An integral formula for large random rectangular matrices and its application to analysis of linear vector channels

TL;DR

This paper introduces a statistical mechanical framework for analyzing large random rectangular matrices in linear vector channels, providing an integral formula for performance characterization and an approximate decoding algorithm.

Contribution

It develops a novel integral formula for large random rectangular matrices and applies it to analyze and decode linear vector channels.

Findings

Characterization of channel performance using the integral formula.

Development of an approximate decoding algorithm.

Framework applicable to large system limits.

Abstract

A statistical mechanical framework for analyzing random linear vector channels is presented in a large system limit. The framework is based on the assumptions that the left and right singular value bases of the rectangular channel matrix $\bH$ are generated independently from uniform distributions over Haar measures and the eigenvalues of $\bH^{\rm T}\bH$ asymptotically follow a certain specific distribution. These assumptions make it possible to characterize the communication performance of the channel utilizing an integral formula with respect to $\bH$, which is analogous to the one introduced by Marinari {\em et. al.} in {\em J. Phys. A} {\bf 27}, 7647 (1994) for large random square (symmetric) matrices. A computationally feasible algorithm for approximately decoding received signals based on the integral formula is also provided.