Classification of Moduli Sets for Residue Number System With Special Diagonal Functions

Abstract

The paper presents algorithms for the generation of Residue Number System (RNS) triples with = 2 − 1 and quadruples with = 2 for some k. Triples and quadruples allow us to design efficient hardware implementations of non-modular operations in RNS such as division, sign detection, comparison of numbers, reverse conversion with using of a diagonal function from requiring division with the remainder by the diagonal module SQ. Division with a remainder in the general case is the most complex arithmetic operation in computer technology. However, the consideration of special cases can significantly simplify this operation and increase the efficiency of hardware implementation. We show that there are 5573 good RNS triples (2301 even and 2372 odd) with elements less than 10 000, as the values of SQ vary from 2 5 − 1 to 2 27 − 1. In contrast, RNS quadruples with = 2 seem to be quite rare. Restricting our search to sums of the elements in a quadruple less than 4000 we find that exactly 31 such quadruples exist. Their values of SQ vary between 2 20 and 2 30 with always even exponent. We suggest the measure of RNS balance and find perfectly balanced RNS among triples according to this measure. We demonstrate the advantages of more balanced quadruples by means of hardware implementation.

Technique to Adjust Adaptive Digital Filter Coefficients in Residue Number System Based Filters

“…Given the advantages and disadvantages of the considered adaptation algorithms, we can conclude that RNS is suitable to improve performance when adjusting the coefficients and reduce the computational complexity of the tuning procedure [22]. These results can be used for building effective parallel computational systems [13] based on computers with parallel structure like FPGA [17,18] and GPU [23,24].…”

“…These limitations exist because the RNS is a nonpositional number system, and numbers magnitude comparison in the RNS form is impossible, so the division operation consists of a magnitude comparison operation that is also a problematic operation. Attempts to solve this problem are still being undertaken by various scientists in different directions [16][17][18], but there is still no universal 2 solution suitable for any tasks. As a result, there are currently no coefficients adjustment algorithms for ADF that are implemented in the RNS.…”

Sign Detection and Signed Integer Comparison for the 3-Moduli Set {2^n±1,2^(n+k)}

“…The absolute value of X equals M − X = 60 − 58 = 2; since X is negative, we have X = (1, 2, 3) = −2. Several approaches have attempted to develop RNS comparison methods (mostly for unsigned numbers) [3,5,10,12,[22][23][24]27]. Some other works compared signed numbers from a different perspective in which the dynamic range included both positive and negative numbers [13,20].…”

“…Such a method exploits the unique number theoretic properties of the moduli. Some other methods have proposed the diagonal function [3,5] to compare RNS numbers. A diagonal number based on (1) is assigned to each number in the dynamic range.…”

“…Difficult operations are often required for several nonlinear procedures, such as median and rank-order filtering [20]. Several RNS comparison [3,5,10,12,13,20,[22][23][24]27], sign-detection [9], and division [25] methods have been proposed over the past two decades. As regarding the role of comparison in other complicated operations such as division, sign, and overflow detection, it is expected that a cost-effective and high-speed implementation of the comparison will assist in improving other complicated operations.…”