Consistency and MSE Performance of MUSIC-Based DOA of a Single Source in White Noise With Randomly Missing Data

MUSIC is a popular algorithm for estimating the direction of arrival (DOA) in array signal processing applications. In this paper, we analyze the performance of the MUSIC algorithm for a single source system, in the presence of noisy and missing data (when only a random subset of the entries in the data matrix are observed). We prove consistency of the DOA estimate when signal from a single source is impinging on low coherence arrays, and derive an analytic expression for the mean-squared-error (MSE) performance of MUSIC for the case of uniform linear arrays, in the large array and relatively large sample setting. Our analysis is mathematically justified in both the sample rich and deficient regimes. The expression for the MSE is a simple function of array geometry, signal-to-noise ratio (SNR), the fraction of entries observed, and the ratio of the number of sensors to number of snapshots. We derive a phase transition threshold which separates a regime where MUSIC is consistent from a regime where MUSIC is inconsistent. This threshold depends upon the SNR, the probability of observing entries in the data matrix, and number of sensors and snapshots in a simple manner which we make explicit.