Computation of the robust symmetrical number system dynamic range

Abstract

The robust symmetrical number system (RSNS) is a number theoretic transform formed using N ⋛ 2 integer sequences and ensures that two successive RSNS vectors (paired terms from all N sequences) differ by only one integer — integer Gray code property. The dynamic range M of the RSNS is defined as the greatest length of combined sequences that contain no ambiguities or repeated paired terms. For all but a select few RSNS sequences there is no closed-form solution to compute the dynamic range and its position. This paper presents an efficient algorithm for computing the dynamic range and its position. The dynamic range is shown to satisfy M < Pf where Pf is the RSNS fundamental period Pf = 2N Πmi. It then follows that M < M where M = Πmi is the dynamic range of the residue number system. An example is presented to demonstrate the algorithm. The efficiency of the algorithm is examined by comparing the speed of computation to a naive search algorithm (using MATLAB on a PC).