On the SIRs (Signal-to-Interference-Ratio) in Discrete-Time Autonomous Linear Networks

In this letter, we improve the results in [5] by relaxing the symmetry assumption and also taking the noise term into account. The author examines two discrete-time autonomous linear systems whose motivation comes from a neural network point of view in [5]. Here, we examine the following discrete-time autonomous linear system: ${\mathbf x}(k+1) = {\mathbf A} {\mathbf x}(k) + {\mathbf b}$ where ${\mathbf A}$ is any real square matrix with linearly independent eigenvectors whose largest eigenvalue is real and its norm is larger than 1, and vector ${\mathbf b}$ is constant. Using the same "SIR" ("Signal"-to-"Interference"-Ratio) concept as in [4] and [5], we show that the ultimate "SIR" is equal to $\frac{a_{ii}}{\lambda_{max} - a_{ii}}$, $i=1, 2, >..., N$, where $N$ is the number of states, $a_{ii}$ is the diagonal elements of matrix ${\bf A}$, and $\lambda_{max}$ is the (single or multiple) eigenvalue with maximum norm.