Analysis of "SIR" ("Signal"-to-"Interference"-Ratio) in discrete-time autonomous linear networks with symmetric weight matrices

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 discretetime autonomous linear systems: 1) The system x(k + 1) = I + α(−rI +W) x(k) which is obtained by discretizing the autonomous continuous-time linear system in [9] using Euler method; where I is the identity matrix, r is a positive real number, and α > 0 is the step size. 2) A more general autonomous linear system descibed by x(k + 1) = −ρI + Wx(k), where W is any real symmetric matrix whose diagonal elements are zero, and I denotes the identity matrix and ρ 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 ρ/λmax where λmax is the maximum (positive) eigenvalue of the matrix W. These results are in line with those of [9] where corresponding continuous-time linear system is examined. 3) The ”Ultimate SIR” in the second system above is equal to ρ/λm where λm is the eigenvalue of W which satisfy |λm − ρ| = max{|λi − ρ|}N i=1 if ρ is accordingly determined from the interval 0 < ρ < 1.