Study of the Reverse Converters for the Large Dynamic Range Four-Moduli Sets

Abstract

The Residue Number System (RNS) is an efficient alternative number system which has been attracted researchers for over three decades. In RNS, arithmetic operations such as addition and multiplication can be performed on residues without carry-propagation between them; resulting in parallel arithmetic and high-speed hardware implementations (Parhami, 2000; Mohan, 2002; Omondi & Premkumar, 2007). Due to this feature, many Digital Signal Processing architectures based on RNS have been introduced in the literature (Soderstrand et al., 1986; Diclaudio et al., 1995; Chaves et al., 2004). In particular, RNS is an efficient method for the implementation of high-speed finite-impulse response (FIR) filters, where dominant operations are addition and multiplication. Implementation issues of RNSbased FIR filters show that performance can be considerably increased, in comparison with traditional two’s complement binary number system (Jenkins et al., 1977; Conway et al., 2004; Cardarilli et al., 2007). As described in (Navi et al., 2011) a typical RNS system is based on a moduli set which is included some pair-wise relatively prime integers. The product of the moduli is defined as the dynamic range, and it denotes the interval of integers which can be distinctively represented in RNS. The main components of an RNS system are a forward converter, parallel arithmetic channels and a reverse converter. The forward converter encodes a weighted binary number into a residue represented number, with regard to the moduli set; where it can be easily realized using modular adders or look-up tables. Each arithmetic channel includes modular adder, subtractor and multiplier for each modulo of set. The reverse converter decodes a residue represented number into its equivalent weighted binary number. The arithmetic channels are working in a completely parallel architecture without any dependency, and this results in a considerable speed enhancement. However; the overhead of forward and reverse converters can counteract this speed gain, if they are not designed efficiently. The forward converters can be designed using efficient methods. In contrast, design of reverse converters have many complexities with many important factors such as conversion algorithm, type and number of moduli. An efficient moduli set with moduli of the form of powers of two can greatly reduce the complexity of the reverse converter as well as arithmetic channels. Due to this, many different moduli sets have been proposed for RNS which can be categorized based on their