Random Matrix Models, Double-Time Painlev\'e Equations, and Wireless Relaying

TL;DR

This paper analyzes the statistical properties of the SNR in a wireless relay system using advanced mathematical models, deriving new differential equations and approximations to better understand system performance.

Contribution

It introduces two novel methods—Painlevé equations and Coulomb Fluid approximation—to characterize the SNR distribution in wireless relaying systems.

Findings

Exact Painlevé V characterization of the Hankel determinant

Closed-form cumulant expressions for SNR

Performance metrics derived from new mathematical models

Abstract

This paper gives an in-depth study of a multiple-antenna wireless communication scenario in which a weak signal received at an intermediate relay station is amplified and then forwarded to the final destination. The key quantity determining system performance is the statistical properties of the signal-to-noise ratio (SNR) \gamma\ at the destination. Under certain assumptions on the encoding structure, recent work has characterized the SNR distribution through its moment generating function, in terms of a certain Hankel determinant generated via a deformed Laguerre weight. Here, we employ two different methods to describe the Hankel determinant. First, we make use of ladder operators satisfied by orthogonal polynomials to give an exact characterization in terms of a "double-time" Painlev\'e differential equation, which reduces to Painlev\'e V under certain limits. Second, we employ Dyson's Coulomb Fluid method to derive a closed form approximation for the Hankel determinant. The two characterizations are used to derive closed-form expressions for the cumulants of \gamma, and to compute performance quantities of engineering interest.