Nonasymptotic Performance Analysis of Direct-Augmentation and Spatial-Smoothing ESPRIT for Localization of More Sources Than Sensors Using Sparse Arrays

TL;DR

This paper provides a finite-snapshot performance analysis of DA-ESPRIT and SS-ESPRIT methods, showing how their localization errors decrease with more snapshots and noise conditions, enabling more sources than sensors with sparse arrays.

Contribution

It offers the first nonasymptotic bounds for DA-ESPRIT and SS-ESPRIT, linking localization accuracy to snapshot number and noise, and demonstrates their resolution improves with more data.

Findings

Localization errors decrease as 1/√L with more snapshots

Exact source localization requires infinite snapshots

Resolution improves with increasing snapshots

Abstract

Direction augmentation (DA) and spatial smoothing (SS), followed by a subspace method such as ESPRIT or MUSIC, are two simple and successful approaches that enable localization of more uncorrelated sources than sensors with a proper sparse array. In this paper, we carry out nonasymptotic performance analyses of DA-ESPRIT and SS-ESPRIT in the practical finite-snapshot regime. We show that their absolute localization errors are bounded from above by $C_1\frac{\max\{\sigma^2, C_2\}}{\sqrt{L}}$ with overwhelming probability, where $L$ is the snapshot number, $\sigma^2$ is the Gaussian noise power, and $C_1,C_2$ are constants independent of $L$ and $\sigma^2$, if and only if they can do exact source localization with infinitely many snapshots. We also show that their resolution increases with the snapshot number, without a substantial limit. Numerical results corroborating our analysis are provided.