Conditions for Identifiability in Sparse Spatial Spectrum Sensing

Abstract

Spatial Spectrum estimation is a key technique used in a wide variety of problems arising in signal processing and communication, particularly those employing multiple antennas. In many scenarios such as direction finding using antenna arrays, it is crucial to estimate which directions in space contribute to active sources (indicated by a non zero power). It has been recently shown that if the sources from different directions are statistically uncorrelated, it is possible to identify as many as O(M2) active sources using only M physical antennas. A sparse representation for the spatial spectrum was further exploited to reconstruct the spectrum using convex optimization techniques. In this paper, we consider the situation when there is non zero cross correlation between the sources impinging from different directions. We investigate if, fundamentally, it still possible to identify more sources than the number of physical sensors and what role the cross correlation terms play. Recovery guarantees are developed to ensure uniqueness of the sparse representation for spectrum sensing. They are further extended to establish conditions under which a greedy heuristic, namely the Orthogonal Matching Pursuit algorithm will successfully recover the sparse spectrum. It is shown that in both cases, it is possible to recover support of larger size provided the correlation terms are small compared to the power of the impinging signals.