A 2 n scaling scheme for signed RNS integers and its VLSI implementation

Abstract

High efficient implementation of scaling in residue number system (RNS) is one of the critical issues for the applications of RNS in digital signal processing (DSP) systems. In this paper, an efficient scaling algorithm for signed integers in RNS is proposed firstly through introducing a correction constant in negative integers scaling procedure. Based on the proposed scaling algorithm, an efficient RNS 2 n scaling implementation method is presented, in which Chinese remainder theorem (CRT) and a redundant modulus are used to perform the base extension to obtain the least significant n bits of RNS integers. With the redundant modulus, the RNS sign detection can be achieved by the parity detection. And then, an approach to update the residue digit of the redundant channel is also proposed. Meanwhile, this paper provides a method of computing the correction constant of the redundant channel in negative integers scaling. The analysis results indicate that the complexity of the proposed scaling algorithm grows linearly with the word-length of the RNS dynamic range without using Look-up Table (LUT). Furthermore, the proposed algorithm is employed for a specific moduli set 2 n scaling. The synthesis results show that the critical path of the proposed algorithm is shortened by 12%, the area and power consumption performance is improved by about 35%, compared to the existing cascading 2 n scaling method for very large scale integration (VLSI) implementation under the same restriction. Besides, the VLSI layout indicates that the parallel structure is simpler. CitationMa S, Hu J H, Ye Y L, et al. A 2 n scaling scheme for signed RNS integers and its VLSI implementation.

A PAPR reduction scheme with residue number system for OFDM

“…The definition of a signed number in RNS is similar to that in two's complement system (TCS) [13,22], that is, an integer R in the legitimate range [ 0, M I ) can be represented as a signed number, R. Thus, if 0 ≤ R < M I /2 or M I /2 ≤ R < M I , R is positive or negative, respectively, where x denotes the smallest integer larger than x.…”

“…Thus, RNS has received wide attention in very large scale integration applications. The activities of RNS focus on RNSto-binary conversion, RNS parity check, and RNS scaling scheme [11][12][13]. Recently, more attention is also paid to RNS in a parallel communication field because of its parallel and fault-tolerant properties [14][15][16][17][18][19].…”

A VLSI Efficient Programmable Power-of-Two Scaler for $\{2^{n}-1,2^{n},2^{n}+1\}$ RNS

“…(20) By substituting from [17] (this expression and (11) of [17] can also be derived directly from mixed radix conversion (MRC) and were embraced in some solutions of the reverse conversion problem [1], [33]) into (20) and using the property , we have (21) It can be easily shown that the multiplicative inverse of modulo is . Hence, .…”

“…Scaling by different power-of-two factors in RNS, also known as truncation in a two's complement number system, has been widely used in different parts of the datapath of a large system to reduce the dynamic range and hardware complexity. It is also commonly used to prevent overflow error in DSP entailing an iterative structure of a large number of multiplications and additions, such as FIR and IIR filters [20]- [22]. To minimize the risk of over-scaling, adaptive scaling is a preferred technique [19] whereby the scaling factor, usually a power of two, is determined at run time.…”

“…Efficient scaling algorithms in RNS have been developed and can be categorized into three main groups [20]: scaling by subset of the moduli set [17], [21], [23]- [26], scaling by a factor that is co-prime with the moduli set [20], [22], [27], and scaling by the power-of-two [17], [28], [29]. Although scaling by a fixed factor in RNS is not without difficulty when it comes to hardware implementation, the problem has been adequately solved for the three moduli set with a scaling factor [17].…”