On the Spectrum of Large Random Hermitian Finite-Band Matrices

TL;DR

This paper discusses the challenge of determining the limiting spectral distribution of large random Hermitian finite-band matrices, which are relevant in modeling certain communication channels, and reviews related recent work.

Contribution

It highlights the open problem of calculating the spectrum of these matrices and reviews recent information-theoretic approaches related to their applications.

Findings

Identifies the open problem of spectrum calculation for large finite-band matrices.

Reviews recent work on communication channels modeled by such matrices.

Lists characteristics and open questions about the limiting spectrum.

Abstract

The open problem of calculating the limiting spectrum (or its Shannon transform) of increasingly large random Hermitian finite-band matrices is described. In general, these matrices include a finite number of non-zero diagonals around their main diagonal regardless of their size. Two different communication setups which may be modeled using such matrices are presented: a simple cellular uplink channel, and a time varying inter-symbol interference channel. Selected recent information-theoretic works dealing directly with such channels are reviewed. Finally, several characteristics of the still unknown limiting spectrum of such matrices are listed, and some reflections are touched upon.