Unitary Precoding and Basis Dependency of MMSE Performance for Gaussian Erasure Channels

TL;DR

This paper analyzes the MMSE performance of Gaussian vector sources over erasure channels under unitary encoding, revealing basis-dependent effects and providing bounds and explicit solutions for various sampling scenarios.

Contribution

It introduces a detailed analysis of basis dependence in MMSE performance with explicit solutions and bounds for unitary encoders in Gaussian erasure channels.

Findings

Eigenvalue distribution influences MMSE bounds.

DFT performance varies with eigenvalue spread.

Sampling rate impacts error in wide-sense stationary signals.

Abstract

We consider the transmission of a Gaussian vector source over a multi-dimensional Gaussian channel where a random or a fixed subset of the channel outputs are erased. Within the setup where the only encoding operation allowed is a linear unitary transformation on the source, we investigate the MMSE performance, both in average, and also in terms of guarantees that hold with high probability as a function of the system parameters. Under the performance criterion of average MMSE, necessary conditions that should be satisfied by the optimal unitary encoders are established and explicit solutions for a class of settings are presented. For random sampling of signals that have a low number of degrees of freedom, we present MMSE bounds that hold with high probability. Our results illustrate how the spread of the eigenvalue distribution and the unitary transformation contribute to these performance guarantees. The performance of the discrete Fourier transform (DFT) is also investigated. As a benchmark, we investigate the equidistant sampling of circularly wide-sense stationary (c.w.s.s.) signals, and present the explicit error expression that quantifies the effects of the sampling rate and the eigenvalue distribution of the covariance matrix of the signal. These findings may be useful in understanding the geometric dependence of signal uncertainty in a stochastic process. In particular, unlike information theoretic measures such as entropy, we highlight the basis dependence of uncertainty in a signal with another perspective. The unitary encoding space restriction exhibits the most and least favorable signal bases for estimation.