Ramasamy Krishnan

Complex digital signal processing using quadratic residue number systems

Recently, the quadratic residue number system (QRNS) has been introduced [6], [7] which allows the multiplication of complex integers with two real multiplications. The restriction is that the number system has either all prime moduli of the form 4K + 1, or composite numbers with prime factors of that form. If an increase in real multiplications from two to three can be tolerated, then the restriction can be lifted to allow moduli of any form: the resulting number system is termed the modified quadratic residue number system (MQRNS). In this paper, the MQRNS is defined, and residue-to-binary conversion techniques in both the QRNS and MQRNS are presented. Hard-ware implementations of multiplication intensive, complex nonrecursive, and recursive digital filters are also presented in this paper where the QRNS and MQRNS structures are realized using a bit-slice architecture.

The modified quadratic residue number system (MQRNS) for complex high-speed signal processing

Recently, the Quadratic Residue Number System ,(QRNS) has been introduced [5], [6], which allows the multiplication of complex integers with two real multiplications. The restriction is that this special residue number system has all moduli with prime factors of the form 4 k + 1 . If an increase in real multiplications from two to three can be tolerated, the restriction can be lifted to allow moduli of any form.

Implementation of complex number theoretic transforms using quadratic residue number systems

Very recently, the Quadratic Residue Number System (QRNS) has been introduced [6], [7]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. The conjugate ring has the remarkable property that both addition and multiplication are performed component-wise. Complex multiplication only requires two base field multiplications and zero additions; this contrasts to the requirement of four multiplications and two additions, or three multiplications and five additions for the Gaussian integer ring. The operations are performed over sub-rings, isomorphic to the conjugate ring, and the results mapped to the conjugate ring via the Chinese Remainder Theorem isomorphism. The primary restriction is the limited form of the moduli set for RNS computations. The QRNS has since been generalized for any type of moduli set with an increase in multiplications from 2 to 3 and the resulting number system has been termed the Modified Quadratic Residue Number System (MQRNS) [3], [4]. In [4] the direct FIR filter architecture and bit-slice architecture for FIR and recursive digital filters have been presented using the QRNS and MQRNS. In this paper, the computation of the Complex Number Theoretic Transform (CNTT) and the hardware implementation of a radix-2 butterfly structure, using high-density ROM arrays, are presented. This paper shows that both the QRNS and MQRNS require almost the same amount of hardware for the implementation of the butterfly structure. The computation of Cyclic Convolution in both the QRNS and MQRNS is discussed.

Computation of complex number theoretic transforms using quadratic residue number systems

Very recently, the Quadratic Residue Number System (QRNS) has been introduced [4,5]. The QRNS is obtained from a mapping of Gaussian integers over a finite ring to a ring of conjugate elements. The conjugate ring has the remarkable property that both addition and multiplication are performed component-wise, therefore complex multiplication only requires two base field multiplications and zero additions. The operations are performed over sub-rings, isomorphic to the conjugate ring via the Chinese Remainder Theorem isomorphism. The primary restriction is the limited form of the moduli set for RNS computations. The QRNS has since been generalized for any type of moduli set with an increase in multiplications from 2 to 3 and the resulting number system has been termed the Modified Quadratic Residue Number System (MQRNS) [1,2]. The direct FIR filter architecture and bit-slice architecture for FIR and recursive digital filters have, been presented using the QRNS and MQRNS [4]. In this paper, the computation of the Complex Number Theoretic Transform(CNTT) and the hardware implementation of a radix-2 butterfly structure, using high-density ROM arrays, are presented. This paper shows that both theQRNS and MQRNS require almost the same amount of hardware for the implementation of the butterfly structure. The computation of Cyclic Convolution in both the QRNS and MQRNS is also discussed.

Computation of generalized FIR filter structure using the modified quadratic residue number system

The systematic approach for generating the generalized number theoretic FIR filter structure based on the complex number theoretic z-transform is presented. A step-by-step computational method that can be used in the implementation and simulation of the filter structure is discussed. The residue number system (RNS) is briefly reviewed. >

A lay-out level comparison of RNS and BNS systolic architectures for complex digital filtering

Algorithms are developed to compute the number of active devices for residue number system (RNS) and binary number system (BNS) signal processing architectures. The precharged CMOS technology is used in order to build the filter architecture. The ROM lookup table and twos complement multiplier are used as the computational cells in the RNS and BNS, respectively. A direct implementation transversal systolic filter architecture is considered in RNS and BNS algorithms.<<ETX>>

Implementation of recursive and nonrecursive digital filters using the single multiplexed ROM in the quadratic residue number system

Recursive and nonrecursive digital filters have been implemented using the proposed single-multiplexed dual-clock computational module (SDCM) in the bit-slice architecture. The amount of memory requirements has been reduced to 50% required for the quadratic-residue-number system (QRNS) or modified-QRNS filter architecture. These filter architectures are suitable for single-chip VLSI implementation of complex digital filters. Though the data rate is reduced by 50%, VLSI implementation of the filter architectures using the RNS principle is certainly possible in this approach.<<ETX>>

Implementation of the generalized FIR filter structure using the residue arithmetic

Very recently, the Quadratic Residue System (QRNS) has been introduced [3,4,5]. Using the QRNS complex multiplication can be performed with two base field multiplication and zero additions. The primary restriction is the limited form of the moduli set for RNS operations. The QRNS has since been geralized for any type of moduli set with an increase in multiplication from 2 to 3 and the resulting number system has been termed Modified Quadratic Residue Number System (MQRNS) [1,2]. In [9] a recursive FIR filter has been developed using the Complex Number Theoretic z-transform (CNT z-transform). Recently, in [6], the implementation of this recursive FIR filter structure has been presented using the QRNS and the MQRNS. Extension of this implementation to generalized FIR filter (Lagrange) has also been briefly presented in [6]. In this paper, we consolidate the implementation aspects of the generalized FIR filter using the MQRNS and also prove that the QRNS is not a suitable medium for the implementation.

VLSI Modular architectures for complex digital signal processing applications

Recently, the Quadratic Residue Number System (QRNS)[3,4] and Modified Quadratic Residue Number System (MQRNS)[1,2] have been introduced to perform complex multiplications efficiently. The growing number of complex digital signal processing applications will be implemented more efficiently and economically by using Very Large Scale Integration (VLSI) technology. In this paper we discuss VLSI implementation of complex multiplication using the QRNS and MQRNS. We also concentrate on the aspects of VLSI implementation of Finite Impulse Response (FIR) filter architectures.

The implementation of the generalized Lagrange FIR filter structure defined over finite fields or rings

This paper discusses the use of the Complex Number Theoretic z-transform in implementing a recursive FIR filter structure for frequency samples spaced around the unit circle in the complex number theoretic z-domain. The complex arithmetic operations have been implemented using the Quadratic Residue Number System (QRNS) and Modified Quadratic Residue Number System (MQRNS) for uniformly spaced frequency samples around the unit circle. We discuss the extension of this technique to non-uniformly spaced samples around the unit circle and the resulting filter structure has been termed the generalized number theoretic FIR filter structure. We demonstrate that the MQRNS is the only suitable tool in implementing this recursive FIR filter structure for non-uniformly spaced frequency samples.