Large Dynamic Range Computations over Small Finite Rings

This paper presents a new multivariate mapping strategy for the recently introduced Modulus Replication Residue Number System (MRRNS). This mapping allows computation over a large dynamic range using replications of extremely small rings. The technique maintains the useful features of the MRRNS, namely: the ease of input coding; the absence of a Chinese Remainder Theorem inverse mapping across the full dynamic range; the replication of identical rings; the natural integration of complex data processing. The concepts are illustrated by a specific example of complex inner product processing associated with a radix-4 decimation in time FFT algorithm. A complete quantization analysis is performed and an efficient scaling strategy chosen based on the analysis. The example processor uses replications of three rings modulo 3, 5 and 7; the effective dynamic range is in excess of 32-bits. The paper also includes VLSI implementation strategies for the processor architecture which consists of arrays of massively parallel linear bit-level pipelines.