Fast complex arithmetic is essential to a wide class of real-time digital transform and filter operations. Traditional methods require a multiplicity of real multiplies and adds be performed on each complex product call. Recently, a new modular (residue) mathematical system has been proposed which more than halves this count. The system developed in this paper overcomes many of the classical weaknesses of a general L-moduli modular system. Such a system is architected, its hardware implementation developed, and complexity-performance metrics analyzed. Both general purpose and array processing forms are advanced and discussed. Special operations, such as complex magnitude squared calculations, are shown to have a particularly efficient implementation in this system. Last, several DSP applications of the developed complex ALU are motivated and experimentally studied.

A single modulus complex ALU for signal processing

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.

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Complex digital signal processing using quadratic residue number systems

Recent advancements in residue arithmetic have given rise to a complex number system variant which better than halves RNS multiplication complexity. This advantage is applied to the problem of implementing a high-speed radix-4 RNS FFT. It is shown that a significant improvement in both complexity and speed can be achieved.

A Radix-4 FFT Using Complex RNS Arithmetic

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 1's complement multioperand adder adding operands of size only 80% of the size of the system's dynamic range.

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

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.

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

The concept of a "Quadratic-Like" Complex Residue Number System is introduced in which certain purely imaginary numbers exist as reals within the system. It is shown that the desirable properties of a Quadratic Complex Residue Number System with regard to extremely simplified addition, subtraction, and multiplication carry over to Quadratic-Like systems. Furthermore, unlike Quadratic systems, there are no restrictions on the moduli for Quadratic-Like systems. Applications of both Quadratic and Quadratic-Like systems to real-time signal processing for ultrasonic nondestructive testing and evaluation are briefly discussed.

Application of Quadratic-Like complex residue number system arithmetic to ultrasonics

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Application of Quadratic-Like complex residue number system arithmetic to ultrasonics