Analysis of "SIR" ("Signal"-to-"Interference"-Ratio) in Discrete-Time Autonomous Linear Networks with Symmetric Weight Matrices

TL;DR

This paper analyzes the Signal-to-Interference Ratio (SIR) in discrete-time autonomous linear systems with symmetric weight matrices, showing that SIR converges to a constant value related to system eigenvalues, extending previous continuous-time results.

Contribution

It introduces a novel analysis of SIR in discrete-time linear systems with symmetric weights, revealing convergence properties and explicit formulas for the ultimate SIR.

Findings

SIR converges to a constant called 'Ultimate SIR' in finite time.

Ultimate SIR equals the ratio of rho to the maximum eigenvalue of W.

Results align with previous continuous-time system analyses.

Abstract

It's well-known that in a traditional discrete-time autonomous linear systems, the eigenvalues of the weigth (system) matrix solely determine the stability of the system. If the spectral radius of the system matrix is larger than 1, then the system is unstable. In this paper, we examine the linear systems with symmetric weight matrix whose spectral radius is larger than 1. The author introduced a dynamic-system-version of "Signal-to-Interference Ratio (SIR)" in nonlinear networks in [7] and [8] and in continuous-time linear networks in [9]. Using the same "SIR" concept, we, in this paper, analyse the "SIR" of the states in the following two $N$-dimensional discrete-time autonomous linear systems: 1) The system ${\mathbf x}(k+1) = \big({\bf I} + \alpha (-r {\bf I} + {\bf W}) \big) {\mathbf x}(k)$ which is obtained by discretizing the autonomous continuous-time linear system in \cite{Uykan09a} using Euler method; where ${\bf I}$ is the identity matrix, $r$ is a positive real number, and $\alpha >0$ is the step size. 2) A more general autonomous linear system descibed by ${\mathbf x}(k+1) = -\rho {\mathbf I + W} {\mathbf x}(k)$, where ${\mathbf W}$ is any real symmetric matrix whose diagonal elements are zero, and ${\bf I}$ denotes the identity matrix and $\rho$ is a positive real number. Our analysis shows that: 1) The "SIR" of any state converges to a constant value, called "Ultimate SIR", in a finite time in the above-mentioned discrete-time linear systems. 2) The "Ultimate SIR" in the first system above is equal to $\frac{\rho}{\lambda_{max}}$ where $\lambda_{max}$ is the maximum (positive) eigenvalue of the matrix ${\bf W}$. These results are in line with those of \cite{Uykan09a} where corresponding continuous-time linear system is examined. 3) The "Ultimate SIR" ...