Diss / 3

Although this definition is commonly used, it should be noted that this metric may only approximate the actual resolution event. Indeed, at Fig. 7, we demonstrate two actual examples from the set of Monte Carlo trials on scenario (7) with an SNR of 15 dB. The resolution definition is compared to a local maxima criterion, with Fig. 7(a) showing a success of the resolution event when local maxima tests fail, and Fig. 7(b) conversely showing a failure of the resolution event when local maxima tests succeed. These variations are statistical however, and on average the probability that (58) is positive should correspond well to the probability of actual resolution. In [33]–[35], analytic approximations of this resolution probability are found for the standard asymptotic conditions constant and proven to be quite accurate for , as in the example with , considered in [35]. However, for , we once again have to apply G-asymptotic analysis. In accordance with the GSA methodology, we have to find a nonrandom (deterministic) function , such that

(59)

under asymptotic condition (1). We can then associate condition with roughly 50% probability of resolution, and

the decision threshold

(60)

with high (and low) resolution probability correspondingly. First of all, at Fig. 8 we introduce sample distributions for the resolution metric calculated for SNR 9, 14, and 25 dB, correspondingly. As usual, we also introduce (for SNR 14 dB) sample distributions averaged over the subsets of improper and proper MUSIC trials (recalling that for SNR 9 dB, practically all MUSIC trials are improper while for SNR 25 dB, all of the trials are proper). While trials with outliers in general have a negative resolution metric (regardless of SNR) and those without outliers have a positive metric, there is some overlap at SNR 14 dB. The overlap between these two distributions is explained by the phenomena illustrated at Fig. 7, since the Monte Carlo simulation DOA estimation algorithm searches for strict maxima in the pseudospectrum rather than applying the subject resolution metric.

To get the analytic GSA result for the function in (59), let us once again consider Theorem 2 by Mestre (48)–(52). According to this theorem, if the splitting condition (16) for

Fig. 7. Resolution of two closely spaced sources in scenario (7), SNR 15 dB. The two example trials illustrate the difference between the resolution event criteria (58) and the local maxima criteria used in the simulation. (a) , but only one maxima; (b) , but two maxima.

According to this result

(63)

and therefore the function in (59) is derived as

(64)

where

and are specified in (51).

•input SNR 14 dB ;

•input SNR 9 dB ; which agree precisely with the mean values of Fig. 8.

(62)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

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.

(“subspace leakage”). Another important conclusion is that the GSA methodology has been proven to be sufficiently accurate for scenarios with surprisingly small

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

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 function that only has the same solution as . The MLE 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 ML analysis as the ML solution.

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

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

the 4th eigenvector projects more onto its proper signal subspace than the noise subspace.

For SNR 0 dB and , the Mestre eigenvalue splitting condition (16) is violated , while the projection of signal eigenvector onto the signal subspace is forecast via (56) as

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.

in (24) is violated, so the projection should converge to zero asymptotically and (56) is no longer valid. Monte Carlo simulations show a mean for of 0.2502. Clearly the 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

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 trials and a training sample size of , the DOA error of a 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.

Fig. 10. Distribution of DOA estimation errors versus projection for a single target scenario on 18-element minimum redundancy array with

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 with closely spaced sources given in (7) with and , 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

Fig. 11. Distribution of DOA estimation errors versus projection for worst error in multiple source scenario on 20-element uniform linear array with 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. Rather than consider a traditional asymptotic analysis, 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)

which differs significantly from the usual constant,

asymptotic assumptions. This analysis, supported by the results of direct Monte Carlo simulations, lead to a number of important observations.

First of all, we demonstrated that while the -consistent 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 confirmation of G-asymptotic derivation accuracy with finite and 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- rated sources, very low SNR, ), MUSIC and MLE 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

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

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.

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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-the- Horizon Operational Radar Network (JORN), first as a Senior Test Engineer and then as Technical Director.

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

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.