− 1 T− K
ˆ
λ
nˆ
λ
n T −μ
I =0 (5) whereλ
ˆn= ˆλ
1 · · ·λ
ˆM−K T,
ˆn=diag
(
λ
ˆn)
andwherethe lastequivalenceisobtainedbymultiplyingbydetˆn−
μ
I.It fol-lowsthat
μ
ˆmform=1,· · · ,M− K areapproximatelytheeigenval-uesof
ˆn−
(
T− K)
−1 ˆλ
n ˆλ
n T .Let us accordingly consider an approximation of the weights wm,m>M− K.Letusintroducethe notation
k=
λ
ˆk− ˆμ
k.Notethat, at high SNR, we have ˆ
λ
kλ
ˆ−1m 1 for k=1,· · · ,M− K and,since
μ
ˆk<λ
ˆk,it followsthat
μ
ˆkλ
ˆ−1m 1. Itthen ensuesthat,form>M− K wm=− M−K k=1
ˆλk
ˆλm
− ˆλk
− ˆμk
ˆλm
− ˆμk
=− M−K k=1 ˆλ
mk ˆ
λ
2 m(
1− ˆλk
ˆλ
−1m)(
1− ˆμk
λ
ˆ−1m)
≈ −ˆλ
−1 m M−K k=1k=−ˆ
λ
−1m M−K k=1 ˆλk
− M−K k=1 ˆμk
≈ −ˆλ
−1 m M−K k=1 ˆλk
− Tr{
ˆn−
(
T− K)
−1ˆ
λ
nˆ
λ
n T}
=−ˆλ
−1 m(
T− K)
−1 M −K k=1 ˆλk
. (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
ˆλk
ˆλm
− ˆλk
− ˆμk
ˆλm
− ˆμk
=1+ M k=M−K+1 ˆ
λm
(
λk
ˆ − ˆμk
)
(
λ
ˆm− ˆλk
)(
λ
ˆm− ˆμk
)
=1+ M k=M−K+1(
λm
ˆλ
ˆ−1 k)(
λk
ˆμ
ˆ−1k − 1)
(
1− ˆλ
mλ
ˆ−1k)(
1− ˆλ
mμ
ˆ−1k)
. (8)UnderthehighSNR assumption,wehavethat
λ
ˆmˆλ
−1k 1form≤M− K and k>M− K.Letus nowshow that
λ
ˆmμ
ˆ−1k 1for m≤M− K andk>M− K.As
μ
ˆk∈ˆ
λ
k−1,ˆλ
k,itfollowsdirectly from the highSNR assumptionthat, for k>M− K+1 andm≤ M− K, ˆ
λ
mμ
ˆ−1k <λ
ˆmˆλ
−1k−1 1. It remains to examine the special caseofkmin=M− K+1, that is of the smallest signal eigenvalue since,
in this case,
μ
ˆkmin lies between the largest noise eigenvalue andthesmallestsignaleigenvalue.Wenowprovethat
μ
ˆkminisclosetoˆ
λ
kmin.Fork≥ kminˆ
λk
ˆλk
− ˆμk
min ≤λk
ˆ min ˆλk
min− ˆμk
min (9)whichimpliesthat
M k=M−K+1 ˆ
λk
ˆλk
− ˆμk
min ≤ Kλk
ˆ min ˆλk
min− ˆμk
min ⇔ M k=1 ˆλk
ˆλk
− ˆμk
min ≤ Kλk
ˆ min ˆλk
min− ˆμk
min + M−K k=1 ˆλk
ˆλk
− ˆμk
min ⇔T≤ Kλk
ˆ min ˆλk
min− ˆμk
min + M−K k=1 ˆλk
ˆλk
− ˆμk
min ⇔Kλk
ˆ min ˆλk
min− ˆμk
min ≥ T− M−K k=1 ˆλk
ˆλk
− ˆμk
min . (10)Theright-handsideoflastequationbeingstrictlygreaterthanT,it followsthat ˆ
μk
min>λk
ˆ min1−K T . (11)
Theprevious equation showsthat
μ
ˆkmin isratherclosetotheup-per bound of the interval
ˆλ
kmin−1,ˆλ
kminto which it belongs. Similar derivations as for (10) can show that, for any k>M− K,
μ
ˆk>λ
ˆk1−M−k+1T
, and hence as the eigenvalues increase, ˆ
μ
k comes closertoλ
ˆk. Furthermore,(11) impliesthatλ
ˆmμ
ˆ−1k min< ˆλ
mˆλ
−1k min 1−K T −11.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 Uˆsˆ −1 s UˆHs =PˆMUSIC−
M−K k=1
λk
ˆ T− K Uˆsˆ −1 s UˆHs (12)
where Uˆs=[uˆM−K+1· · · ˆuM] and
ˆs=diag
(
λ
ˆs)
withλ
ˆs= ˆλ
M−K+1 · · · ˆλ
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
−1m(
T− K)
−1M−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=iλi
λ
j T(
λi
−λ
j)
2 ujuHj − uiuHi . (13)Wecanrewritethepreviousequationas
E
PˆMUSIC− UnUH n =− M i=M−K+1 M−K j=1λ
iλ
j T(
λi
−λ
j)
2 ujuHj − uiuHi =− M i=M−K+1 M−K j=1λ
iσ
2 T(
λ
i−σ
2)
2 ujuHj− uiuHi = − M i=M−K+1λi
σ
2 T(
λi
−σ
2)
2 UnUHn+(
M− K)
σ
2 T M i=M−K+1λi
uiuHi(
λi
−σ
2)
2 (14)where
σ
2 standsfortheWNP.Therefore,whenevaluatedatatrueDoA
θ
k,the averagevalueofMUSIC localizationfunction isgivenby E
LˆMUSIC(
θk
)
=(
M− K)
σ
2 T M i=M−K+1λ
i|
aH(
θk
)
ui|
2(
λ
i−σ
2)
2 ≈(
M− K)
σ
2 T M i=M−K+1λ
−1 i|
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.Inother 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
2and we define the signal to noise ratio as SNR= P
σ2. We
con-sider asfigure ofmerit the meansquare errordefined asMSE=
P p=1E
(
θ
ˆp−θ
p)
2.5000Monte-Carlosimulations areruntoes-timate MSE. In Figs. 1–3 we plot MSE as a function of SNR for variousvaluesofT,namelyT=15,T=25andT=75.Ascan be
Fig. 1. MSE of MUSIC, G-MUSIC and sG-MUSIC DoA estimates versus SNR. T = 15 .
Fig. 2. MSE of MUSIC, G-MUSIC and sG-MUSIC DoA estimates versus SNR. T = 25 .
Fig. 3. MSE of MUSIC, G-MUSIC and sG-MUSIC DoA estimates versus SNR. T = 75 .
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.
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