The Performance of MUSIC in White Noise withLimited Samples and Missing Data

Raj Tejas Suryaprakash and Raj Rao NadakuditiElectrical Engineering and Computer Science

University of Michigan, Ann ArborAnn Arbor, Michigan 48109–2122

Email: rajtejas,rajnrao@umich.edu

Abstract—The Multiple Signal Classification (MUSIC) algo-rithm is widely used to estimate the direction of arrival (DOA)of signals impinging on a sensor array. In this work, we analyzethe performance of the MUSIC algorithm in the presence of whitenoise, and when only a random, sample independent subset of theentries in the data matrix are observed, in both the sample richand deficient regimes. We derive a simple, closed form expressionfor the mean-squared-error (MSE) performance of MUSIC for asingle source system, in the asymptotic regime and validate ouranalysis with simulations.

I. INTRODUCTION

Many signal processing applications involve estimating thedirection of arrival (DOA) of signals impinging on a sensorarray. Subspace based techniques are widely used in DOAestimation due to their high resolution capabilities, low com-plexity and robustness to modeling errors [2]. The MultipleSignal Classification (MUSIC) [3] algorithm is among themost widely used and studied subspace based DOA algo-rithms, due to its generality in application [1]. Given a datamatrix containing samples, or snapshots, of signals impingingon the array, MUSIC estimates a projection matrix onto thesignal subspace from the data, and returns at DOA estimate viaminimization of an objective function involving this projectionmatrix.The performance of MUSIC when we have a large numberof snapshots relative to the number of sensors is well un-derstood (e.g. [2], [4], [5]). However, less is known aboutthe performance of MUSIC in the sample deficient regime.Further, it may not be possible to collect all measurementsdue to resource constraints or sensor faults [6], such as inthe distributed array setting, where it might be prohibitiveto transmit all but a subset of the observed data values, dueto channel interference when all the sensors simultaneouslytransmit data. Alternatively, in the presence of temporal pro-cessing constraints, would it possible for MUSIC to achieveacceptable performance with a random subset of the datawithout complete performance breakdown? We consider onepossible mechanism of data loss or data compression in suchscenarios - that of sample loss, or ‘missing data’. In this setting

Work supported by ONR Young Investigator Award N00014-11-1-0660, USArmy Research Office (ARO) under grant W911NF-11-1-0391, NSF grantCCF-1116115, and AFOSR Young Investigator Award FA9550-12-1-0266.

we assume that we observe a random, sample independentfraction p of the entries in the data matrix. In this work,we present a unified analysis of the performance of MUSICalgorithm in the presence of noise and missing data, which ismathematically justified in both the sample rich and deficientregimes, for a system with a single source, for uniform lineararrays.We derive a simple, closed form expression for the mean-squared-error (MSE) performance of MUSIC for a singlesource system, in the setting where the number of arrayelements is large. This work extends the minimal stochasticequivalent model for MUSIC performance analysis in theasymptotic regime proposed by the authors in [8].The derived expression for MSE of MUSIC accurately quan-tifies the DOA estimation performance loss due to noise andmissing data, and facilitates system design via a trade-offbetween the number of sensors, snapshots and data com-pression to achieve required MSE performance. Our analysisbrings into sharp focus a phase transition phenomenon, thatgeneralizes the phase transition behavior of MUSIC previouslyanalyzed in the literature [9]. Using the analysis presentedhere, one can predict this phase transition boundary as afunction of the system parameters, viz. the number of sensors,snapshots, the eigen signal-to-noise ratio (SNR), fraction ofdata observed, and the ‘aspect ratio’: the ratio of number ofsensors to the number of snapshots. The algorithm suffersfrom a performance breakdown below this phase transitionboundary, when the proportion of observed data entries dropsbelow a critical value at a given SNR and aspect ratio, or whenthe eigen-SNR drops below a critical value at a fixed datacompression level and aspect ratio, and the results presentedhere may be used to design the system parameters to ensurethat we apply MUSIC in a regime where the DOA estimateis reliable.Our analysis draws upon recent results in random matrixtheory (RMT) [7], which precisely quantify the bias of thesignal subspace components estimated from the data, relativeto the true underlying subspace components. Since the MUSICalgorithm relies on these estimates to form the signal subspaceprojection matrix, the performance loss of MUSIC can becharacterized by drawing upon the RMT results to characterizetheir accuracy.

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The rest of the paper is organized as follows. Section IIdescribes the system model and the MUSIC algorithm. SectionIII describes pertinent results from random matrix theory. Thetheoretical MSE of DOA estimation performance is describedin Section IV. The expression is validated with numerical sim-ulations in Section V. Concluding remarks appear in SectionVI.

II. SYSTEM MODEL AND MUSIC ALGORITHM

Let k narrowband, uncorrelated signals from sources locatedat θ1, . . . , θk be incident on a uniform linear array having nsensors. Each of the m received snapshots of the signals ismodeled as [2], [3]

yi =1√nAsi + zi for i = 1, . . . ,m, (1)

where A =[a(θ1), . . . , a(θk)

]∈ Cn×k is the matrix

containing the steering vector a(θi) associated with the sourceat θi. si is the ith snapshot of the signal vector, whoseamplitude we assume is (complex) Gaussian distributed sothat si ∼ CN (0, diag(σ2

1 , . . . , σ2k)). We assume that the data

is corrupted by additive white Gaussian noise with zero meanand identity covariance, so that zi ∼ CN (0, In), where In isthe n× n identity matrix. We assume si and zi are mutuallyindependent for all i = 1, . . . ,m. In this work, since weanalyze a single source system, we assume that k = 1.The received snapshot vectors are stacked in a data matrix,Y =

[y1, . . . , ym

]. In the missing or compressed data

setting, we assume that we observe a sample independentfraction p the entries of the noisy data matrix Y , and themissing entries are set to 0. Therefore, the observed matrix,with both noisy and missing data, may be modeled as:

Yij =

Yij with probability p0 with probability 1− p .

(2)

Concisely, we may express the noisy data matrix Y , formedby stacking the noisy data snapshots yi, i = 1, . . . ,m givenby (1) as its columns, as

Y = S + Z, (3)

where S = σu(θ)sH , and σ, u(θ), s =[s1, . . . , sm

]and zi,

the columns of Z, are defined by (1). The observed data matrixmodel given by (2) is equivalent to

Y = Y Mask, (4)

where Mask is a matrix populated by Bernoulli randomvariables, which take value 1 with probability p and 0 withprobability 1 − p. Let U ΣV H be the singular value decom-position of Y , such that its singular values are ordered, withσ1 ≥ σ2 ≥ . . . ≥ σmin(m,n) > 0. Let U =

[u1 . . . un

].

When k is known, we form an estimate of the basis of thesignal subspace by selecting the left singular vectors associatedwith the k largest singular values, thus: Us =

[u1 . . . uk

].

The estimate of the projection matrix onto the signal subspaceis then PUs

= UsUHs .

When the system has a single source, Us = u1 := u, where wedrop the subscript for notational simplicity. Then PUs

= uuH .This is input to the MUSIC algorithm which returns the DOAestimate θ1 := θ by solving the optimization problem

θ = arg minφ−u(φ)H uuHu(φ) := arg min

φP(φ, u) , (5)

where u(φ) := a(φ1)√n

, u is the estimate of the signal subspacebasis, and P(φ, u) is the MUSIC objective function. We seethat the performance of MUSIC is determined by the accuracyof the subspace component estimate u. In the absence of noise,with completely observed data, u = u(θ), and hence θ = θ,corresponding to the objective value P(θ, u) = −1. Let ∆θ =θ − θ be the DOA estimation error. The goal of this work isto predict E

[∆θ2

], given n,m, p, σ and θ.

III. PERTINENT RESULTS FROM RANDOM MATRIXTHEORY

From (5) and the subsequent discussion, we see that theperformance loss of MUSIC is characterized by the accuracyof u, the estimate of u(θ). We quantify this accuracy interms of the inner product 〈u(θ), u〉, using recent results fromrandom matrix theory, as described below. We omit the detailsof the proofs due to space constraints. The results below extendthe results of [7] to the missing data setting.

Theorem 1: If u(θ) ∈ Cn×1 and v ∈ Cm×1 be the largestleft and right singular vectors of the matrix S in (3), andlet ||u(θ)||∞ < ηu

logCu (n)√n

, and ||v||∞ < ηvlogCv (m)√

m, for

some ηu, ηv, Cu, Cv > 0, where ||.||∞ denotes the maximumnorm. Let u be the estimate of u, found by computing theleft singular vector of Y as in (4), corresponding to its largestsingular value. Then, as n,m −→∞ and n

m −→ c ∈ (0,∞),

|〈u(θ), u〉|2 := α2 a.s.−→

1− c(1 + pσ2)

pσ2(pσ2 + c)if σ >

c14

√p,

0 otherwise.

Corollary 1: Assume that k = 1, and si, zi are as definedin Theorem 1. Then, for large enough n,

E[∣∣∣∣(In − uuH)u

∣∣∣∣2

]:= β2 = 1− α2 + higher order terms,

where α2 is given by Theorem 1, and In is the n×n identitymatrix.The key insight from Theorem 1 is that the sample sin-gular vector u is an informative estimate of the noise-free,underlying subspace component u(θ) only if the eigen-SNRσ is greater than a critical value, σcrit = c1/4

/√p. In this

informative regime, the estimate u lies on a cone around thetrue vector u(θ), and the angle between u(θ) and u is adeterministic quantity given by Theorem 1. The critical SNRσcrit is determined by σ, the fraction of observed entries p,and the ‘aspect ratio’, or the ratio of number of sensors tonumber of samples, c = n

m . When operating in the informativeregime, we expect the MUSIC algorithm to localize thesource accurately, and its MSE to decrease as the accuracyof the estimate u increases, i.e. as α2 −→ 1. Below this

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critical SNR, |〈u(θ), u〉|2| = o(1), so that the estimate uis essentially random. Therefore, below the phase transitionboundary, given by σcrit, u is noise-like and is uninformative,and thus we expect the performance of MUSIC to break down.Furthermore, when the eigen-SNR and the aspect ratio is heldfixed, there is a critical fraction of data, pcrit =

√c

σ2 that mustbe observed so that u is informative. Compression of the databeyond this critical value will also lead to a performancebreakdown.The conditions ||u(θ)||∞ < ηu

logCu (n)√n

, and ||v||∞ <

ηvlogCv (m)√

m, may be called ‘low coherence’ conditions, a term

from recent matrix completion and compressed sensing litera-ture [10]. They appear in, for example, [6]. The low coherenceconditions ensure that the entries of the matrix u(θ)u(θ)H

have about the same magnitude. If u(θ) is aligned with thecanonical basis, i.e. is sparse, random sampling will result ina loss of information. u(θ) whose energy (in the sense of l-2norm) is not concentrated in a few entries will behave betterunder random sampling. The uniform linear array satisfies thelow coherence condition. Other geometries do as well, but acomplete analysis is out of the scope of this work. Corollary1 provides insight about the overlap between singular vectorsestimated from the data and components orthogonal to thesignal subspace basis.

IV. THEORETICAL MSE DERIVATION

Armed with Theorem 1 and Corollary 1, we may nowobtain an asymptotic closed form MSE expression for theperformance of MUSIC for a single source system. We followthe approach in [2] to get a handle on the estimation error term∆θ. Since MUSIC minimizes the objective function P(φ, u)in (5), the derivative of the objective function evaluated at theDOA estimate θ must be 0. In other words,

∂P (φ, u)

∂φ

∣∣∣∣φ=θ

:= P (1)(θ, u) = 0,

where the superscript indicates derivative with respect to θ.Expanding the first derivative of the objective function aroundthe true DOA θ using Taylor Series yields,

P (1)(θ, u) = P (1)(θ, u)+(∆θ)P (2)(θ, u)+ h. o. t. = 0, (6)

where we neglect the higher order terms (h. o. t.), to obtainthe following expression for the estimation error ∆θ:

∆θ = −P(1)(θ, u)

P (2)(θ, u). (7)

Using the quadratic form of the objective function from (5)gives us the following expressions for the numerator anddenominator of ∆θ:

P (1)(θ, u) = 2<u(θ)H uuHu(1)(θ), and (8)

P (2)(θ, u) = 2<u(θ)H uuHu(2)(θ)+ 2|u(1)(θ)H u|2, (9)

where u(1)(θ) and u(2)(θ) are, respectively, the first andsecond derivatives of the manifold vector u(θ), evaluated atthe true DOA.

We notice that the inner product terms between the sampleand population singular vectors that appear in (8) and (9) arethe very objects which are characterized by the random matrixtheory results in Theorem 1 and Corollary 1.By applying those theorems to analyze the MSE, we obtain

E[∆θ2

]=

1− α2

2nγ2α2+ higher order terms,

where we may neglect the higher order terms to obtain

E[∆θ2

]≈ 1− α2

2nγ2α2(10)

which is the theoretical expression for the MSE of ∆θ, whereγ2 = ||u(1)(θ)||22 − |u(θ)Hu(1)(θ)|2 and α2 is given byTheorem 1.The utility of this simple expression lies in its applicationto design systems that operate above the phase transitionregime to ensure MUSIC performance does not break down.The expression tells us that when the estimate u is moreinformative, as α −→ 1, the MSE of the DOA estimate tendsto 0, as expected. As the accuracy of the estimate decreases,the MSE increases as 1−α2

α2 . When the compression (1 − p)is too high, or the eigen-SNR σ is too low, so that thesystem is operating below the phase transition region, then theexpression for theoretical MSE evaluates to infinity, indicatingcomplete performance breakdown.Thus, we may now design systems to ensure operation abovethe performance breakdown region. For instance, given arequired performance level, and number of sensors, we mayvary the number of snapshots collected, or extent of datacompression to achieve required performance using (10).Li et al. derived the following expression for the MSEperformance of MUSIC using perturbation theory [2,p. 1178, eq. (81)].

MSELLV =1

2nσ2γ2, (11)

where γ2 = ||a(1)(θ)||22−|a(1)(θ)u(θ)|2, and a(1)(θ) is the firstderivative of the manifold vector. In their notation, γk = γ2;(11) may be obtained by using the definition of γk from TableII on p. 1178 in [2], recognizing that Ω0 = I − u(θ)u(θ)H ,and appropriately substituting for for σn, the noise variance,M , the number of sensors, and Rs, the signal covariancematrix. We see that (11) captures the component of MSE dueto array geometry, but does not account for the accuracy ofthe singular vector estimates, which we gain from the randommatrix theory based approach of this work. We note here thatthe analysis in [2] was not carried out in the large array regime.However, even for moderately sized arrays, (10) providesimproved accuracy in predicting the MSE at low SNRs, asshown in Figure 1.

V. NUMERICAL SIMULATIONS

Figure 1 plots the MSE versus SNR for two different valuesof n and m, both sets chosen so that n/m = 1.5. There isno missing data, i.e. p = 1. This plot illustrates the validity

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0 1 2 3 4 5 6 7−50

−40

−30

−20

−10

0

10

20

30

SNR

MS

E (

dB

)MSE vs SNR, n = 150,60, m = 100,40, p = 1

Empirical

Theoretical, Closed Form

LLV

n = 150

n = 60

σcrit

Fig. 1. Empirical MSE of DOA estimate versus SNR = σ for n = 150 andn = 60. Here we set p = 1, and operate in a sample deficient regime, wherethe number of snapshots m = n/1.5 so that c = n/m = 1.5 in Theorem 1.The theoretical prediction for the MSE was plotted by using the closed formexpression in (10). The empirical values correspond to generating the fulldata matrix and running MUSIC on it; the regions where there is excellentagreement with the theoretical predictions are where the high-dimensionalrandom matrix theoretic predictions are accurate. The predictions are lessaccurate as one nears the breakdown point which corresponds to values nearSNR ≈ c1/4/√p. ‘LLV predictions’ refers to the MSE predicted by (11), asderived in [2].

of our theoretical expression and the improving accuracyof the underlying random matrix theoretic predictions withincreasing system size. As predicted, below critical SNR, theMUSIC algorithm breaks down completely. We also compareour predictions to the previous predictions in the literature, byLi et. al., which derived using perturbation theory in [2].Figure 2 plots the MSE versus the fraction of data observed(p) for fixed system size of 150 sensors and 100 snapshots(i.e., sample deficient regime), chosen so that n/m = 1.5.This plot illustrates the validity of our analysis of MUSICwith missing data. At low SNR, the prediction is inaccurateclose to the phase transition boundary due to the finite size ofthe simulated system, as opposed to the asymptotic analysis.Figure 3 plots the Empirical MSE the MUSIC DOA estimateversus σ and fraction of data observed p. This plot illustratesthe accuracy in predicting the phase transition boundary σcritas p is varied.

VI. CONCLUSIONS

In this paper, we presented a unified analysis of the MUSICalgorithm in the presence of white noise and missing data, fora single source system using uniform linear arrays, which wasmathematically justified in both the sample rich and deficientregimes. Our analysis revealed a phase transition phenomenonwhich generalizes the phase transition behavior of MUSICpreviously analyzed in the literature [9]. We characterizedthe MSE performance of MUSIC DOA estimation in closedform, and predicted the phase transition boundary of MUSIC

0 0.2 0.4 0.6 0.8 1−45

−40

−35

−30

−25

−20

−15

−10

−5

0

5

% Data Observed

MS

E (

dB

)

MSE vs p, n = 150, m = 100, SNR = 5,2.5

Empirical

Theoretical, Closed Form

σ = 2.5

σ = 5

pcrit, σ = 5

pcrit, σ = 2.5

Fig. 2. Empirical MSE of DOA estimate versus fraction of data observed= p, for two different SNRs σ = 2.5 and 5. System was moderately sizedat n = 150 and m = 100 so that c = n/m = 1.5 in Theorem 1. The twovertical lines correspond to theoretical breakdown points pcrit =

√σ

σ2 . We seethat σ = 5, prediction of MSE is accurate throughout, but at low SNR, σ =2.5, close to the phase transition region, prediction becomes inaccurate. Thisis due to the finite dimensions of the simulated system, while the theoreticalprediction was derived for the asymptotic regime.

0.2 0.4 0.6 0.8 1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

p

σ

MSE (dB), k = 1, c = 1.5, n = 600

−60

−50

−40

−30

−20

−10

0

10

20

σcrit

(p)

Fig. 3. Empirical MSE of DOA estimation for a system with 600 sensors and400 snapshots, plotted as SNR varies from 0.1− 5 and fraction of observedentries varies from 0.1 − 1. The white line is a plot of the phase transitionboundary σcrit as a function of p. The phase transition boundary prediction isaccurate for the entire range of SNR and p. Due to the finite dimensions of thesystem, the we see the phase transition boundary in this plot has some width,but asymptotically, it will reduce to a line which coincides with σcrit(p).

performance, facilitating system design to operate in a regimewhere the DOA estimates are informative and reliable.

REFERENCES

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Signal Processing, IEEE Transactions on, vol. 40, no. 7, pp. 1758,1774,Jul 1992.

[2] F. Li, H. Liu and R. J. Vaccaro, “Performance analysis for DOA estimationalgorithms: unification, simplification, and observations,” Aerospace andElectronic Systems, IEEE Transactions on, vol. 29, no. 4, pp. 1170,1184,Oct 1993.

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[4] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramer-Rao bound,” Acoustics, Speech and Signal Processing, IEEE Transactionson, vol. 37, no. 5, pp. 720,741, May 1989.

[5] P. Stoica and A. Nehorai, ”MUSIC, maximum likelihood, and Cramer-Rao bound: further results and comparisons,” Acoustics, Speech andSignal Processing, IEEE Transactions on, vol. 38, no. 12, pp. 2140,2150,Dec 1990.

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[7] F. Benaych-Georges and R. R. Nadakuditi, “The singular values andvectors of low rank perturbations of large rectangular random matrices,”J. Multivar. Anal., 111, 120-135, Oct 2012.

[8] R. T. Suryaprakash and R. R. Nadakuditi, ”The performance of music-based DOA in white noise with missing data,” Statistical Signal Process-ing Workshop (SSP), 2012 IEEE , pp.800,803, 5-8 Aug. 2012.

[9] B. A. Johnson, Y. I. Abramovich and X. Mestre, “Music, G-Music, andMaximum-likelihood performance breakdown,” IEEE Trans. Sig. Proc.,vol. 56, no. 8, pp. 3944–3958, 2008.

[10] E. J. Candes and B. Recht, “Exact matrix completion via convexoptimization,” Foundations of Computational Mathematics, vol. 9, no. 6,pp. 717–772, 2009.

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