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

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" ...Comment: 35 pages, 4 figures, submitted to IEEE Trans. on Circuits and Systems 1 (TCAS1) in February 200

A Boolean Hebb rule for binary associative memory design

A binary associative memory design procedure that gives a Hopfield network with a symmetric binary weight matrix is introduced in this paper. The proposed method is based on introducing the memory vectors as maximal independent sets to an undirected graph, which is constructed by Boolean operations analogous to the conventional Hebb rule. The parameters of the resulting network is then determined via the adjacency matrix of this graph in order to find a maximal independent set whose characteristic vector is close to the given distorted vector. We show that the method provides attractiveness for each memory vector and avoids spurious memories whenever the set of given memory vectors satisfy certain compatibility conditions, which implicitly imply sparsity. The applicability of the design method is finally investigated by a quantitative analysis of the compatibility conditions

A Boolean Hebb rule for binary associative memory design

We propose a binary associative memory design method to be applied to a class of dynamical neural networks. The method is based on introducing the memory vectors as maximal independent sets to an undirected graph and on designing a dynamical network in order to find a maximal independent set whose characteristic vector is close to the given distorted vector. We show that our method provides the attractiveness for each memory vector and avoids the occurance of spurious states whenever the set of given memory vectors satisfies certain compatibility conditions. We also analyze the application of this design method to the discrete Hopfield network