Performance of Uniform and Sparse Non-Uniform Samplers In Presence of Modeling Errors: A Cramér-Rao Bound Based Study

Abstract

This paper evaluates the performance of uniform and sparse nonuniform sampling techniques (namely nested and coprime sampling) for line spectrum estimation, in presence of nonideal conditions such as perturbation in sampling instants, and limited data for computing statistical averages. Coprime and nested sampling are well-known deterministic sampling techniques that operate at rates significantly lower than Nyquist, and yet allow perfect reconstruction of the spectra of wide sense stationary signals. However, theoretical guarantees for these samplers assume ideal conditions such as synchronous sampling, and ability to perfectly compute statistical expectations. This paper studies the performance of coprime and nested samplers when these assumptions are violated. Using a general grid-based signal model that applies to both spatial and temporal line spectrum estimation, the effect of perturbations in sampling instants is evaluated by deriving fundamental Cramer-Rao Bounds (CRB) for line spectrum estimation with perturbed samplers. For the first time, simplified expressions for the Fisher Information matrix for perturbed coprime and nested samplers are derived, which explicitly highlight the role of coarray. Even in presence of perturbations, it is possible to resolve O(M <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) spectral lines under appropriate conditions on the size of the grid. The effect of finite data on the CRB is also studied, and necessary and sufficient conditions are derived to ensure that the CRB decreases monotonically to zero with the number of measurements, even when there are more sources than sensors. Finally, the theoretical results derived in this paper are supported by extensive numerical experiments.