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

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.