On the Sample Complexity of Multichannel Frequency Estimation via Convex Optimization

TL;DR

This paper demonstrates that increasing the number of channels in multichannel frequency estimation reduces the required sample size per channel for exact recovery using atomic norm minimization, under mild conditions.

Contribution

It provides a theoretical analysis showing that the sample complexity per channel decreases as the number of channels increases in multichannel frequency estimation.

Findings

Sample size per channel decreases with more channels for exact estimation.

Order $Kig( ext{log }Kig)ig(1+rac{1}{L} ext{log }Nig)$ samples suffice.

Numerical results support the theoretical analysis.

Abstract

The use of multichannel data in line spectral estimation (or frequency estimation) is common for improving the estimation accuracy in array processing, structural health monitoring, wireless communications, and more. Recently proposed atomic norm methods have attracted considerable attention due to their provable superiority in accuracy, flexibility and robustness compared with conventional approaches. In this paper, we analyze atomic norm minimization for multichannel frequency estimation from noiseless compressive data, showing that the sample size per channel that ensures exact estimation decreases with the increase of the number of channels under mild conditions. In particular, given $L$ channels, order $K\left(\log K\right) \left(1+\frac{1}{L}\log N\right)$ samples per channel, selected randomly from $N$ equispaced samples, suffice to ensure with high probability exact estimation of $K$ frequencies that are normalized and mutually separated by at least $\frac{4}{N}$. Numerical results are provided corroborating our analysis.