Asymptotic Analysis of Multidimensional Jittered Sampling

Abstract

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 Marčenko-Pastur law as d 0 ! 1. 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.

Field Reconstruction in Sensor Networks With Coverage Holes and Packet Losses

“…Next, we approximate the MSE of the finite size system in (4) through an asymptotic model, which assumes the size of V to grow to infinity while the ratio of its number of rows to its number of columns tends to a finite limit, β, greater than zero, i.e., we assume lim n,m→∞ β n,m = β Indeed, in our recent works [8]- [10] it was shown that this asymptotic model provides a tight approximation of the MSE of the finite size system, already for small values of n and m. Under these conditions, we therefore define the asymptotic expression of the MSE as [10]:…”

“…In the plots, we compare the asymptotic empirical spectral distribution (AESD) f LSDs f λ,x ′ and f λ,u since an analytic expression of f λ,u is still unknown. However, in [8]- [10] it is shown that, already for small values of n, the AESD f (n)…”

“…We take as performance metric the mean square error (MSE) on the reconstructed field. As a reconstruction technique, we use linear filtering and we adopt the filter that minimizes the MSE (i.e., the LMMSE filter) [8]- [10]. The matrix representing the sampling system, in the following denoted by V, results to be a d-fold Vandermonde matrix 1 .…”

Bayesian Post-Processing Methods for Jitter Mitigation in Sampling

“…Due to the lack of attention to nonlinear postprocessing, it is not readily apparent from the literature that these linear estimators are far from optimal. The effects of jitter on linear MMSE reconstruction of bandlimited signals are discussed in [10] and extended to the asymptotic case and multidimensional signals in [11]. More recently, [12] uses a second-order Taylor series approximation to perform weighted least-squares fitting of a jittered random signal.…”

“…The MSE performance of the Bayes MMSE estimator as computed using the Gibbs/slice sampler is compared against both the unbiased linear MMSE estimator (20) and the no-jitter linear MMSE estimator (22), as well as the EM algorithm approximation to the ML estimator from [4]. The values of , , , , , and are determined for the average , , and using (10) and (11). The EM algorithm uses the true values of , , and , while the linear estimators and the Gibbs/slice sampler treat , , and as random variables.…”