Robust Estimates of Covariance Matrices in the Large Dimensional Regime and Application to Array Processing

0.23 MB·English

by  Romain Couillet

1
Robust Estimates of Covariance Matrices in the
Large Dimensional Regime
and Application to Array Processing
1 2 3
Romain Couillet , Fre´de´ric Pascal , and Jack W. Silverstein
1
3 Telecommunication department, Supe´lec, Gif sur Yvette, France.
1
2
0 SONDRA Laboratory, Supe´lec, Gif sur Yvette, France.
2
3
Department of Mathematics, North Carolina State University, NC, USA.
n
a
J

8

]
T
Abstract
I
.
s
c Thisarticle studies the limitingbehavior of a class of robust population covariance matrix estimators, originally
[
due to Maronna in 1976, in the regime where both the number of available samples and the population size grow

2
large.Usingtoolsfromrandommatrixtheory,weprovethatthedifferencebetweenthesamplecovariancematrixand
v
0 (ascaledversionof) such robust estimatortends tozeroinspectral norm, almost surely, thisbeingvalidirrespective
2
ofthedistributionofthepopulationvectorentries.Thisresultisappliedtoprovethatrecentsubspacemethodsarising
3
5 from random matrix theory can be made robust without altering their first order behavior.
.
4
0
2 I. INTRODUCTION
1
: Many multi-variate signal processing detection and estimation techniques are based on the empirical covariance
v
Xi matrix of a sequence of samples x1,...,xn from a random population vector x CN. Assuming E[x] = 0 and
∈
r E[xx∗] = C , the strong law of large numbers ensures that, for independent and identically distributed (i.i.d.)
a N
samples,
n
1
Sˆ = x x∗ C
N n i i → N
i=1
X
almost surely (a.s.), as the number n of samples increases. Many subspace methods, such as the multiple signal
classifier (MUSIC) algorithm and its derivatives [1], [2], heavily rely on this property by identifying C with Sˆ ,
N N
leadingto appropriateapproximationsof functionalsof C in the large n regime.However,this standardapproach
N
has two major limitations: the inherentinadequacyto small sample sizes (when n is not too large compared to N)
and the lack of robustnessto outliers or heavy-taileddistribution of x. Althoughthe formerissue was probablythe
Silverstein’s work is supported by the U.S. Army Research Office, Grant W911NF-09-1-0266. Couillet’s work is supported by the ERC
MOREEC–120133.
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firsthistoricallyrecognized,itisonlyrecentlythatsignificantadvanceshavebeenmadeusingrandommatrixtheory
[3]. As for the latter, it has spurred a strong wave of interest in the seventies, starting with the works from Huber
[4] on robust M-estimation. The objective of this article is to provide a first bridge between the two disciplines
by introducing new fundamental results on robust M-estimates in the random matrix regime where both N and n
grow large at the same rate.
Aside from its obvious simplicity of analysis, the sample covariance matrix (SCM) Sˆ is an object of primal
N
interest since it is the maximum likelihood estimator of C for x Gaussian. When x is not Gaussian, the SCM
N
as an approximationof C may howeverperform very poorly. This problem was identified in multiple areas such
N
as multivariate signal processing or financial asset management, but was particularly recognized in adaptive radar
and sonar processing where the signals understudy are characterized by impulsive noise and outlying data. Robust
estimationtheoryaimsattacklingthisproblem[5].Amongothersolutions,theso-calledrobustM-estimatorsofthe
populationcovariance matrix, originally introducedby Huber[4] and investigatedin the seminal work of Maronna
[6], have imposed themselves as an appealing alternative to the SCM. This estimator, which we denote Cˆ , is
N
defined implicitly as a solution of1
n
1 1
Cˆ = u x∗Cˆ−1x x x∗ (1)
N n N i N i i i
i=1 (cid:18) (cid:19)
X
for u a nonnegative function with specific properties. These estimators are particularly appropriate as they are the
maximum likelihood estimates of C for specific distributions of x and some specific choices of u, such as the
N
family of elliptical distributions [7]. For any such u, Cˆ is, up to a scalar, a consistent estimate for C for N
N N
fixed and n . The robust estimators are also used to cope with distributions of x with heavy tails or showing
→∞
a tendency to produce outliers, such as when x 2 has a K-distribution often met in the context of adaptive radar
k k
processing with impulsive clutter [8].
A second angle of improvement of subspace methods has recently emerged due to advances in random matrix
theory. The latter aims at studying the statistical properties of matrices in the regime where both N and n grow
large. It is known in particular that, if x = A y with y CM, M N, a vector of independent entries with
N
∈ ≥
zero mean and unit variance, then, under some conditions on C = A A∗ and y, in the large N,n (and M)
N N N
regime, the eigenvalue distribution of (almost every) Sˆ converges weakly to a limiting distribution described
N
implicitly byits Stieltjes transform[9]. When C is the identitymatrixforall N, this distributiontakesan explicit
N
form known as the Marc˘enko-Pastur law [10]. Under some additional moment conditions on the entries of y, it
has also been shown that the eigenvalues of Sˆ cannot lie infinitely often away from the support of the limiting
N
distribution[11].Inthe pasttenyears,thesetwo resultsandsubsequentworkshavebeenappliedtorevisitclassical
signalprocessing techniquessuch as signal detectionschemes[12] orsubspace methods[13], [14]. In these works,
traditionaln-consistentdetectionandestimationmethodswereimprovedinto(N,n)-consistentapproaches,i.e.they
1Ourexpressiondiffersfromthestandardconventionwhere x∗iCˆN−1xiistraditionally notscaledby1/N.Thecurrentformishowevermore
convenient foranalysis inthelarge N,nregime.
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provideestimates that are consistent in the large N,n regime ratherthan in the fixed N and large n regime. These
improved estimators are often referred to as G-estimators.
In this article, we study the asymptotic first order properties of the robust M-estimate Cˆ of C , given by (1),
N N
in the regime where N, n (and M) grow large simultaneously, hereafter referred to as the random matrix regime.
Although the study of the SCM Sˆ for vectors x with rather general distributions is accessible to random matrix
N
theory, as in e.g. the case of elliptical distributions [15], the equivalent analysis for Cˆ is often very challenging.
N
In the presentarticle, we restrictourselvesto vectors x ofthe type x=A y with y havingindependentzero-mean
N
entries.One importanttechnicalchallengebroughtby thematrix Cˆ ,usually notmetinrandommatrixtheory,lies
N
in the dependence structure between the vectors {u(N1x∗iCˆN−1xi)12xi}ni=1 (as opposed to the independent vectors
x n forthe matrix Sˆ ). We fundamentallyrely on the set of assumptionson the functionu taken by Maronna
{ i}i=1 N
in [6] to overcome this difficulty. Our main contribution consists in showing that, in the large N,n regime, and
under some mild assumptions, Cˆ αSˆ 0, a.s., for some constant α>0 dependent only on u. This result
N N
k − k→
is in particular in line with the conjecture made in [16] according to which Cˆ αSˆ a.s. 0 for the function
N N
k − k −→
u(s) = 1/s studied extensively by Tyler [17], [18]; however, the function u(s) = 1/s does not enter our present
scheme as it creates additional difficulties which leave the conjecture open.
A major practical consequence of our result is that the matrix Sˆ , at the core of many random matrix-based
N
estimators, can be straightforwardly replaced by Cˆ without altering the first order properties of these estimators.
N
We generically call the induced estimators robust G-estimators. As an application example, we provide in this
article a robust direction-of-arrival estimator, referred to as robust G-MUSIC, based on the G-MUSIC estimator
from Mestre [19].
The remainder of the article is structured as follows. Section II provides our theoretical results. Section III
introduces the robust G-MUSIC estimator. Section IV then concludes the article. All technical proofs are detailed
in the appendices.
Notations:Thearrow‘a.s.’denotesalmostsureconvergence.For A CN×N Hermitian,λ (A) ... λ (A)
1 N
−→ ∈ ≤ ≤
are its ordered eigenvalues. The norm is the spectral norm for matrices and the Euclidean norm for vectors.
k·k
For A,B Hermitian, A B means that A B is nonnegative definite. The notation A∗ denotes the Hermitian
(cid:23) −
transpose of A. We also write ı=√ 1.
−
II. MAIN RESULTS
Let X =[x ,...,x ] CN×n, where x = A y CN, with y =[y ,...,y ]T CM having independent
1 n i N i i i1 iM
∈ ∈ ∈
entries with zero mean and unitvariance, A CN×M, and C ,A A∗ CN×N be a positive definite matrix.
N ∈ N N N ∈
We denote c ,N/n, c¯ ,M/N, and define the sample covariance matrix Sˆ of the sequence x ,...,x by
N N N 1 n
n
1 1
Sˆ , XX∗ = x x∗.
N n n i i
i=1
X
Let u:R+ R+ (R+ =[0, )) be a function fulfilling the following conditions:
→ ∞
(i) u is nonnegative, nonincreasing, and continuous on R+;
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(ii) the function φ:R+ R+, s su(s) is nondecreasing and bounded, with sup φ(x)=φ >1. Moreover,
→ 7→ x ∞
φ is increasing in the interval where φ(s)<φ .
∞
Classical M-estimators Cˆ defined by (1) for such function u include the Huber estimator, with φ(s)= φ∞ s
N φ∞−1
fors [0,φ 1],φ >1,and φ(s)=φ for s φ 1.Since u(s) isconstantfor s φ 1anddecreases
∞ ∞ ∞ ∞ ∞
∈ − ≥ − ≤ −
fors φ 1,this estimatorweightsthe majorityof the samples x ,...,x by a commonfactorandreducesthe
∞ 1 n
≥ −
impact of the outliers. The widely used function u(s)=(1+t)(t+x)−1 for some t>0 shows similar properties,
here with φ =1+t. In either scenario, robustness can be controlled by properly setting φ .
∞ ∞
To pursue, we need the following statistical assumptions on the large dimensional random matrices under study.
A1. The random variables y , i n, j M, are independent either real or circularly symmetric complex
ij
≤ ≤
(i.e. E[y2] = 0) with E[y ] = 0 and E[y 2] = 1. Also, there exists η > 0 and α > 0, such that, for all i,j,
ij ij | ij|
E[y 8+η]<α.
ij
| |
A2. c¯ 1 and, as n ,
N
≥ →∞
0<liminfc limsupc <1, limsupc¯ < .
N N n
n ≤ n n ∞
A3. There exists C ,C >0 such that
− +
C <liminf λ (C ) limsup λ (C ) <C .
− 1 N N N +
N { }≤ N { }
Note that the assumptions neither request the entries of y to be identically distributed nor impose the existence
of a continuous density. This assumption is adequate for a large range of application scenarios such as factor
models in finance or general signal processing models with independent entry-wise non-Gaussian noise, although
the requirement of independence in the entries of y is somewhat uncommon in the classical applications of robust
estimationtheory.Theentry-wiseindependenceishowevercentralinthisarticlefortheemergenceofaconcentration
of the quadratic forms 1x∗Cˆ−1x , i = 1,...,n. Further generalizations, e.g. to elliptical distributions for x,
N i N i
would break this effect and would certainly entail a much different asymptotic behavior of Cˆ . These important
N
considerations are left to future work.
Technically,A1–A3 mainly ensure that the eigenvaluesof Sˆ and Cˆ lie within a compactset away from zero,
N N
a.s., forall N,n large,which is a consequence(althoughnon immediate) of [11], [14]. Note also that A2 demands
liminf c > 0, so that the following results do not contain the results from [6], [18], in which N is fixed and
N N
n , as special cases. With these assumptions, we are now in position to provide the main technical result of
→∞
this article.
Theorem 1: Assume A1–A3 and consider the following matrix-valued fixed-point equation in Z CN×N,
∈
n
1 1
Z = u x∗Z−1x x x∗. (2)
n N i i i i
i=1 (cid:18) (cid:19)
X
Then, we have the following results.
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(I) There exists a unique solution to (2) for all large N a.s. We denote Cˆ this solution, defined as
N
Cˆ = lim Z(t)
N
t→∞
where Z(0) =I and, for t N,
N
∈
n
1 1
Z(t+1) = u x∗(Z(t))−1x x x∗.
n N i i i i
i=1 (cid:18) (cid:19)
X
(II) Defining Cˆ =I when (2) does not have a unique solution, we also have
N N
φ−1(1)Cˆ Sˆ a.s. 0.
N N
− −→
(cid:13) (cid:13)
Proof: The proof is provided in Appendix(cid:13)A. (cid:13)
(cid:13) (cid:13)
An immediate corollary of Theorem 1 is the asymptotic closeness of the ordered eigenvalues of φ−1(1)Cˆ and
N
Sˆ .
N
Corollary 1: Under the assumptions of Theorem 1,
max φ−1(1)λ (Cˆ ) λ (Sˆ ) a.s. 0.
i N i N
i≤N − −→
(cid:12) (cid:12)
Proof: The proof is provided in Appe(cid:12)ndix A. (cid:12)
(cid:12) (cid:12)
Some comments are called for to understand Theorem 1 in the context of robust M-estimation.
Theorem 1–(I) can be first compared to the result from Maronna [6, Theorem 1] which states that a solution
to (2) exists for each set x ,...,x under certain conditions on the dimension of the space spanned by the n
1 n
{ }
vectors, as well as on u(s), N, and n (in particular u(s) must satisfy φ > n/(n N) in [6]). Our result may
∞
−
be considered more interesting in practice in the sense that the system sizes N and n no longer condition φ and
∞
therefore do not constrain the definition of u(s). Theorem 1–(I) can also be compared to the results on uniqueness
[6], [18] whichhold forall N,n undersome furtherconditionson u(s), such as φ(s) is strictly increasing [6]. The
latter assumption is particularly demanding as it may reject some M-estimators such as the Huber M-estimator for
whichφ(s) isconstantforlarges.Theorem1–(I)tradestheseassumptionsagainstarequirementforN andn tobe
“sufficiently large” and for x ,...,x to belong to a probability one sequence. Precisely, we demand that there
1 n
{ }
exists an integer n dependingon the randomsequence (x ,...,x ) ∞ , such that forall n n , existence and
0 { 1 n }n=1 ≥ 0
uniqueness are established under no further condition than the definition (i)–(ii) of u(s) and A1–A3.
Theorem 1–(II), which is our main result, states that, as N and n grow large with a non trivial limiting ratio,
the fixed-point solution Cˆ (either always defined under the assumptions of [6], [18] or defined a.s. for large
N
enough N) is getting asymptotically close to the sample covariance matrix, up to a scaling factor. This implies in
particular that, while Cˆ is an n-consistent estimator of (a scaled version of) C for n and N fixed, in the
N N
→∞
large N,n regime it has many of the same first order statistics as Sˆ . This suggests that many results holding for
N
Sˆ in the large N,n regime should also hold for Cˆ , at least concerning first order convergence. The estimators
N N
Cˆ , parametrizable through u, should then be seen as a class of alternatives for Sˆ . Note also that Theorem 1 is
N N
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independentof the choice of the distribution of the entries of y or of the choice of the function u, which is similar
to the equivalent result in the classical fixed-N large-n regime.
In terms of applications to signal processing, recall first that the n-consistency results on robust estimation [6],
[18] imply that many metrics based on functionals of C can be consistently estimated by replacing C by Cˆ .
N N N
Theinconsistencyofthesamplecovariancematrixtothepopulationcovarianceintherandommatrixregime,along
with Theorem 1, suggest instead that this approachwill lead in general to inconsistentestimators in the large N,n
regime, and therefore to inaccurate estimates for moderate values of N,n,M. However, any metric based on C ,
N
andforwhichan (N,n)-consistentestimatorinvolvingSˆ exists, is verylikely tobe (N,n)-consistentlyestimated
N
by replacing Sˆ by φ−1(1)Cˆ . The interest of this replacement obviously lies in the possibility to improve the
N N
metric through an appropriate choice of u, in particular when y exhibits outlier behavior or has heavy tails. In the
following section, we give a concrete example in the context of MUSIC-like estimation in array processing [19].
Remark 1: In a similar context, it is shown in [11] and [20] that the eigenvalues of Sˆ are asymptotically
N
contained in the support of their limiting compactly supported distribution if and only if the entries of y have
finite fourthordermoment.Thisfirstsuggeststhatthe technicalassumptionA1 whichrequires y to haveuniformly
bounded 8+η moment may be relaxed to y having only finite fourth order moments for Theorem 1 to hold.
ij
This being said, since most of the aforementioned(N,n)-consistent estimatorsinvolving Cˆ or Sˆ rely on a non-
N N
degeneratebehaviorof these eigenvalues(see e.g. [21, Chapters 16–17]fordetails), the finite fourth ordermoment
conditioncannotpossiblybefurtherrelaxedfortheseestimatorstobeusable.Asaconsequence,althoughA1might
seem very restrictive in a robust estimation framework as it discards the possibility to consider distributions of x
with heavy tail behavior, it is a close to necessary condition for robust estimation in the random matrix regime to
be meaningful.
III. APPLICATION:ROBUST G-MUSIC
Consider K signal sources impinging on a collection of N collocated sensors with angles of arrival θ ,...,θ .
1 K
The data x CN received at time t at the array is modeled as
t
∈
K
x = √p s(θ )z +σw
t k k k,t t
k=1
X
wheres(θ) CN isthedeterministicunitnormsteeringvectorforsignalsimpingingthesensorsatangleθ,z C
k,t
∈ ∈
is the signal source modeled as a zero mean, unit variance, and finite 8+η order moment random variable, i.i.d.
across t and independent across k, p >0 is the transmit power of source k (p <p for some p >0) and
k k max max
σw CN is the received noise at time t, independent across t, with i.i.d. zero mean, variance σ2 >0, and finite
t
∈
8+η order moment entries.
We can write
xt =ANyt, AN , S(Θ)P21 σIN
h i
where S(Θ)=[s(θ ),...,s(θ )], P =diag(p ,...,p ), and y =(z ,...,z ,wT)T CN+K.
1 K 1 K t 1,t K,t t ∈
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Taking n independent observations x ,...,x of the process x and assuming n,N, and M = N +K large
1 n t
accordingly to Assumption A2, Assumptions A1–A3 are met and Theorem 1 can be applied. This yields the
following result.
Theorem 2 (Robust G-MUSIC): Under the current model, denote E CN×(N−K) a matrix containing in
W
∈
columns the eigenvectors of C with eigenvalue σ2. Also denote eˆ the eigenvector of Cˆ with eigenvalue λˆ ,
N k N k
λ (Cˆ ) (recall that λˆ ... λˆ ), with Cˆ defined as in Theorem 1 (with Cˆ =I when (2) does not have a
k N 1 N N N N
≤ ≤
unique solution). Then, as N,n in the regime of Assumption A2, and K fixed,
→∞
a.s.
γ(θ) γˆ(θ) 0
− −→
where
γ(θ)=s(θ)∗E E∗ s(θ)
W W
N
γˆ(θ)= β s(θ)∗eˆeˆ∗s(θ)
i i i
i=1
X
and
1+ N λˆk µˆk , i N K
β = k=N−K+1 λˆi−λˆk − λˆi−µˆk ≤ −
i  PN−K λˆk (cid:16) µˆk (cid:17) , i>N K
 − k=1 λˆi−λˆk − λˆi−µˆk −
P (cid:16) T(cid:17)
with µˆ ... µˆ the eigenvalues of diag(λˆ) 1 λˆ λˆ , λˆ =(λˆ ,...,λˆ )T.
1 ≤ ≤ N − n 1 N
Proof: The proof can be found in Appendix E. p p
The function γ(θ) is the defining metric for the MUSIC algorithm [1], the zeros of which contain the θ ,
i
i 1,...,K . Theorem 2 proves that the N,n-consistent G-MUSIC estimator of γ(θ) proposed by Mestre in
∈ { }
[13] can be extendedinto a robustG-MUSICmethod.The latter consistsin replacingthe sample covariancematrix
Sˆ as in [13] by the robust estimator Cˆ . The angles θ are then estimated as the deepest minima of γˆ(θ). This
N N i
new technique is expected to perform better than either MUSIC or G-MUSIC in the finite (N,n) regime in the
caseofnon-Gaussiannoise,foranappropriatechoiceofthe functionu.Provingsorequiresthestudyofthesecond
order statistics of γ(θ), which is left to future work. Note also that our result does not prove the N,n-consistency
in the estimates of θ ,...,θ , which would demand to show
1 K
a.s.
sup γ(θ) γˆ(θ) 0.
| − |−→
θ∈[−π,π]
Proving this convergencerequires more advanced results; see [22, Section 4.3.2] for a discussion on this topic.
In the following,we providecomparative performanceresults between the classical MUSIC, the robust MUSIC,
the G-MUSIC, and the robust G-MUSIC algorithms. We recall that the MUSIC algorithm consists in determining
the deepest local minima of the function
N−K
γˆ∞(θ)= s(θ)∗eˆSeˆS∗s(θ)
i i
i=1
X
January9,2013 DRAFT
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101
100
10−1
E
S
M 10−2
10−3
MUSIC
RobustMUSIC
10−4 G-MUSIC
RobustG-MUSIC
10−5
−10 −5 0 5 10 15 20
SNR[dB]
Fig.1. MSEperformance ofthevarious MUSICestimators forK=1,Gaussiannoise, N =10,andn=50.
where eˆS is the eigenvector associated with the i-th smallest eigenvalue of Sˆ (the notation recalls the fact
i N ∞
that n for N fixed in this setting). Robust MUSIC is equivalent to MUSIC but uses eˆ instead of eˆS in the
→∞ i i
expression of γˆ∞(θ). G-MUSIC determines the local minima of γˆ(θ) but with eˆS instead of eˆ. Finally, robust
i i
G-MUSIC seeks the minima of γˆ(θ), as described in Theorem 2.
We take z standard Gaussian, independent across k and t, and w a vector with independent zero-mean unit
k,t t
variance entries w , such that w 2 has a K-distribution with shape parameter ν and scale parameter 1/ν, i.e.
it it
| |
w =w˜ √g
it it it
where w˜ CN(0,1) and g Γ(ν,ν−1) are independent random variables. Note that the entries of w have
it it t
∼ ∼
moments of all orders, with in particular a fourth order moment given by
E w 4 =2ν−1(1+ν).
it
| |
(cid:2) (cid:3)
The case where w has purely Gaussian entries, i.e. when ν , will be used as a referencescenario. When ν is
t
→∞
large,w modelsthe realistic scenarioof a sensorarraywith close-to-Gaussiannoise.Forν small, andin particular
t
forν <1(resultingintoanoisedistributionwithlargehighordermoments),thescenariocanbeeitherusedtoreflect
independentantennareadingerrorsin a sensorarrayorto modela distributedsensornetworkin which each sensor
faces independentimpulsive noise (e.g. in a MIMO-STAP setting [23], [24]). We choose u(s)=(1+ν′)/(ν′+s),
forsome ν′ >0 which controlsthe degreeof robustnessof the estimator(ν′ brings u(s)=1, hence reduced
→∞
robustness). We set here ν′ = 2.5 in all simulations. We model the steering vectors by [s(θ)] = exp(ıπksin(θ))
k
as in a uniform linear array of N elements with half wavelength inter-element spacing. We take N =10, n=50,
and p =1 for all k. Under these conditions, Cˆ satisfies [6, Assumption (E)] for ν′ 2.5, implying that Cˆ is
k N N
≥
well defined for each x ,...,x and not only for all large n a.s.
1 n
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101
100
10−1
E
S
M 10−2
10−3
MUSIC
RobustMUSIC
10−4 G-MUSIC
RobustG-MUSIC
10−5
−10 −5 0 5 10 15 20
SNR[dB]
Fig.2. MSEperformance ofthevarious MUSICestimators forK=1,noisewithν=0.1,N =10,andn=50.
101
100
10−1
E
S
M 10−2
10−3
MUSIC
RobustMUSIC
10−4 G-MUSIC
RobustG-MUSIC
10−5
−10 −5 0 5 10 15 20
SNR[dB]
Fig.3. MSEperformance ofthevarious MUSICestimators forK=1,noisewithν=0.02,N =10,andn=50.
We first consider K = 1 with θ = 18◦. Figure 1, Figure 2, and Figure 3 depict the mean-square error (MSE)
1
performance E[γ(θ ) γˆ(θ )2] of the above estimators, as a function of the signal-to-noise ratio (SNR) σ−2.
1 1
| − |
In Figure 1, we take w Gaussian, i.e. with ν for which E[w 4] = 2. In Figure 2, we set ν = 0.1,
t it
→ ∞ | |
implying E[w 4]=22, and call this case the intermediate noise scenario. Finally, in Figure 3, we take ν =0.02,
it
| |
and then E[w 4] = 102, and refer to this case as the impulsive noise scenario. We naturally expect the robust
it
| |
techniquesto bring largerperformancegains in the latter scenario than in the Gaussian and intermediate ones. The
simulationsarebasedon 50000MonteCarlosimulationsperSNR value.We firstobservethatbothrobustmethods
perform very similar to their non-robustcounterpartsin a Gaussian noise setting. In the intermediate noise setting,
January9,2013 DRAFT
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1
MUSIC
RobustMUSIC
0.8 G-MUSIC
RobustG-MUSIC
y
abilit 0.6
b
o
pr
n
o
uti 0.4
ol
s
e
R
0.2
0
−5 0 5 10 15 20
SNR[dB]
Fig.4. Resolution performance ofthevarious MUSICestimators, θ1=10◦,θ2=15◦,noisewithν=0.1,N =10,andn=50.
the robust approaches then overcome the non-robust ones, especially in the low-to-medium SNR region where we
see a significant performance advantage for robust G-MUSIC methods against G-MUSIC. In the impulsive noise
scenario, we then see both robust methods show a large gain compared to the non-robust ones. In this regime,
the random matrix advantage of G-MUSIC versus MUSIC disappears completely, while being largely favorable to
the robust scheme. The latter two results translate the fact that, if the noise non-Gaussianity and the small sample
size are not both appropriately controlled, one of the two will overtake the other, making G-MUSIC or robust
MUSIC inefficient. On the contrary, robust G-MUSIC, which controls both problems, always brings a significant
performance advantage.
In Figure4, we depictthe performanceof resolutionof two close sourcesof the MUSICestimators. Forthis, we
take K = 2, θ = 10◦, θ = 15◦, and ν = 0.1, i.e. E[w 4] = 22. The curves show the probability of detecting
1 2 it
| |
exactly two local minima of γˆ (or γˆ∞) within [5◦,20◦], based on 50000 Monte Carlo simulations for each SNR
value. In this close-to-Gaussian noise setting, the robust G-MUSIC algorithm shows a much stronger resolution
power than the G-MUSIC, although both operate at similar MSE.
The robust G-MUSIC example is an illustrative application of Theorem 1 demonstrating the strong advantage
broughtbya jointrobustandrandommatrix-basedsignalprocessingframework.Thetheoreticalperformancegains
are howevernot easy to obtain as they would require the elaborationof central limit theorems(CLT). In the robust
G-MUSIC example, this demands a CLT for the quantity n(θˆ θ), which requires more advancedtools than these
−
presented in this article.
January9,2013 DRAFT