Average Case Analysis of Compressive Multichannel Frequency Estimation Using Atomic Norm Minimization

Compressive multichannel frequency estimation refers to the process of retrieving the frequency profile shared by multiple signals from their compressive samples. A recent approach to this problem relies on atomic norm minimization which exploitsjoint sparsity among the channels, is solved using convex optimization, and has strong theoretical guarantees. We provide in this paper an average-case analysis for atomic norm minimization by assuming proper randomness on the amplitudes of the frequencies. We show that the sample size per channel required for exact frequency estimation from noiseless samples decreases as the number of channels increases and is on the order of $K\displaystyle \log K\left(1+\frac{1}{L}\log N\right)$, where K is the number of frequencies, L is the number of channels, and N is a fixed parameter proportional to the sampling window size and inversely proportional to the desired resolution.