Subspace Leakage Analysis and Improved DOA Estimation with Small Sample Size

Classical methods of DOA estimation such as the MUSIC algorithm are based on estimating the signal and noise subspaces from the sample covariance matrix. For a small number of samples, such methods are exposed to performance breakdown, as the sample covariance matrix can largely deviate from the true covariance matrix. In this paper, the problem of DOA estimation performance breakdown is investigated. We consider the structure of the sample covariance matrix and the dynamics of the root-MUSIC algorithm. The performance breakdown in the threshold region is associated with the subspace leakage where some portion of the true signal subspace resides in the estimated noise subspace. In this paper, the subspace leakage is theoretically derived. We also propose a two-step method which improves the performance by modifying the sample covariance matrix such that the amount of the subspace leakage is reduced. Furthermore, we introduce a phenomenon named as root-swap which occurs in the root-MUSIC algorithm in the low sample size region and degrades the performance of the DOA estimation. A new method is then proposed to alleviate this problem. Numerical examples and simulation results are given for uncorrelated and correlated sources to illustrate the improvement achieved by the proposed methods. Moreover, the proposed algorithms are combined with the pseudo-noise resampling method to further improve the performance.Comment: 37 pages, 10 figures, Submitted to the IEEE Transactions on Signal Processing in July 201

Regularized Covariance Matrix Estimation in Complex Elliptically Symmetric Distributions Using the Expected Likelihood Approach - Part 1: The Over-Sampled Case

In \cite{Abramovich04}, it was demonstrated that the likelihood ratio (LR) for multivariate complex Gaussian distribution has the invariance property that can be exploited in many applications. Specifically, the probability density function (p.d.f.) of this LR for the (unknown) actual covariance matrix $\R_{0}$ does not depend on this matrix and is fully specified by the matrix dimension $M$ and the number of independent training samples $T$. Since this p.d.f. could therefore be pre-calculated for any a priori known $(M,T)$, one gets a possibility to compare the LR of any derived covariance matrix estimate against this p.d.f., and eventually get an estimate that is statistically ``as likely'' as the a priori unknown actual covariance matrix. This ``expected likelihood'' (EL) quality assessment allows for significant improvement of MUSIC DOA estimation performance in the so-called ``threshold area'' \cite{Abramovich04,Abramovich07d}, and for diagonal loading and TVAR model order selection in adaptive detectors \cite{Abramovich07,Abramovich07b}. Recently, a broad class of the so-called complex elliptically symmetric (CES) distributions has been introduced for description of highly in-homogeneous clutter returns. The aim of this series of two papers is to extend the EL approach to this class of CES distributions as well as to a particularly important derivative of CES, namely the complex angular central distribution (ACG). For both cases, we demonstrate a similar invariance property for the LR associated with the true scatter matrix \mSigma_{0}. Furthermore, we derive fixed point regularized covariance matrix estimates using the generalized expected likelihood methodology. This first part is devoted to the conventional scenario ($T \geq M$) while Part 2 deals with the under-sampled scenario ($T \leq M$)

hpDJ: An automated DJ with floorshow feedback

Many radio stations and nightclubs employ Disk-Jockeys (DJs) to provide a continuous uninterrupted stream or “mix” of dance music, built from a sequence of individual song-tracks. In the last decade, commercial pre-recorded compilation CDs of DJ mixes have become a growth market. DJs exercise skill in deciding an appropriate sequence of tracks and in mixing 'seamlessly' from one track to the next. Online access to large-scale archives of digitized music via automated music information retrieval systems offers users the possibility of discovering many songs they like, but the majority of consumers are unlikely to want to learn the DJ skills of sequencing and mixing. This paper describes hpDJ, an automatic method by which compilations of dance-music can be sequenced and seamlessly mixed by computer, with minimal user involvement. The user may specify a selection of tracks, and may give a qualitative indication of the type of mix required. The resultant mix can be presented as a continuous single digital audio file, whether for burning to CD, or for play-out from a personal playback device such as an iPod, or for play-out to rooms full of dancers in a nightclub. Results from an early version of this system have been tested on an audience of patrons in a London nightclub, with very favourable results. Subsequent to that experiment, we designed technologies which allow the hpDJ system to monitor the responses of crowds of dancers/listeners, so that hpDJ can dynamically react to those responses from the crowd. The initial intention was that hpDJ would monitor the crowd’s reaction to the song-track currently being played, and use that response to guide its selection of subsequent song-tracks tracks in the mix. In that version, it’s assumed that all the song-tracks existed in some archive or library of pre-recorded files. However, once reliable crowd-monitoring technology is available, it becomes possible to use the crowd-response data to dynamically “remix” existing song-tracks (i.e, alter the track in some way, tailoring it to the response of the crowd) and even to dynamically “compose” new song-tracks suited to that crowd. Thus, the music played by hpDJ to any particular crowd of listeners on any particular night becomes a direct function of that particular crowd’s particular responses on that particular night. On a different night, the same crowd of people might react in a different way, leading hpDJ to create different music. Thus, the music composed and played by hpDJ could be viewed as an “emergent” property of the dynamic interaction between the computer system and the crowd, and the crowd could then be viewed as having collectively collaborated on composing the music that was played on that night. This en masse collective composition raises some interesting legal issues regarding the ownership of the composition (i.e.: who, exactly, is the author of the work?), but revenue-generating businesses can nevertheless plausibly be built from such technologies

Performance analysis of an improved MUSIC DoA estimator

This paper adresses the statistical performance of subspace DoA estimation using a sensor array, in the asymptotic regime where the number of samples and sensors both converge to infinity at the same rate. Improved subspace DoA estimators were derived (termed as G-MUSIC) in previous works, and were shown to be consistent and asymptotically Gaussian distributed in the case where the number of sources and their DoA remain fixed. In this case, which models widely spaced DoA scenarios, it is proved in the present paper that the traditional MUSIC method also provides DoA consistent estimates having the same asymptotic variances as the G-MUSIC estimates. The case of DoA that are spaced of the order of a beamwidth, which models closely spaced sources, is also considered. It is shown that G-MUSIC estimates are still able to consistently separate the sources, while it is no longer the case for the MUSIC ones. The asymptotic variances of G-MUSIC estimates are also evaluated.Comment: Revised versio

Space Time MUSIC: Consistent Signal Subspace Estimation for Wide-band Sensor Arrays

Wide-band Direction of Arrival (DOA) estimation with sensor arrays is an essential task in sonar, radar, acoustics, biomedical and multimedia applications. Many state of the art wide-band DOA estimators coherently process frequency binned array outputs by approximate Maximum Likelihood, Weighted Subspace Fitting or focusing techniques. This paper shows that bin signals obtained by filter-bank approaches do not obey the finite rank narrow-band array model, because spectral leakage and the change of the array response with frequency within the bin create \emph{ghost sources} dependent on the particular realization of the source process. Therefore, existing DOA estimators based on binning cannot claim consistency even with the perfect knowledge of the array response. In this work, a more realistic array model with a finite length of the sensor impulse responses is assumed, which still has finite rank under a space-time formulation. It is shown that signal subspaces at arbitrary frequencies can be consistently recovered under mild conditions by applying MUSIC-type (ST-MUSIC) estimators to the dominant eigenvectors of the wide-band space-time sensor cross-correlation matrix. A novel Maximum Likelihood based ST-MUSIC subspace estimate is developed in order to recover consistency. The number of sources active at each frequency are estimated by Information Theoretic Criteria. The sample ST-MUSIC subspaces can be fed to any subspace fitting DOA estimator at single or multiple frequencies. Simulations confirm that the new technique clearly outperforms binning approaches at sufficiently high signal to noise ratio, when model mismatches exceed the noise floor.Comment: 15 pages, 10 figures. Accepted in a revised form by the IEEE Trans. on Signal Processing on 12 February 1918. @IEEE201

Multi-Step Knowledge-Aided Iterative ESPRIT for Direction Finding

In this work, we propose a subspace-based algorithm for DOA estimation which iteratively reduces the disturbance factors of the estimated data covariance matrix and incorporates prior knowledge which is gradually obtained on line. An analysis of the MSE of the reshaped data covariance matrix is carried out along with comparisons between computational complexities of the proposed and existing algorithms. Simulations focusing on closely-spaced sources, where they are uncorrelated and correlated, illustrate the improvements achieved.Comment: 7 figures. arXiv admin note: text overlap with arXiv:1703.1052