On the d-dimensional Quasi-Equally Spaced Sampling

TL;DR

This paper analyzes the eigenvalue distribution of certain random matrices arising in signal processing, showing convergence to the Marcenko-Pastur law as the dimension increases, with implications for signal reconstruction error estimation.

Contribution

It derives the moments of the eigenvalue distribution for d-dimensional quasi-equally spaced matrices and proves their convergence to the Marcenko-Pastur law as dimension grows.

Findings

Eigenvalue distribution moments derived for large matrices

Distribution converges to Marcenko-Pastur law as d->infinity

Applications in estimating mean square error in signal reconstruction

Abstract

We study a class of random matrices that appear in several communication and signal processing applications, and whose asymptotic eigenvalue distribution is closely related to the reconstruction error of an irregularly sampled bandlimited signal. We focus on the case where the random variables characterizing these matrices are d-dimensional vectors, independent, and quasi-equally spaced, i.e., they have an arbitrary distribution and their averages are vertices of a d-dimensional grid. Although a closed form expression of the eigenvalue distribution is still unknown, under these conditions we are able (i) to derive the distribution moments as the matrix size grows to infinity, while its aspect ratio is kept constant, and (ii) to show that the eigenvalue distribution tends to the Marcenko-Pastur law as d->infinity. These results can find application in several fields, as an example we show how they can be used for the estimation of the mean square error provided by linear reconstruction techniques.