Loading the bases: a new number representation with applications

Abstract

A number system has recently been introduced that uses two orthogonal bases (Double-base Number System-DBNS). In its direct form the system provides a very sparse two-dimensional number representation which appears, initially, to be a curiosity. After some research by our group, however, the number system has proved to have some interesting and potentially far-reaching applications. The number system has been extended to more than 2 bases and a logarithmic version, which we refer to as the Multi-dimensional Logarithmic Number System (MDLNS), has also proved useful for implementing digital filters. An important property of the MDLNS, that the computational complexity associated with each base reduces both as the number of bases and as the number of digits (or logarithmic components) increase, gives rise to some simple implementation procedures. In this article we will explore some of the theory of the linear and logarithmic systems associated with this new representation, and provide examples of applications in cryptography and digital filter implementation.