Alexander Skavantzos

Implementation issues of the two-level residue number system with pairs of conjugate moduli

One of the most important considerations when designing residue number systems (RNSs) is the choice of the moduli set; this is due to the fact that the dynamic range of the system, its speed, as well as its hardware complexity, depend on both the forms as well as the number of moduli chosen; In this paper, a new class of multimoduli RNSs based on sets of forms {2/sup n(1)/-1, 2/sup n(1)/+1, 2/sup n2/-1, 2/sup n(2)/+1, /spl middot//spl middot//spl middot/, 2/sup n(L)/-1, 2/sup n(L)/+1} is presented. The moduli 2/sup n(i)/-1 and 2/sup n(i)/+1 are called conjugates of each other. The new RNSs that rely on pairs of conjugate moduli result in hardware-efficient two-level implementations for the weighted-to-RNS and RNS-to-weighted conversions, achieve very large dynamic ranges, and imply fast and efficient RNS processing. When compared with conventional systems of the same number of moduli and the same dynamic range, the proposed new systems offer the following benefits: (1) hardware savings of 25 to 40% for the weighted-to-RNS conversion and (2) a reduction of over 80% in the complexity of the final Chinese remainder theorem (CRT) involved in the RNS-to-weighted conversion. Thus, significant compromises between large dynamic ranges, fast internal processing, and low complexity are achieved by the new systems.

An efficient residue to weighted converter for a new residue number system

The Residue Number System (RNS) is an integer system appropriate far implementing fast digital signal processors since it can support parallel, carry-free, highspeed arithmetic. In this paper a new RNS system and an efficient implementation of its residue-to-weighted converter are presented. The new RNS is a balanced 5-moduli system appropriate for large dynamic ranges. The new residue-to-binary converter is very fast and hardware-efficient and is based on a 1s complement multioperand adder adding operands of size only 80% of the size of the systems dynamic range.

New multipliers modulo 2/sup N/-1

Techniques for computing the product of two N-bit integers modulo 2/sup N/-1 from their k-bit byte decompositions are presented. A modulus 2/sup N/-1 is chosen, as multiplication performed in this modulus can be reconstructed from the cyclic convolution between the sequences of the k-bit bytes of the decomposed numbers. It is shown that cyclic convolutions can be computed using only additions and squaring operations but not two-operand multiplications. Since the squaring operation is a one-operand operation, significant savings in ROM bits can be obtained if look-up tables are used. >

On the polynomial residue number system (digital signal processing)

The theory of the polynomial residue number system (PRNS), a system in which totally parallel polynomial multiplication can be achieved provided that the arithmetic takes place in some carefully chosen ring, is examined. Such a system is defined by a mapping which maps the problem of multiplication of two polynomials onto a completely parallel scheme where the mapped polynomial coefficients are multiplied pairwise. The properties of the mapping and the rules of operations in the PRNS are proven. Choices of the rings for which the PRNS is defined are also studied. It is concluded that the PRNS can offer significant advantages in those digital signal processing (DSP) applications that involve multiplication-intensive algorithms like convolutions and one-dimensional or multidimensional correlation. >

Full adder-based arithmetic units for finite integer rings

Most implementations of accumulators, multipliers, or multiplier-accumulator units, operating in a finite integer ring, R(m), are based on ROMs or PLAs. This paper proposes a full adder-based arithmetic unit, called an (FA)-based AU/sub m/, capable of performing both addition and general multiplication at the same time, in R(m). For all moduli, FA-based AU/sub m/s are shown to execute much faster and have much less hardware complexity and smaller time-complexity products than ROM-based AU/sub m/s. For large values of m, they are also shown to be less complex and have smaller time-complexity products than ROM-based units, which are capable of performing multiplication only by a constant. Since the proposed units use full adders as the basic building block, they result in easy-to-design, modular, and regular VLSI implementations. >

Grouped-moduli residue number systems for fast signal processing

In this paper a new class of multi-moduli residue number systems (RNS) and their efficient RNS-to-weighted converters are presented. The new RNS systems are based on sets consisting of two groups of moduli with the moduli-product within one group being a form 2/sup a/(2/sup b/-1) with the product of the moduli within the other group being a form 2/sup c/-1. The new RNS-to-weighted converters are based on efficient combinations of the Chinese remainder theorem and mixed radix conversion (MRC) decoding techniques. Systems based on four, five and seven moduli are studied. The new systems allow very efficient implementations for their RNS-to-weighted decoders, imply fast and balanced RNS arithmetic and achieve large dynamic ranges.

Polynomial residue complex signal processing

The polynomial residue number system (PRNS) is a system in which the product of two polynomials can take place in parallel and with the minimum number of multiplications. The system is an extension of the quadratic residue number system (QRNS) which has been successfully used in complex digital signal processing. It is shown that an N-point complex linear convolution can be computed with 4N real multiplications when using the PRNS instead of 22 real multiplications when using the QRNS. The savings in the number of multiplications occur if some restrictions are placed on the modular ring used for performing the complex residue number system operations. >

A systematic approach for selecting practical moduli sets for residue number systems

The residue number system (RNS) is a useful tool for digital signal processing (DSP) since it can support parallel, carry free, high speed arithmetic. An RNS is defined by a set of relatively prime integers called the moduli set. The most important consideration when designing RNS systems is the choice of the moduli set. In order to maintain simple arithmetic, several example cases of moduli sets containing numbers of the forms 2(/sup k/1)+1, 2(/sup k/2)-1 and 2(/sup k/3) have been considered and studied by RNS researchers in the past. However, there is a lack of a comprehensive theory of properties of numbers of the forms 2(/sup k/1)+1 and 2(/sup k/2)-1 and of how these numbers can be used as moduli choices for RNS systems. A detailed and comprehensive theoretical study of properties of numbers of the forms 2(/sup k/1)+1 and 2(/sup k/2)-1 is presented. This study will enable RNS researchers and engineers to make the very best moduli selections for RNS systems.<<ETX>>

Parallel Decomposition Of Complex Multipliers

This paper discusses the mathematical basis and hardware implementations of new complex multipliers. They are based on polynomial approximations of complex numbers and their processing via a recently developed parallel arithmetic system. The new multipliers allow a variety of implementation options and are shown to exhibit performance up to 3.5 times better than traditional techniques in a multiplicative intensive environment.

On the binary quadratic residue system with noncoprime moduli

The residue number system (RNS) appropriate for implementing fast digital signal processors since it can support parallel, carry-free, high-speed arithmetic. A development in residue arithmetic is the quadratic residue number system (QRNS), which can perform complex multiplications with only two integer multiplications instead of four. An RNS/QRNS is defined by a set of relatively prime integers, called the moduli set, where the choice of this set is one of the most important design considerations for RNS/QRNS systems. In order to maintain simple QRNS arithmetic, moduli sets with numbers of forms 2/sup n/+1 (n is even) have been considered. An efficient such set is the three-moduli set (2/sup 2k-2/+1.2/sup 2k/+1.2/sup 2k+2/+1) for odd k. However, if large dynamic ranges are desirable, QRNS systems with more than three relatively prime moduli must be considered. It is shown that if a QRNS set consists of more than four relatively prime moduli of forms 2/sup n/+1, the moduli selection process becomes inflexible and the arithmetic gets very unbalanced. The above problem can be solved if nonrelatively prime moduli are used. New multimoduli QRNS systems are presented that are based on nonrelatively prime moduli of forms 2/sup n/+1 (n even). The new systems allow flexible moduli selection process, very balanced arithmetic, and are appropriate for large dynamic ranges. For a given dynamic range, these new systems exhibit better speed performance than that of the three-moduli QRNS system.