MUSIC for Multidimensional Spectral Estimation: Stability and Super-Resolution

This paper presents a performance analysis of the MUltiple SIgnal Classification (MUSIC) algorithm applied on D dimensional single-snapshot spectral estimation while s true frequencies are located on the continuum of a bounded domain. Inspired by the matrix pencil form, we construct a D-fold Hankel matrix from the measurements and exploit its Vandermonde decomposition in the noiseless case. MUSIC amounts to identifying a noise subspace, evaluating a noise-space correlation function, and localizing frequencies by searching the s smallest local minima of the noise-space correlation function. In the noiseless case, (2s)D measurements guarantee an exact reconstruction by MUSIC as the noise-space correlation function vanishes exactly at true frequencies. When noise exists, we provide an explicit estimate on the perturbation of the noise-space correlation function in terms of noise level, dimension D, the minimum separation among frequencies, the maximum and minimum amplitudes while frequencies are separated by 2 Rayleigh Length (RL) at each direction. As a by-product the maximum and minimum non-zero singular values of the multidimensional Vandermonde matrix whose nodes are on the unit sphere are estimated under a gap condition of the nodes. Under the 2-RL separation condition, if noise is i.i.d. Gaussian, we show that perturbation of the noise-space correlation function decays like √(log(#(N))/#(N)) as the sample size #(N) increases. When the separation among frequencies drops below 2 RL, our numerical experiments show that the noise tolerance of MUSIC obeys a power law with the minimum separation of frequencies.