MUSIC for joint frequency estimation: Stability with compressive measurements

This paper studies the application of MUtiple Signal Classification (MUSIC) algorithm on Multiple Measurement Vector (MMV) problem for the purpose of frequency parameter estimation while s true frequencies are located in the continuum of a bounded domain and sensors are randomly selected from a Uniform Linear Array (ULA). The MUSIC algorithm amounts to identifying a noise subspace from measurements, forming a noise-space correlation function and searching the s smallest local minima of the noise-space correlation function. Under the assumption that the true frequencies are separated by at least one Rayleigh Length (RL), we show that with high probability the noise-space correlation function is stably perturbed by noise if the number of sensors n ~ O(s) up to a logarithmic factor by means of a compressive version of discrete Ingham inequalities. As the theory implies, our numerical experiments demonstrate that the reconstruction error of MUSIC with n random sensors makes little difference once n is above a point of transition.