Theoretical Aspects of a Design Method for Programmable NMR Voters

TL;DR

This paper analyzes the mathematical properties of a matrix-based design method for programmable NMR voters, providing insights into their eigenvalues, eigenvectors, and system characteristics to enhance fault-tolerance and scalability.

Contribution

It presents a detailed mathematical analysis of the matrix properties underlying a scalable, programmable NMR voter design, including eigenvalues, eigenvectors, and their relation to voter outputs.

Findings

Explicit characteristic polynomials for correct and erroneous matrices.

Eigenvalues and eigenvectors reveal system information.

Relations established between voter outputs and eigenpairs.

Abstract

Almost all dependable systems use some form of redundancy in order to increase fault-tolerance. Very popular are the $N$-Modular Redundant (NMR) systems in which a majority voter chooses the voting output. However, elaborate systems require fault-tolerant voters which further give additional information besides the voting output, e.g., how many module outputs agree. Dynamically defining which set of inputs should be considered for voting is also crucial. Earlier we showed a practical implementation of programmable NMR voters that self-report the voting outcome and do self-checks. Our voter design method uses a binary matrix with specific properties that enable easy scaling of the design regarding the number of voter inputs N. Thus, an automated construction of NMR systems is possible, given the basic module and arbitrary redundancy $N$. In this paper we present the mathematical aspects of the method, i.e., we analyze the properties of the matrix that characterizes the method. We give the characteristic polynomials of the properly and erroneously built matrices in their explicit forms. We further give their eigenvalues and corresponding eigenvectors, which reveal a lot of useful information about the system. At the end, we give relations between the voter outputs and eigenpairs.