Fast scaling in the residue number system

Copyright © 2009 IEEEA new scheme for precisely scaling numbers in the residue number system (RNS) is presented. The scale factor K can be any number coprime to the RNS moduli. Lookup table implementations are used as a basis for comparisons between the new scheme and scaling schemes from the literature. It is shown that new scheme decreases hardware complexity compared to previous schemes without affecting time complexity.Yinan Kong and Braden Phillip

Pipelined Two-Operand Modular Adders

Pipelined two-operand modular adder (TOMA) is one of basic components used in digital signal processing (DSP) systems that use the residue number system (RNS). Such modular adders are used in binary/residue and residue/binary converters, residue multipliers and scalers as well as within residue processing channels. The design of pipelined TOMAs is usually obtained by inserting an appriopriate number of latch layers inside a nonpipelined TOMA structure. Hence their area is also determined by the number of latches and the delay by the number of latch layers. In this paper we propose a new pipelined TOMA that is based on a new TOMA, that has the smaller area and smaller delay than other known structures. Comparisons are made using data from the very large scale of integration (VLSI) standard cell library

Systematic redundant residue number system codes: analytical upper bound and iterative decoding performance over AWGN and Rayleigh channels

The novel family of redundant residue number system (RRNS) codes is studied. RRNS codes constitute maximum–minimum distance block codes, exhibiting identical distance properties to Reed–Solomon codes. Binary to RRNS symbol-mapping methods are proposed, in order to implement both systematic and nonsystematic RRNS codes. Furthermore, the upper-bound performance of systematic RRNS codes is investigated, when maximum-likelihood (ML) soft decoding is invoked. The classic Chase algorithm achieving near-ML soft decoding is introduced for the first time for RRNS codes, in order to decrease the complexity of the ML soft decoding. Furthermore, the modified Chase algorithm is employed to accept soft inputs, as well as to provide soft outputs, assisting in the turbo decoding of RRNS codes by using the soft-input/soft-output Chase algorithm. Index Terms—Redundant residue number system (RRNS), residue number system (RNS), turbo detection

Efficient convolvers using the Polynomial Residue Number System technique

The problem of computing linear convolution is a very important one because with linear convolution we can mechanize digital filtering. The linear convolution of two N-point sequences can be computed by the cyclic convolution of the following 2N-point sequences. The original sequence padded with N zero’s each. The cyclic convolution of two N-point sequences requires multiplications and additions for its computation. A very efficient way of computing cyclic convolution of two sequences is by using the Polynomial Residue Number System (PRNS) technique. Using this technique the cyclic convolution of two N-point sequences can be computed using only N multiplications instead of N2 multiplications. This can be achieved based on some forward and inverse PRNS transformation mappings. These mappings rely on additions, subtractions and many scaling operations (multiplications by constants). The PRNS technique would lose a lot in value if these many scaling operations were difficultly implemented. In this thesis we will show how to calculate cyclic convolution of two sequences using the PRNS technique based on forward and inverse transformation mapping which rely on complement operations (negations), additions and rotation operations. These rotation operations do not require any computational hardware. Therefore the complicated hardware required for the scaling operations has now been substituted by rotators, which do not require any computational hardware

On the range of a covering function

Let {a_s(mod n_s)}_{s=1}^k (k>1) be a finite system of residue classes with the moduli n_1,...,n_k distinct. By means of algebraic integers we show that the range of the covering function w(x)=|{1\le s\le k: x=a_s (mod n_s)}| is not contained in any residue class with modulus greater one. In particular, the values of w(x) cannot have the same parity.Comment: 7 pages; to appear in J. Number Theor