Efficient scaling in the residue number system

Abstract

In this note a unified Residue Number System scaling technique is presented which allows the designer a great deal of flexibility in choosing the scale factor. The technique is based on the L(E + 6)-CRT , a generalization of the work descibed in [7], [9], and [l-41. By embedding the scaling process in the Chinese Remainder Theorem, the L(e + 6)-CRT can also be used to simplify the residue-te analog conversion problem. The flexibility in choosing the scale factor and new reduced system modulus comes at the cost of potentially large errors, however. We show that the errors induced by the L(e + 6)-CRT can be divided into two distinct bands; a, band of small errors and a band of errors on the order of the reduced system modulus. For a given scale factor, we give inequalities that allow us to choose a reduced system modulus so that the large error band is avoided.

The MasPar MP-1 as a computer arithmetic laboratory

“…A sample of these are listed in the Residue Number Systems section of the Bibliography, Refs. [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

RNS Scaling Algorithm for a New Moduli Set {2^(2n+1) +1, 2^(2n+1), 2^(2n+1)-1}

“…In other words, their design provided more accurate scaled residues at the cost of more clock cycles. Two scaling algorithms for the moduli set (2 n -1, 2 n , 2 n + 1) using CRT were proposed in 1989 [15]. The first algorithm was based on the L-CRT algorithm for L-tuple RNS.…”

“…It was suitable for small-word-length applications and performed scaling directly on the residue digits rather than relying on residue-tobinary conversion. From the implementation point of view, scaling algorithms are implemented either in LUT (look up table) based approaches [7], [15], [16], [17], [21], [22], [23], or adder-based approaches [5]. Generally, all the LUT-based designs in the literature are subject to poor pipeline-ability and high hardware complexity when the number of moduli increases.…”

RNS-based Watermarking for IP Cores

“…Thus, this FIR filter includes 2C-to-index conversion and eight index-based Galois field parallel RNS channels, including index-to-RNS conversion tables. Efficient block decomposition 2C-to-RNS [19] and £-CRT-based [15] RNS-to-2C conversions are used to optimize area and performance of both filters.…”