A Grid-Less Approach to Underdetermined Direction of Arrival Estimation Via Low Rank Matrix Denoising

The problem of direction of arrival (DOA) estimation of narrowband sources using an antenna array is considered where the number of sources can potentially exceed the number of sensors. In earlier works, the authors showed that using a suitable antenna geometry, such as the nested and coprime arrays, it is possible to localize O(M²) sources using M sensors. To this end, two different approaches have been proposed. One is based on an extension of subspace based methods such as MUSIC to these sparse arrays, and the other employs l1 norm minimization based sparse estimation techniques by assuming an underlying grid. While the former requires the knowledge of number of sources, the latter suffers from basis mismatch effects. In this letter, a new approach is proposed which overcomes both these weaknesses. The method is hybrid in nature, using a low rank matrix denoising approach followed by a MUSIC-like subspace method to estimate the DOAs. The number of sources is revealed as a by-product of the low rank denoising stage. Moreover, it does not assume any underlying grid and thereby does not suffer from basis mismatch. Numerical examples validate the effectiveness of the proposed method when compared against existing techniques.

I. Introduction

Direction of Arrival (DOA) estimation using an array of antennas is a central problem in array processing [3]. The DOA estimation problem is closely related to that of harmonic retrieval, which is solved by classical subspace-based methods such as MUSIC [1], ESPRIT [2] etc. These algorithms use the key assumption that the number of active sources is less than the number of sensors , which manifests itself in the form of a low rank signal subspace component. In a series of recent papers [14]–[16], [18], the authors have demonstrated that this restriction is often a consequence of using sub optimal array geometries (spatial sampling schemes), such as the uniformly spaced antenna arrays (ULA). A new array geometry based on non uniform sampling, viz., the nested array was proposed which is inherently capable of identifying narrowband sources, which is order-wise optimal in , if we only use the second order statistics (covariance) of the measurements. However, the conventional harmonic retrieval algorithms can no longer be used when , since the data covariance matrix no longer contains a low rank component spanning the signal subspace. To this end, the authors proposed two techniques for DOA estimation based on novel ways to process the data covariance to localize source directions. The first is based on a suitable extension of the MUSIC algorithm, called SS-MUSIC [16] while the other [14], [15], [18] uses compressive sensing (CS) inspired norm minimization techniques by assuming a sparse representation of the incoming DOAs on a pre specified discrete frequency grid. While SS-MUSIC requires the model order (number of sources) to be pre specified or estimated, the CS based approach automatically finds the number of sources. However, the CS-based methods suffers from basis mismatch effects [13] since the true DOAs are unlikely to lie on the pre specified grid, no matter how finely it is chosen. In contrast, once the model order is correctly identified, SS-MUSIC can theoretically exhibit infinite angle-precision in identifying the DOAs.