MUSIC, G-MUSIC, and Maximum-Likelihood Performance Breakdown

MUSIC, G-MUSIC, and Maximum-Likelihood Performance Breakdown B.A. Johnson, Y.I. Abramovich, X. Mestre

Publication: IEEE Transactions on Signal Processing Vol.: 56 No.: 2 Date: Aug. 2008

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 8, AUGUST 2008

MUSIC, G-MUSIC, and Maximum-Likelihood Performance Breakdown Ben A. Johnson, Student Member, IEEE, Yuri I. Abramovich, Senior Member, IEEE, and Xavier Mestre, Member, IEEE Abstract—Direction-of-arrival estimation performance of MUSIC and maximum-likelihood estimation in the so-called “threshold” area is analyzed by means of general statistical analysis (GSA) (also known as random matrix theory). Both analytic predictions and direct Monte Carlo simulations demonstrate that the well-known MUSIC-specific “performance breakdown” is associated with the loss of resolution capability in the MUSIC pseudo-spectrum, while the sample signal subspace is still reliably separated from the actual noise subspace. Significant distinctions between (MUSIC/G-MUSIC)-specific and MLE-intrinsic causes of “performance breakdown,” as well as the role of “subspace swap” phenomena, are specified analytically and supported by simulation. Index Terms—Array signal processing, generalized likelihoodratio tests, signal detection and estimation, G-estimation.

I. INTRODUCTION

I

T has been known for a long time that when the sample support and/or signal-to-noise ratio (SNR) on an -variate antenna array is insufficient, MUSIC performance “breaks down” and rapidly departs from the CRB [1], [2]. In most studies, the phenomenon blamed for such performance breakdown in subspace-based methods is the so-called subspace swap when the “measured data is better approximated by some components of the orthogonal (“noise”) subspace than by the components of the signal subspace” [3]. Analytical studies of this phenomenon usually rely upon the traditional asymptotic assumptions and associated perturbation analysis of a sample covariance matrix eigendecomposition (see [4] for example). While maximum-likelihood estimation (MLE) does not require signal eigenspace to be split into “signal” and “noise” subspaces, it has been known for a long time that under certain “threshold” conditions, MLE may also experience “performance breakdown” and generate severely erroneous estimates (“outliers”) not consistent with the CRB predictions (see [5, pp. 278–286]).

Manuscript received November 16, 2006; revised January 8, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Sven Nordebo. This work was funded under DSTO/RLM R&D Collaborative Agreement 290905. B. A. Johnson is with RLM, Pty., Ltd., Edinburgh, SA, 5111, Australia, and also with the Institute for Telecommunications Research, University of South Australia, Mawson Lakes, SA, 5095 (e-mail: [email protected]). Y. I. Abramovich is with the Defence Science and Technology Organisation (DSTO), ISR Division, Edinburgh SA 5111, Australia (e-mail: [email protected]). X. Mestre is with the Centre Tecnològic de Telecomunicacions de Catalunya (CTTC), Castelldefels, 08860 Barcelona, Spain (e-mail: xavier.mestre@cttc. cat). Digital Object Identifier 10.1109/TSP.2008.921729

Historically, analytical studies of MLE “breakdown” have been performed for a single signal in noise [6]–[9], or occasionally for multiple sources [10]–[12]), but almost always relying perturbation on traditional asymptotic analysis (with some notable exceptions, such as [13]). Since for a single source, MLE may be implemented via one-dimensional search (similar to MUSIC) over the traditional matched filter (beamformer) output, comparison with the MUSIC threshold condition is straightforward. For multiple sources, analysis of MLE threshold conditions is not as simple, primarily because the globally optimal ML solution often cannot be easily identified. Yet, recent investigations for multiple source scenarios, conducted primarily by Monte Carlo simulations [14], demonstrated a “gap” in the minimum sample support and/or SNR between the MUSIC-specific and ML-intrinsic threshold conditions. In fact, it was demonstrated that for the considered multisource scenarios, MLE breakdown occurs at a significantly lower SNR than for MUSIC. It is therefore clear that for multiple-source scenarios, different mechanisms are responsible for MLE and MUSIC “breakdowns” which have not been thoroughly investigated. In [15] and [16], an improvement in MUSIC “threshold performance” has been derived by X. Mestre, based on recent findings of the general statistical analysis (GSA) approach (also known as random matrix theory) that considers different asymptotic conditions constant

(1)

i.e., where both the array dimension and the number of snapshots grow without bound, but at the same rate. While it was long known that for a finite , sample eigenvectors in the covariance matrix eigendecomposition (2) are biased estimates of the true eigenvectors , GSA methodconsistent) ology allowed Mestre to specify the ( such that (under certain condiG-MUSIC function tions)

(3) where is the MUSIC pseudospectrum of the true covariance matrix. Specifically

1053-587X/$25.00 © 2008 IEEE

(4)

JOHNSON et al.: MUSIC, G-MUSIC, AND ML PERFORMANCE BREAKDOWN

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where is the -variate (unity norm) array steering vector are the eigenvectors of the sample in the direction and , associated with eigenvalues covariance matrix . Note that for , the last eigenof point sources, values are zero. For the known number , is [16] the G-weighting function

(5)

with tions of

denoting the

real valued solu-

(6) The weighting in (4) allows the parenthesized term to be treated as the consistent (under G-asymptotics) estimate of of the actual covariance matrix the noise subspace (where the subspace is an eigenvector matrix ). of the individual eigenvectors While Mestre demonstrated some improvement in threshold conditions with respect to conventional MUSIC, he noted that “it was rather disappointing to observe that the use of -consistent estimates does not cure the breakdown effect of subspace-based techniques (in MUSIC) and it merely moves to a lower SNR” [17], indicating that G-MUSIC is not able to avoid the fundamental phenomena that separates MUSIC breakdown conditions from MLE ones. This was surprising given that the G-MUSIC derivations (3)–(6) and actual Monte Carlo simulations were conducted under conditions which “guarantee separation of the noise and first signal eigenvalue cluster of the asymptotic eigenvalue distribution of ” [15]. Therefore, G-asymptotically the “subspace swap” phenomenon is precluded by these conditions, and yet MUSIC and G-MUSIC breakdown was regularly observed in the conducted Monte Carlo trials under these conditions. Clearly, the connections between “subspace swap” in MUSIC, G-MUSIC, and MLE “performance breakdown,” as well as the relevance of the GSA methodology for practically and values, needs to be clarified. In this paper, we limited introduce results of our attempts to do so. To this purpose, the paper is organized in a slightly unusual way, starting with simulation results and then examining some underlying theoretical considerations. We start in Section II with the results of Monte Carlo trials for a typical multisource scenario that on one hand illustrates significantly better MLE performance in the “threshold” area compared with MUSIC and G-MUSIC, but more importantly serves as a test-case for GSA prediction accuracy assessment. In Section III, we derive G-asymptotic “subspace swap” conditions and compare them with results of direct Monte Carlo trials. We demonstrate that for the considered scenarios, MUSIC-specific breakdown is associated with intersubspace “leakage” (rather than full subspace swap) whereby a small portion of the true signal eigenvector resides in the sample

noise subspace (and visa-versa). In Section IV, we show that this leakage is sufficient for loss of source resolution and associated MUSIC breakdown, and demonstrate a G-asymptotic prediction of the loss of resolution. In Section V, we demonstrate that unlike MUSIC, the MLEintrinsic breakdown is directly associated with severe “subspace swap” in the sample covariance matrix, and therefore can be quite accurately predicted (in terms of SNR and sample support) for a given scenario. This is shown with both multisource and more traditional single source scenarios. In Section VI, we summarize and conclude the paper. II. “PERFORMANCE BREAKDOWN” IN DOA ESTIMATION: SIMULATION RESULTS In this section, we illustrate MUSIC, G-MUSIC and MLE performance in the threshold region (of parameters) that spans the range from “proper” MUSIC behavior (no outliers) to MLE complete “performance breakdown.” For this reason, we once again consider the scenario used in [15], [18], and [19], with a -element uniform linear array (ULA), training and samples, array element spacing of independent equal power Gaussian sources (stochastic source model) located at azimuth angles 20

10 35 37

(7)

immersed in white noise, with various per-element source SNRs (ranging from 15 to 25 dB, or set to specific SNRs for more detailed investigation). for this mixture is The covariance matrix (8)

where the noise power is ; source SNR is given by , and is the DOA -dependent -variate “steering” (antenna manifold) vector. in our Monte Carlo simulaThe number of sources tions are assumed to be known a priori. MUSIC and G-MUSIC algorithms are implemented as usual by selecting the largest maxima of the MUSIC and G-MUSIC pseudo-spectra:

(9)

(10)

as specified in (5)–(6). respectively, with In the Gaussian case, MLE is theoretically obtained by the selection of the single largest maxima of the multivariate likelihood function (LF) [20]

(11)

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where represents the parameters power and angle of arfor the sources. However, since the actual global rival extremum of the LF cannot be guaranteed in practice, MLE performance is assessed using an MLE-proxy algorithm [14]. The essence of this algorithm is to first find a local extremum of the likelihood function in the vicinity of the actual parameters for every Monte Carlo trial. We then make an initial “seed” estimate of the actual parameters using MUSIC or G-MUSIC to derive the DOAs and power estimation such as in for a given trial is treated as [21]. This set of DOA estimates representative of MLE performance if (12) In the event that the likelihood threshold is not exceeded, we use an iterative process to replace the MUSIC or G-MUSIC DOA estimates one by one with univariate LF searches. As opposed to approaches that are looking only for a local extremum in vicinity of the a priori known source location, this approach may initialize the ML search with a solution that is “far away” from the true one (when a “improper” MUSIC or G-MUSIC solution is used as the initial point). It is thus able to uncover “far away” maxima with a sufficient LF value to pass the threshold in (12) (i.e., MLE breakdown). If, however, the ML search results fail to meet the threshold condition (12) via whatever ML optimization approach we have chosen, then we treat this case as a failure of the optimization search routine to find a global maxima rather than an MLE failure and discard this result. Finally, the solutions are then evaluated for DOA estimation error. This ensures that, to the extent possible, even in the presence of MLE breakdown we are evaluating the underlying MLE performance. While use of the threshold in (12) is not a practical approach, it adopts the same clairvoyant knowledge of the true solution as does the Cramér–Rao lower bound for MLE performance assessment. Also, while outside the scope of this paper, it should be noted that there are practical versions of this MLE-proxy algorithm [22], [23] which rely on statistical invariance of a modified likelihood ratio and have a quite reasonable threshold performance compared with the “clairvoyant” MLE-proxy algorithm used here. Fig. 1(a) shows the mean-square error (MSE), averaged over 300 trials, for DOA estimates of the two closely spaced sources (at 35 and 37 ). The figure demonstrates the familiar “threshold effect” in MSE for the DOA estimation process, with the sudden degradation in DOA accuracy (due to outliers) as the SNR is decreased. The MLE breakdown is demonstrated with the MLE-proxy algorithm discussed above, using two different “seeding” solutions produced by MUSIC and G-MUSIC correspondingly. Also shown is the stochastic Cramér-Rao bound (CRB) for the two sources at 35 and 37 (averaged together). One can observe the improvement in threshold performance delivered by G-MUSIC compared with MUSIC, as demonstrated in [15]. The improvement is more dramatic when examining the percentage of solutions that contain an outlier [Fig. 1(b)], as MUSIC deteriorates much more rapidly than G-MUSIC with decreasing SNR, but both algorithms are still outperformed by the MLE-proxy in this scenario. Based on these introduced results, one has to conclude that for the considered multisource scenario, completely different

Fig. 1. Multiple-source estimation on a 20-element uniform linear array training samples for MUSIC, G-MUSIC, and MLE. The SNR with T breakpoint (the “threshold”) decreases from around 20 dB for MUSIC to 17 dB for G-MUSIC, but is still dramatically greater than the MLE-proxy (LF-PAC) threshold observed at around 0 dB. Note that the invariance of the MLE-proxy results with respect to the “seeding” solution (MUSIC or G-MUSIC) indicates the reliable association of the results with true MLE performance in the threshold area. (a) Mean-square error. (b) Outlier production rate.

= 15

mechanisms drive (MUSIC/G-MUSIC)-specific and MLE-intrinsic breakdown. This difference is the primary topic of our paper. III. “SUBSPACE SWAP” AND MUSIC “PERFORMANCE BREAKDOWN” The subspace swap phenomenon has often been treated as the sole apparent mechanism “responsible” for performance breakdown in subspace-based techniques. This phenomenon is specified [3] as a case when the estimates of the noise subspace eigenwith increasing probvalues ability become larger than the estimates of the signal subspace . “More precisely, in such eigenvalues actua case one or more pairs in the set ally estimate noise (subspace) eigenelements instead of signal elements,” Hawkes, Nehorai and Stoica note in [24]. The fact that subspace swap is associated with MUSIC breakdown has


been well demonstrated in the literature, with several (only partially) successful attempts undertaken to analytically specify the threshold conditions for a given scenario. In [24], for example, it was admitted that analytical predictions “grossly underestimate probability of subspace swap in and below the threshold region.” This lack of complete success may be attributed to the constant asymptotic perturbation traditional eigendecomposition analysis adopted for these derivations. Acconditions tual breakdown is of course observed in finite is quite far from classical and the scenario in (7) with asymptotic assumptions. Alternatives to the traditional asymptotic assumptions for the ratio may be considered type of scenarios with a limited served by the G-asymptotic assumption (1). Yet, it could well be and values, analytic derivations that for practical (finite) based on this asymptotic assumption also lead to erroneous results. In order to investigate this matter further, let us briefly introduce some asymptotic convergence results for the eigenvalues and eigenvectors of the sample covariance matrix. The first nonobvious property of the eigenvalues of the sample covariance matrix is the fact that their empirical distribution tends almost surely to a deterministic probability density G-asymptotically. It turns out that this asymptotic density of sample eigenvalues becomes organized in clusters located around the positions of the true eigenvalues. For the covariance matrix model considered here (8), one can easily identify a cluster associated with the single noise eigenvalue, and a cluster (or a set of clusters) associated with the signal subspace. The number and position of these clusters depends strongly on the asymptotic number of samples per antenna element that are available in order to construct the sample covariance matrix. If the number of samples per antenna element is too low, the asymptotic sample eigenvalue distribution remains in a single cluster. As the number of samples per antenna is increased, the asymptotic sample eigenvalue distribution breaks off into distinct clusters, each one potentially associated with a single (possibly repeated) true eigenvalue. It has been shown [25] that of the distinct for the th eigenvalue ) to be true eigenvalues (which occur with multiplicity to be well separated from estimated (i.e., for the cluster of clusters), that the neighboring

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is sepawith the noise subspace eigenvalue rated from the rest of the eigenvalue distribution (the “subspace splitting condition”) is given by (16) denotes the minimum real-valued solution to the where (15), considering multiplicity. is greater If the number of samples per antenna element than the right-hand side of (16), one can ensure that signal and noise sample eigenvalues will be separated in the asymptotic sample eigenvalue distribution, and a subspace swap will occur with probability zero. It turns out that whenever there is asymptotic separation between signal and noise subspaces, one can effectively describe the behavior of the sample eigenvalues and eigenvectors using the following result from [15]: Let be independent and identically distributed (i.i.d.) -variate districomplex-valued column vectors from the bution with circularly symmetric complex random variables , that has the having zero mean and covariance matrix following eigendecomposition (17) (18) where are the true signal eigenvalues is the corresponding eigenvector matrix. Let the sample and be specified as matrix (19) Consider the th signal sample eigenvector (assumed to be associated with a sample eigenvalue with multiplicity one) and a deterministic column vector . One can try to analyze the behavior of the sample eigenvector by studying the behavior of , and relate it somehow to the deterthe scalar product . It turns out that, as at the ministic quantity same rate under a satisfied “eigenvalue splitting condition” (14) for all eigenvalues, we get

(20) where

(13) (14)

for condition”). The factor real-valued solutions of

almost surely, where the weights orem 2]

are defined as [26, The-

(the “eigenvalue splitting denotes the (21) and where

are the real-valued solutions to

(15) (22) ordered as . Specifically, the ratio of the number of training samples to the dimension of the array necessary to guarantee that the eigenvalue cluster associated

of the corresponding repeated according to the multiplicity . This result is powerful, but allows for little interpretation. In

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order to simplify the analysis it is common practice to consider the particular case of the so-called “spiked population covariance matrix model.” This class of covariance matrix was introduced by Johnstone [27], and it describes the asymptotic behavior of a class of covariance matrices obtained from plane waves in noise (8). This model is essentially a particularization of the general one, based on the simplifying assumption that the contribution of the signal subspace is negligible in the asymptotic regime, in the sense that only the dimension of the noise subspace scales up with the number of antenna elements, whereas the dimension of the signal subspace remains fixed. Under this simplification of the original model (which implies for fixed in the above formulas), we see that letting the asymptotic subspace splitting condition in (16) becomes

(29) or, equivalently, as (30)

we see that (using

) (31)

(23) which can also be expressed as (24) . where Let us now investigate the behavior of the solutions to (22) under the spiked population covariance model. Note first that (22) can equivalently be written as

With all this, we are now able to investigate the behavior of the weights in (21) under the spiked population model simplification. Indeed, let us first concentrate on the case (this corresponds to the convergence of a particular signal as in (32), sample eigenvector). Expressing the weights shown at the bottom of the page, and using the above limits on , we obtain the

(33) (25) Now, let us first consider can never go to zero for any term of (25) will go to zero as will converge to the solution of

. By definition, we have . Hence, . Consequently, the first for a fixed , and

(26) namely, (27) . We observe Let us now consider the convergence of , so that, by examining the first term in that (25), the only possibility is that (28) (otherwise, the first term of (25) would go to zero, and we would end up with the solution to (26), which is not in the interval of interest). Furthermore, by expressing (25) in the following way:

Hence, one can ensure that, under the spiked population covari, one has ance matrix, and assuming (34) where is a deterministic column vector with uniformly bounded norm regardless of the number of antenna elements. This is precisely the result introduced by Paul [28] for the specific class of spiked population covariance matrices, where again a fixed limited number of eigenvalues is greater than the smallest one, whose multiplicity grows with the number of antennas. If we replace with an eigenvector of the true covariance matrix , we observe that (under the spiked population covariance matrix model, and assuming asymptotic subspace separation) the projection of a sample eigenvector onto the linear space spanned by an eigenvector associated with a different eigenvalue converges to zero, i.e., (35)

(32)


In addition, Paul also studied the convergence of the sample eigenvector when there is no asymptotic separation between . In signal and noise subspaces, namely particular, he established that, under the spiked population covariance matrix model (36) almost surely as at the same rate (in fact, Paul proved this for real-valued Gaussian observations with a diagonal covariance matrix, but we conjecture that the result is also valid for the observation model considered here). In [28], Paul admitted that a “crucial aspect of the work [29], [30] is the discovery of a phase transition phenomenon,” which is clearly analogous to the subspace swap phenomena known in the signal processing literature for 20 years [31]. Here, we have shown that the condition (16), or the simplified one for the spiked population matrix (24), which asymptotically prevents the phase transition phenomenon from occurring, is in fact the condition which guarantees the separability of the signal and noise subspaces in the asymptotic sample eigenvalue distribution. Note that the subspace splitting condition (16) may be satisfied, preventing “inter-subspace” swap, while for some signal subspace eigenvalues, the similar eigenvalue splitting condition (14) may not be met. In that latter case, those sample signal subspace eigenvalues collapse into a single cluster, and the expressions (20)–(21) must be modified. Yet, this “intra-subspace” swap is not important for subspace techniques, where only intersubspace swap matters. Tufts et al. [1] stated that the threshold effect was associated with the probability that the measured data is better approximated by some components of the orthogonal subspace than by some components of the signal subspace. A narrow investigation of the relationship between these G-asymptotic phase transitions and subspace swap would therefore pinpoint the conditions under which the norm of the scalar product between the true and estimated eigenvectors in the signal subspace fall below 0.5, in which case Tufts description of the subspace swap becomes clearly equivalent. Condition (24), obtained under the spiked population covariance model, implies that in our source scenario, when the eigenvalue is below the threshold , the projection of the fourth eigenvector onto the sample signal subspace (37) Furthermore, the “signal processing” subspace swap definition implies

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scenario (7) considered in Section II for the following four SNR values: no MUSIC outliers; • Input SNR 25 dB 50% MUSIC outliers; • Input SNR 14 dB almost100% MUSIC outliers; • Input SNR 9 dB onset of ML breakdown. • Input SNR 0 dB First, at Fig. 2 for each of these four SNR values we separately show the sample distributions of the four eigenvalues in the signal subspace, along with the distribution of all nonzero eigenvalues in the noise subspace. As expected, the separation between the noise and signal subspace eigenvalues decrease as the source SNR decreases, but one can see that even for the SNR of 9 dB with practically 100% MUSIC breakdown, the cluster of nonzero noise subspace eigenvalues is still well separated from the minimal signal subspace eigenvalue . It is only at the lowest plotted SNR of 0 dB that we see significant overlap between the noise and signal subspace eigenvalues. The eigenvalues for the underlying true covariance matrices (deSNR are noted Eig (40) (41) (42) (43)

Eig Eig Eig Eig which means that the subspace splitting condition from (24)

(44) is satisfied for all four SNR values (although the 0 dB SNR case is marginal). The subspace splitting condition given in (16) can be computed for the transition from the signal subspace to the noise subspace which is the splitting condition between the 4th and 5th eigenvalues. This gives a value of 0.21, 0.24, 0.29, and 1.08 for 25, 14, 9, and 0 dB, respectively. This value is clearly for all but the last SNR value. Based on less than these GSA metrics only, one would conclude that the noise and signal subspace eigenvalues are distinct for all but the last case at 0 dB SNR, and therefore subspace techniques should operate robustly at the higher SNRs. Yet significant MUSIC breakdown occurs in the 9- and 14-dB SNR case. It is also important to conditions examined here do not establish that the finite change dramatically asymptotically (in the GSA sense (1)), so scenario considered above, in addition to the let us examine the following three scenarios with an increased and dimension, but with the ratio held constant. The original scenario:

(38) SNR

or, equivalently

14 dB (45)

Eig (39)

i.e., the last signal eigenvector is better represented by the noise subspace than the signal subspace. Therefore, the behavior of is of prime importance for our analthe projection ysis. In order to explore and validate these GSA analytic predictions with respect to MUSIC breakdown, let us analyze the

A 200-element array:

SNR Eig

4 dB (46)

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Fig. 3. Projection of 4th true eigenvector onto the sample signal subspace (scenario (45)–(47)). All scenarios show significant MUSIC breakdown ( 60 %; 30 %, and 15 %, respectively), but the mean projection is converging to the predicted value in (56), whose high value indicates little subspace “leakage.”

A 400-element array with:

20 Eig

Fig. 2. Eigenvalue Distributions for Scenario (7). Even in the presence of significant MUSIC breakdown for scenario SNR of 9 and 14 dB, the signal subspace eigenvalues remain well separated from the noise subspace eigenvalues. (a) 25-dB SNR—No MUSIC Breakdown; (b) 14-dB SNR— 50% MUSIC outliers; (c) 9-dB SNR— 100 % MUSIC outliers; and (d) 0-dB SNR—Start of ML breakdown.

10 35 35.1 SNR

1 dB (47)

Inter-source separations and SNR in the increased array size scenarios (46) and (47) have been chosen to produce essentially the same signal eigenvalues as per the original scenario (45) with . All three eigenspectra have minimal signal subspace eigenvalues in the range of 64–66, which would allow us to expect the same G-asymptotic behavior under the spiked population covariance model. At Fig. 3, we introduce sample dis, calculated for all three tributions of the projection in all cases). First scenarios (45), (46), (47) (with of all, we can clearly observe that in full agreement with GSA, conthe projection is converging as stant) to a non-statistical deterministic value. Indeed, one can observe quite a consistent convergence of the sample distributo a delta-function, whether the scenario tions for contains a MUSIC outlier or not. Furthermore, while the results are converging asymptotically, the mean values observed at our modest array dimension of 20 elements are already quite accurate (to within 0.5% of the mean observed with 400 elements). In order to predict these asymptotic deterministic values, let us consider the following Theorem 2 of Mestre [26].


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Theorem 2: If the splitting condition (16) for the smallest is satisfied, the random value signal subspace eigenvalue

(48) where is a deterministic (unity norm) column vector, asymptends to the nonrandom totically value , i.e., as

(49)

where (50) Fig. 4. Eigenvalue distributions for first 4 eigenvalues and noise eigenvalues 4 dB. Note significant overlap between second and for scenario (46), SNR third eigenvalue distribution.

=

and are the eigenvectors of the matrix arranged in descending order and

(51) where lution to

is the minimal (potentially negative) real-valued so-

(52) assuming that . This theorem allows us to find the asymptotic MUSIC pseuis an antenna steering vector. If dospectrum (9) if (the th eigenvector of the actual covariance mainstead ), from (20), we get trix (53) where

is specified by (51). Therefore, we get (for

) (54)

For the “spiked population covariance matrix”, when (27) (and ), we finally get (55) and for our specific scenario with the minimal signal subspace eigenvalue associated with

(56) One can see that we get the same asymptotic expression as in (35), but now for the projection onto the entire sample subspace. This means that when the “intra-subspace swap” is precluded by

having the eigenvalue splitting condition (14) satisfied for all signal subspace eigenvectors, the power (35) of the true eigenasymptotically resides in the fourth sample subspace vector eigenvector , while the remaining power resides in the sample noise subspace. If instead only the subspace splitting condition (16) is satisfied , then the same power (56) is distributed across multiple sample signal subspace eigenvectors. As can be observed in Fig. 3, the discrepancy between the and the prediction (56) estimated mean values for Monte Carlo is within the fourth decimal point for a set of elements. While the match for trials and an array of (56) is quite good even for small arrays, we separately observe that the projections of the sample eigenvectors onto the indican deviate vidual true signal subspace eigenvectors, significantly from (35) for even large arrays, under some circumstances. The problem occurs when the eigenvalue splitting condition (14) is not met for all signal subspace eigenvalues and “intra-subspace” swap within the signal subspace precludes individual projections from presenting values close to those predicted by (35). This difference between observation and predicgrow, source separation tion persists for large arrays if as is decreased, as we have done in (45)–(47). For the scenario (46) elements, samples, and a very small with and third difference between the second eigenvalue (see Fig. 4), intra-subspace swap for eigenvectors 2 and 3 was frequently observed with and the sample distribution of distributed widely over the [0 1] interval, as seen in Fig. 5. While this intra-subspace swap phenomena will not persist G-asymptotically as the relative dimension of the signal subspace vanishes, it still indicates that for breakdown analysis on finite arrays, the projection onto the entire subspace rather than individual eigenvectors is the appropriate metric. Finally, the most important observation from the MUSIC breakdown standpoint is that for both “proper” trials with no outliers and “improper” MUSIC trials with at least one outlier, the minimal signal subspace eigenvector still resides in the sample signal subspace with more than 95% of its power, converging asymptotically to 98%. This convergence is accurately

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M = 200

Fig. 5. Inner product of sample and true eigenvectors for element array and scenario (46). Correlation of the second and third sample eigenvectors with their associated true eigenvectors jhe e ij is poor because of frequent eigenvector swap, as is the match between the observed mean and the prediction from (35). Such “intra-subspace” swap should not affect MUSIC or any other subspace-based technique.

^

predicted by (56) for multiple SNR values, as indicated in Fig. 6. The main conclusion that is now supported both by GSA theory and direct Monte Carlo simulations is that for the considered scenario, subspace swap is not responsible for the MUSIC breakdown phenomenon observed, and the underlying mechanism requires further exploration. IV. SOURCE RESOLUTION AND MUSIC “PERFORMANCE BREAKDOWN” Careful examination of the pseudospectrum produced during trials with MUSIC outliers, such as in our scenario (7) with SNR dB, show that in many trials, the MUSIC algorithm selected an erroneous peak at despite the fact that the pseudo-spectrum value at that peak was significantly smaller than the pseudo-spectrum values at any true source direction

(57) This happened only because MUSIC was unable to resolve the third and the fourth closely located sources 35 37 and instead found a single maxima in their vicinity. This well-known phenomena of loss of MUSIC resolution capability [4] is not directly associated with the “subspace swap” phenomenon and in fact is associated with a significantly smaller portion of sample signal subspace energy residing in the noise subspace than is required for subspace swap as defined in (38). This fact has been already demonstrated by the experimental data in Figs. 3 –6 as well as the GSA prediction (56). To examine the effect of this loss of resolution further, we need to define a “resolution event.” In [32], Cox defines a res-

Fig. 6. Comparison of predicted and observed projection of the fourth sample eigenvector onto the true signal subspace. The correspondence between the obp servations and the predictions above
0, but ^ < 0, but two maxima. only one maxima; (b)

According to this result (63) and therefore the function

in (59) is derived as

(64) are the eigenvectors of the matrix . According to where this expression, we get the following G-asymptotic values • input SNR 25 dB ; • input SNR 14 dB ; • input SNR 9 dB ; which agree precisely with the mean values of Fig. 8. Consistent with our earlier conclusion that subspace swap was not the sole reason for MUSIC outlier production, we can now see that MUSIC performance breakdown can occur when MUSIC is unable to resolve some closely spaced sources

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(“subspace leakage”). Another important conclusion is that the GSA methodology has been proven to be sufficiently accurate for scenarios with surprisingly small and , compared with the G-asymptotic requirement ( constant), and can provide accurate predictions of the MUSIC resolution performance. The introduced analysis sheds some light on the reasons for the disappointing threshold performance improvement delivered by G-MUSIC, as observed in Section II. While G-MUSIC is indeed able to counter the bias in noise subspace estimation caused by the low sample support, the almost-sure convergence to the accurate zeros of the cost function (3) still may possess sufficient variance such that the value at a midpoint between the two closely spaced sources is even smaller, leading to a single minima (or maxima for the inverse pseudospectrum) and corresponding loss of resolution. V. SUBSPACE SWAP AND MLE PERFORMANCE BREAKDOWN

Fig. 8. Resolution metric ^ for closely spaced sources = 35 ; = 37 in (7). For SNR = 14 dB, the sample distributions are practically symmetric with regard to zero, with trials with outliers shifted into the negative domain and trials without outliers shifted into the positive domain. For SNR = 9 dB with 100 % MUSIC breakdown, the sample distribution entirely resides in the negative domain, while for SNR = 25 dB with no MUSIC breakdown, it resides entirely in the positive domain. (a) Complete MUSIC breakdown; (b) 50 % MUSIC breakdown; and (c) no MUSIC breakdown.

In a way, the improved performance of MLE in the threshold region relative to MUSIC (and G-MUSIC) is reflected in the fact that MUSIC breaks down significantly earlier than estimation of the number of sources by information theoretic criteria (ITC) [36]. Both the MLE criterion and the ITC approach test the entire covariance matrix model to fit the training data, while MUSIC (and G-MUSIC) selects each DOA estimate independently, with no respect as to how the entire set of produced estimates fits the input data. Therefore, the significant difference between the MUSIC-specific and the ML-intrinsic threshold conditions is the penalty one has to pay for replacing the multivariate ML optimization problem that finds the set of estimates that jointly best-fit the input data, by the univariate search of the . The MLE function that only has the same solution as breakdown is observed under conditions when a set of DOA estimates that contains a severely erroneous estimate (an outlier) generates a LF value that exceeds the local extremum in the vicinity of the true solution. In other words, there are maxima of the LF (including the global maxima) that exceed the local maximum of the LF considered by the traditional asymptotic as the ML solution. ML analysis For a solution that contains an outlier to be “more likely” than the actual covariance matrix, the training data should indeed generate a sample signal subspace with some of its elements better represented by the true noise subspace. Therefore, the subspace swap phenomenon is more likely to be associated with the MLE breakdown rather than with the breakdown in subspace techniques. In order to demonstrate this, let us analyze MLE performance in scenario (7) at three SNR values: 2 ; 0 dB with [again as in (43)]; and dB with 4 dB with . 2 dB and , the Mestre eigenvalue For SNR splitting condition (16) for the fourth eigenvalue is just satisfied , indicating that we have (almost-sure) separation of the signal and noise subspaces asymptotically (in the GSA sense) while according to (56) for this SNR, we get (65)

and for this to happen, quite negligible power of the actual signal subspace has to reside in the sample noise subspace

Monte Carlo simulations show a mean for of 0.6815, agreeing well with the prediction and indicating that


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the 4th eigenvector projects more onto its proper signal subspace than the noise subspace. , the Mestre eigenvalue splitFor SNR 0 dB and , while ting condition (16) is violated the projection of signal eigenvector onto the signal subspace is forecast via (56) as (66) Monte Carlo simulations show a mean for of 0.5098. This indicates that the subspace swap condition given in (39) is essentially satisfied and subspace swap is statistically likely, consistent with the observation in Fig. 1 that MLE breakdown starts to occur at 0-dB input SNR. 4 dB, , and the condition for For SNR given G-asymptotic convergence in (24) is violated, so the projection should converge to zero asymptotically and (56) is no longer valid. Monte Carlo simof 0.2502. Clearly the ulations show a mean for scenario is experiencing significant subspace swap, once again consistent with results given in Fig. 1. Having established that MLE performance breakdown in the examined multiple source scenario is indeed reliably associated with subspace swap (whereas MUSIC breakdown in the same scenario is associated with subspace leakage, as shown in Section III, and resultant loss of resolution, as shown in Section IV), we next examine a single source scenario, such as studied in [2], [37], where we expect that subspace swap will indeed be the sole mechanism responsible for both MLE and MUSIC DOA estimation performance breakdown. To this end, we introduce a second scenario with a single target, based, as in Athley [2], [37], on a sparse minimum redundancy array (MRA) [38], where the generation of outliers is more likely due to poor sidelobe performance. We configuration use the following specific , suggested for the MRA context in [39], and confirmed to be minimally redundant [40]. The threshold effect of MLE estimation in this scenario [provided by the Barlett spectrum or conventional beamforming (CBF)] can be observed in Fig. 9 to occur around 5 dB for . In (38), we defined subspace swap as occurring when the projection of last true eigenvector into the underlying sample noise subspace was higher than into the sample signal subspace. To examine whether this subspace swap is the sole mechanism for MLE breakdown, we can plot for each of 1000 Monte Carlo , the DOA error of a trials and a training sample size of single source estimated with the MRA versus the correlation between the “maximal” sample and true eigenvector. These plots are shown in Fig. 10 for source SNRs ranging from very low values which result in complete MLE breakdown [input SNR of 18 dB, as shown in Fig. 10(a)] to values where there is no MLE breakdown [input SNR of 0 dB, as shown in Fig. 10(d)]. Fig. 10 clearly demonstrates that when the projection of the signal true eigenvector onto the sample signal subspace is high, there is no MLE breakdown [i.e., the upper right quadrant of Figs. 10(a)–(d) are all free of any Monte Carlo trials]. Interestingly, however, the converse is not true. When the projection of the signal true eigenvector onto the sample signal subspace

Fig. 9. MSE for MUSIC, G-MUSIC and MLE (CBF) DOA estimation on a 18-element minimum redundancy array with 1000 trials/SNR step. Note that MUSIC and G-MUSIC estimators deliver essentially the same performance in this circumstance, as expected for single sources.

^ E^ e Fig. 10. Distribution of DOA estimation errors versus projection e E for a single target scenario on 18-element minimum redundancy array with T = 14 training samples. As SNR is increased, the projection approaches unity, and the estimation accuracy improves. Note, however, that trials with low projection values still frequently have low estimation errors. (a) 100 % MLE breakdown; (b) 50 % MLE breakdown; (c) rare MLE breakdown; and (d) no MLE Breakdown.

is low, a DOA outlier estimate may or may not be produced. Thus, subspace swap is a necessary but not sufficient condition for MLE breakdown to occur. Turning our attention back to the uniform line array scenario and with closely spaced sources given in (7) with , we now conduct a similar examination, using DOA estimates provided by the MLE-proxy algorithm. Because there are multiple sources, but usually only one outlier, we plot the worst observed error versus the projection of the 4th true eigenvector onto the sample signal subspace in Fig. 11. The behavior is remarkably similar to the single source performance shown earlier, and therefore the observation that subspace swap is a

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^ E^ e Fig. 11. Distribution of DOA estimation errors versus projection e E for worst error in multiple source scenario on 20-element uniform linear array with T = 15 training samples. As with the single source case, the projection of the true eigenvector onto the sample signal subspace is a necessary but not sufficient condition for breakdown to occur. (a) 100 % MLE breakdown; (b) 50 % MLE breakdown; (c) rare MLE breakdown; and (d) no MLE breakdown.

necessary but not sufficient condition for MLE breakdown is strongly reinforced. VI. SUMMARY AND CONCLUSION In this paper we investigated the well-known performance breakdown phenomenon in DOA estimation techniques, which manifests as a dramatic and rapid departure of estimation accuracy from the CRB due to the increasing probability of erroneous “outlier” estimates as the SNR or number of training samples is decreased below certain threshold values. We analyzed this phenomenon for conventional MUSIC, the recently developed G-MUSIC [15], [16], and MLE for multiple and single Gaussian source scenarios with i.i.d. sample support. asymptotic analRather than consider a traditional ysis, we specifically considered parameters far removed from the traditional asymptotic regime, focusing on under-sampled scenarios with the number of training samples less than the antenna dimension . To provide theoretical analysis of this small-sample regime, we employed the so-called General Statistical Analysis (GSA) methodology that considers the asymptotic regime (67) constant, which differs significantly from the usual asymptotic assumptions. This analysis, supported by the results of direct Monte Carlo simulations, lead to a number of important observations. -consistent First of all, we demonstrated that while the G-MUSIC DOA estimator outperforms, as expected, MUSIC in the threshold region, this improvement is marginal compared

with the clearly superior threshold performance of the ML estimator for the considered scenarios. Such a significant distinction in the threshold conditions clearly indicates that performance breakdown in subspace-based techniques is caused by a phenomenon which differs from the one that causes MLE performance breakdown. In this regard, the most controversial observation gained was that for multiple-source scenarios, MUSIC and G-MUSIC performance breakdown frequently takes place for SNR and sample support conditions that (according to GSA predictions) should almost surely preclude the “subspace swap” phenomenon. Since traditionally subspace swap has been associated with performance breakdown in subspace DOA estimation techniques, detailed analysis of the actual reasons for breakdown and confirand mation of G-asymptotic derivation accuracy with finite values was necessary to verify the observation. Customarily, subspace swap is described as an event where a particular (minimal) signal subspace eigenvector is better represented (expanded) by the noise subspace of the sample covariance matrix, rather than by the sample signal subspace. Our analysis demonstrated that the GSA methodology very accurately predicts the subspace swap conditions, even for antenna dimensions and sample volume which are far from the G-asymptotic regime. We thus observed that both theoretical predictions and simulation results show that MUSIC (and G-MUSIC) performance breakdown can take place when less than 5% of the minimal signal subspace eigenvector’s power residing in the sample noise subspace, leaving more than 95% of this power residing in the sample signal subspace. Clearly such insignificant inter-subspace “leakage” is far from the subspace swap condition, which has been defined here and elsewhere as the point where more than 50% of the true eigenvector power resides in the wrong sample subspace. We then demonstrated that this small subspace “leakage” is sufficient for MUSIC to lose its capability to resolve poorly separated sources. In that case, MUSIC will “pick” a completely erroneous DOA estimate in addition to the single unresolved peak. Once again, GSA methodology was found to be able to predict this breakdown condition using the clairvoyant DOAs and asymptotically justified weighting factors, allowing us to find the threshold SNR and/or sample volume required for reliable resolution for a given array configuration and scenario. MLE performance breakdown takes place when a set of estimates that contain an outlier is “more likely” than the true parameters or even the local LF maximum in their vicinity. For this to happen, the input data should be insufficient, and therefore with no surprise we established that MLE breakdown is indeed reliably associated with the subspace swap phenomena, well predicted by the GSA methodology. It is obvious that scenarios where the MUSIC (and G-MUSIC) pseudo-spectrum does not differ significantly from the conventional Barlett spectrum (single or well-sepa), MUSIC and MLE rated sources, very low SNR, techniques will demonstrate similar threshold performance, with full subspace swap becoming the common reason for breakdown in both techniques. For single source cases, the similar breakdown point for MLE and MUSIC is well correlated with GSA-derived eigenvalue splitting and subspace swap


predictions. It was noted, however, that while high projection values of the minimal signal eigenvector onto the sample signal subspace precludes the formation of outliers leading to performance breakdown, low projection values (indicating in some cases almost complete subspace swap) did not always lead to performance breakdown. Therefore, subspace swap is a necessary, but not sufficient, condition for DOA estimation breakdown, with other factors such as statistical variations of the source power and manner of distribution of the signal subspace power across the sample noise subspace also influencing outlier production. It is clear that performance breakdown caused by subspace swap always causes unrecoverable performance breakdown regardless of the DOA estimation technique. In contrast, MUSICspecific performance breakdown caused by the loss of resolution capability (and less severe subspace leakage) is recoverable, as previously demonstrated [14], [23]. Both conditions (subspace swap and subspace leakage leading to loss of resolution) are shown to be accurately predicted based on GSA-derived analytical conditions. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers which pointed out a number of suggested improvements. REFERENCES [1] D. Tufts, A. Kot, and R. Vaccaro, “The threshold effect in signal processing algorithms which use an estimated subspace,” in SVD and Signal Processing II: Algorithms, Analysis and Applications, R. Vaccaro, Ed. New York: Elsevier, 1991, pp. 301–320. [2] F. Athley, “Performance analysis of DOA estimation in the threshold region,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Orlando, FL, 2002, vol. 3, pp. 3017–3020. [3] J. Thomas, L. Scharf, and D. Tufts, “The probability of a subspace swap in the SVD,” IEEE Trans. Signal Process., vol. 43, no. 3, pp. 730–736, Mar. 1995. [4] M. Kaveh and A. Barabell, “The statistical performance of the MUSIC and the minimum-norm algorithms in resolving plane waves in noise,” IEEE Trans. Signal Process., vol. 34, no. 2, pp. 331–341, 1986. [5] H. L. van Trees, Detection, Estimation, and Modulation Theory: Part I. New York: Wiley, 1968. [6] D. Rife and R. Boorstyn, “Single tone parameter estimation from discrete-time observations,” IEEE Trans. Inf. Theory, vol. 20, no. 5, pp. 591–598, Sep. 1974. [7] A. Steinhardt and C. Bretherton, “Thresholds in frequency estimation,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Apr. 1985, vol. 10, pp. 1273–1276. [8] B. G. Quinn and P. J. Kootsookos, “Threshold behavior of the maximum likelihood estimator of frequency,” IEEE Trans. Signal Process., vol. 42, no. 11, pp. 3291–3294, Nov. 1994. [9] B. James, B. D. O. Anderson, and R. C. Williamson, “Characterization of threshold for single tone maximum likelihood frequency estimation,” IEEE Trans. Signal Process., vol. 43, no. 4, pp. 817–821, Apr. 1995. [10] D. C. Rife and R. R. Boorstyn, “Multiple tone parameter estimation from discrete time observations,” Bell Syst. Tech. J., vol. 55, pp. 1389–1410, Nov. 1976. [11] F. Athley, “Threshold region performance of deterministic maximum likelihood DOA estimation of multiple sources,” in Proc. 37th Asilomar Conf. Signals, Systems, Computers, Nov. 2002, vol. 2, pp. 1283–1287. [12] H. Wang and M. Kaveh, “On the performance characterization of signal-subspace processing,” in Proc. 19th Asilomar Conf. Signals, Systems, Computers, Nov. 1985, pp. 73–77. [13] E. Boyer, P. Forster, and P. Larzabal, “Nonasymptotic performance analysis of beamforming with stochastic signals,” IEEE Signal Process. Lett., vol. 11, no. 1, pp. 23–25, Jan. 2004.

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[14] Y. I. Abramovich, N. K. Spencer, and A. Y. Gorokhov, “GLRT-based threshold detection-estimation preformance improvement and application to uniform circular antenna arrays,” IEEE Trans. Signal Process., vol. 55, no. 1, pp. 20–31, Jan. 2007. [15] X. Mestre, “An improved weighted MUSIC algorithm for small sample size scenarios,” in Proc. 14th Workshop on Adaptive Sensor Array Processing, Lexington, MA, 2006 [Online]. Available: www.ll.mit.edu/ asap/asap_06/pdf/Papers/19_mestre_pa.pdf, MIT-LL [16] X. Mestre, “An improved subspace-based algorithm in the small sample size regime,” presented at the IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP) Toulouse, France, 2006. [17] X. Mestre and M. A. Lagunas, “Modified subspace algorithms for DoA estimation with large arrays,” IEEE Trans. Signal Process., 2006, submitted for publication. [18] Y. I. Abramovich, B. A. Johnson, and X. Mestre, “Performance breakdown in MUSIC, G-MUSIC and maximum likelihood estimation,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Honolulu, HI, Apr. 15–20, 2007, vol. 2, pp. 1049–1052, IEEE. [19] A. Steyerl and R. Vaccaro, “Performance of direction-of-arrival estimators at low signal-to-noise ratios,” presented at the Adaptive Sensor and Array Processing Workshop, Lexington, MA, Jun. 2007, MIT-LL. [20] T. Anderson, An Introduction to Multivariate Statistical Analysis. New York: Wiley, 1958. [21] B. Ottersten, M. Viberg, P. Stoica, and A. Nehorai, “Exact and large sample maximum likelihood techniques for parameter estimation and detection in array processing,” in Radar Array Processing. Berlin, Germany: Springer-Verlag, 1993, pp. 99–151. [22] N. K. Spencer and Y. I. Abramovich, “Performance analysis of DOA estimation using uniform circular antenna arrays in the threshold region,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Montreal, QC, Canada, 2004, vol. 2, pp. 233–236. [23] B. A. Johnson and Y. I. Abramovich, “GLRT-based outlier prediction and cure in under-sampled training conditions using a singular likelihood ratio,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Honolulu, HI, Apr. 15–20, 2007, vol. 2, pp. 1129–1132, IEEE. [24] M. Hawkes, A. Nehorai, and P. Stoica, “Performance breakdown of subspace-based methods: Prediction and cure,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Salt Lake, UT, 2001, vol. 6, pp. 4005–4008. [25] X. Mestre, “Estimating the eigenvalues and associated subspaces of correlation matrices from a small number of observations,” presented at the 2nd Int. Symp. Comm. Control, Signal Processing, Marrakech, Morocco, 2006. [26] X. Mestre, “Improved estimation of eigenvalues of covariance matrices and their associated subspaces using their sample estimates,” IEEE Trans. Inf. Theory 2006 [Online]. Available: http://www.cttc.cat/drafts/ cttc-rc-2006-004.pdf, submitted for publication [27] I. M. Johnstone, “On the distribution of the largest eigenvalue in principal component analysis,” Ann. Stat., vol. 29, no. 2, pp. 295–327, 2001. [28] D. Paul, “Asymptotics of the leading sample eigenvalues for a spiked covariance model,” Stanford Univ., Stanford, CA, Tech. Rep., Dec. 2004 [Online]. Available: http://anson.ucdavis.edu/~debashis/techrep/ eigenlimit.pdf [29] J. Baik, G. B. Arous, and S. Péché, “Phase transition of the largest eigenvalue for non-null complex covariance matrices,” Ann. Probab., vol. 33, no. 5, pp. 1643–1697, 2005. [30] J. Baik and J. W. Silverstein, “Eigenvalues of large sample covariance matrices of spiked population models,” J. Multivar. Anal., vol. 97, no. 6, pp. 1382–1408, Jul. 2006. [31] D. W. Tufts, A. C. Kot, and R. J. Vaccaro, “The threshold analysis of SVD-based algorithms,” in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), New York, Apr. 1988, pp. 2416–2419. [32] H. Cox, “Resolving power and sensitivity to mismatch of optimum array processors,” J. Acoust. Soc. Amer., vol. 54, pp. 771–785, 1973. [33] H. B. Lee and M. S. Wengrovitz, “Resolution threshold of beamspace MUSIC for two closely spaced emitters,” IEEE Trans. Signal Process., vol. 38, no. 9, pp. 1545–1559, Sep. 1990. [34] W. Xu and M. Kaveh, “Alternatives lor the definition and evaluation of resolution thresholds of signal-subspace parameter estimators,” presented at the IEEE Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Adelaide, SA, Australia, Apr. 1994. [35] Q. Zhang, “Probability of resolution of the MUSIC algorithm,” IEEE Trans. Signal Process., vol. 43, no. 4, pp. 978–987, Apr. 1995. [36] F. Li and R. J. Vaccaro, “Unified analysis for DOA estimation algorithms in array signal processing,” Signal Process., vol. 25, no. 2, pp. 147–169, 1991.

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Ben A. Johnson (S’04) received the B.S. (cum laude) degree in physics from Washington State University, Pullman, WA, in 1984 and the M.S. degree in digital signal processing from the University of Southern California, Los Angeles, in 1988. He is currently working towards the Ph.D. degree at the Institute of Telecommunications Research, University of South Australia, Mawson Lakes, South Australia, focusing on application of spatio–temporal adaptive processing in high-frequency radar. From 1984 to 1989, he was a Systems Engineer in airborne radar at Hughes Aircraft Company (now Raytheon), El Segundo, CA. From 1989 to 1998, he was a Senior Radar Engineer in ground-based surveillance systems with Sensis Corporation, DeWitt, NY. Since 1998, he has been with Lockheed Martin, Bethesda, MD (assigned to a joint venture defense contractor, RLM, Pty. Ltd., Edinburgh, South Australia) on the Jindalee Over-theHorizon Operational Radar Network (JORN), first as a Senior Test Engineer and then as Technical Director.


Yuri I. Abramovich (M’96–SM’06) received the Dipl.Eng. (Hons.) degree in radio electronics and the Cand.Sci. degree (Ph.D. equivalent) in theoretical radio techniques, both from the Odessa Polytechnic University, Odessa, Ukraine, in 1967 and 1971, respectively, and the D.Sc. degree in radar and navigation from the Leningrad Institute for Avionics, Leningrad, Russia, in 1981. From 1968 to 1994, he was with the Odessa State Polytechnic University, Odessa, Ukraine, as a Research Fellow, Professor, and ultimately as Vice-Chancellor of Science and Research. From 1994 to 2006, he was at the Cooperative Research Centre for Sensor Signal and Information Processing (CSSIP), Adelaide, Australia. Since 2000, he has been with the Australian Defence Science and Technology Organisation (DSTO), Adelaide, as Principal Research Scientist, seconded to CSSIP until its closure. His research interests are in signal processing (particularly spatio–temporal adaptive processing, beamforming, signal detection and estimation), its application to radar (particularly over-the-horizon radar), electronic warfare, and communication. Dr. Abramovich is currently an Associate Editor of the IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS and previously served as Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2002 to 2005.

Xavier Mestre (S’96–M’04) received the M.S. and Ph.D. degrees in electrical engineering from the Universitat Politècnica de Catalunya (UPC), Spain, in 1997 and 2003, respectively. From January 1998 to December 2002, he worked as a Research Assistant for UPC’s Communications Signal Processing Group. In January 2003, he joined the Telecommunications Technological Center of Catalonia (CTTC), Barcelona, Spain, where he currently holds a position as a Senior Research Associate in the area of radio communications. Dr. Mestre was recipient of a 1998–2001 Ph.D. scholarship (granted by the Catalan Government) during the pursuit of the Ph.D. degree. He was also awarded the 2002 Rosina Ribalta second prize for the Best Doctoral Thesis Project within areas of information technologies and communications by the Epson Iberica foundation.