Nonasymptotic performance analysis of ESPRIT and spatial-smoothing ESPRIT

TL;DR

This paper provides a nonasymptotic analysis of ESPRIT and spatial-smoothing ESPRIT, establishing bounds on frequency estimation errors with finite snapshots and SNR, supported by new matrix perturbation bounds and numerical validation.

Contribution

It introduces a novel nonasymptotic performance bound for ESPRIT methods with finite data and noise levels, extending classical matrix perturbation theory.

Findings

Frequency estimation error bounded by $C\frac{\max(\sigma, \sigma^2)}{\sqrt{L}}$ with high probability.

Results hold for finite snapshots and SNR, not just asymptotic regimes.

Extensions to MUSIC and SS-MUSIC are also demonstrated.

Abstract

This paper is concerned with the problem of frequency estimation from multiple-snapshot data. It is well-known that ESPRIT (and spatial-smoothing ESPRIT in presence of coherent sources or given limited snapshots) can locate the true frequencies if either the number of snapshots or the signal-to-noise ratio (SNR) approaches infinity. In this paper, we analyze the nonasymptotic performance of ESPRIT and spatial-smoothing ESPRIT with finitely many snapshots and finite SNR. We show that the absolute frequency estimation error of ESPRIT (or spatial-smoothing ESPRIT) is bounded from above by $C\frac{\max(\sigma, \sigma^2)}{\sqrt{L}}$ with overwhelming probability, where $\sigma^2$ denotes the Gaussian noise variance, $L$ is the number of snapshots and $C$ is a coefficient independent of $L$ and $\sigma^2$, if and only if the true frequencies can be localized by ESPRIT (or spatial-smoothing ESPRIT) without noise or with infinitely many snapshots. Our results are obtained by deriving new matrix perturbation bounds and generalizing the classical Schur product theorem, which may be of independent interest. Extensions to MUSIC and SS-MUSIC are also made. Numerical results are provided corroborating our analysis.