The history and fundamentals of distributed arithmetic (DA) are presented. Ways to increase the speed of DA multiplication are described. DA is applied to a biquadratic digital filter, providing an example of vector dot-product and vector-matrix-product mechanization. Applications to transformers and nonlinear and/or nonstationary processing with DA are discussed. It is seen that DA is a very efficient means to mechanize computations that are dominated by inner products. >

Applications of distributed arithmetic to digital signal processing: a tutorial review

The ancient study of the residue numbering system, or RNS, begins with a verse from a third-century book, Suan-ching, by Sun Tzu:* We have things of which we do not know the number, If we count them by threes, the remainder is 2, If we count them by fives, the remainder is 3, If we count them by sevens, the remainder is 2, How many things are there? The answer, 23. How to get the answer 23 is outlined in Sun Tzu's historic work. He presents a formula for manipulating remainders of an integer after division by 3, 5, and 7. We commemorate this contribution today by referring to one of the common rules of converting remainders, or residues, into integers as the Chinese Remainder Theorem, or CRT. This theorem, as well as the theory of residue numbers, was set forth in the 19th century by Carl Friedrich Gauss in his celebrated Disquisitiones Arithmetical.-t This 1700-year-old number system has been attracting a great deal of attention recently. Digital systems structured into residue arithmetic units may play an important role in ultra-speed, dedicated, real-time systems that support pure parallel processing of integer-valued data. It is a \"carry-free\" system that performs addition, subtraction, and multiplication as concurrent (parallel) operations, side-stepping one of the principal arithmetic delays managing carry information. The first attempt to use some of the unique RNS properties was made by D. H. Lenhmet who, in 1932, built a special-purpose machine he called the \"photo-electric sieve.\" This electro-mechanical device factored Mersenne *L. Dickson in the History of the Theorv of Numbers attributes the origin of the RNS to Sun Tsu (not Tzu) in the first centurv AD, but most scholars accept the Sun Tzu origin. numbers. Then in the mid-50's, the Czech researchers Svaboda and Valach conducted experiments on a hard-wired, small moduli RNS machine, which they used to study error codes. I The same idea apparently occurred to Aiken and Garner. 2 From the late 50's to mid-60's, the Department of Defense supported the RNS research by Szabo and Tanaka at Lockheed. They worked on a special-purpose digital correlator while a team from RCA looked into designing a general-purpose machine. Experimentally , these early efforts met with little success because winding the custom core memory required specialized residue mappings. The most tangible result of these early efforts was a comprehensive text written by Szabo and Tanaka, which survived only …

Residue Arithmetic A Tutorial with Examples

Residue generator is an essential building block of encoding/decoding circuitry for arithmetic error detecting codes and binary-to-residue number system (RNS) converter. In either case, a residue generator is an overhead for a system and as such it should be built with minimum amount of hardware and should not compromise the speed of a system. Multioperand modular adder (MOMA) is a computational element used to implement various operations in digital signal processing systems using RNS. A comprehensive study of new residue generators and MOMA's is presented. The design methods given here take advantage of the periodicity of the series of powers of 2 taken module A (A is a module). Four design schemes of the n-input residue generators mod A, which are best suited for various pairs of n and A, are proposed. Their pipelined versions can be clocked with the cycle determined by the delay of a full-adder and a latch. A family of design methods for parallel and word-serial, using similar concepts, is also given. Both classes of circuits employ new highly-parallel schemes using carry-save adders with end-around carry and a minimal amount of ROM and are well-suited for VLSI implementation. They are faster and use less hardware than similar circuits known to date. One of the MOMA's can be used to build a high-speed residue-to-binary converter based on the Chinese remainder theorem. >

Design of residue generators and multioperand modular adders using carry-save adders

A technique to extend the base of a residue number system (RNS) based on the Chinese remainder theorem (CRT) and the use of a redundant modulus, is proposed. The technique obtains the residue(s) of a given number in the extended moduli without resorting to the traditional mixed-radix conversion (MRC) algorithm. The base extension can be achieved in log/sub 2/n table lookup cycles, where n is the number of moduli in the RNS. The superiority of the technique, compared in terms of latency and hardware requirements to the traditional Szabo-Tanaka method is demonstrated. >

Fast base extension using a redundant modulus in RNS

Efficient signal processing algorithms are important for embedded and power-limited applications since, by reducing the number of computations, power consumption can be reduced significantly. Similarly, efficient algorithms are also critical to very large scale applications such as video processing and four-dimensional medical imaging. This self-contained guide, the only one of its kind, enables engineers to find the optimum fast algorithm for a specific application. It presents a broad range of computationally-efficient algorithms, describes their structure and implementation, and compares their relative strengths for given problems. All the necessary background mathematics is included and theorems are rigorously proved, so all the information needed to learn and apply the techniques is provided in one convenient guide. With this practical reference, researchers and practitioners in electrical engineering, applied mathematics, and computer science can reduce power dissipation for low-end applications of signal processing, and extend the reach of high-end applications.

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Fast Algorithms for Signal Processing

A technique is presented for implementing a finite impulse response (FIR) digital filter in a residue number system (RNS). For many years residue number coding has been recognized as a system which provides a capability for the implementation of high speed multiplication and addition. The advantages of residue coding for the design of high speed FIR filters result from the fact that an FIR requires only the high speed residue operations, i.e., addition and multiplication, while not requiring the slower RNS operations of division or sign detection. A new hardware implementation of the Chinese Remainder Theorem is proposed for an efficient translation of residue coded outputs into natural numbers. A numerical example illustrates the principles of residue encoding, residue arithmetic, and residue decoding for FIR filters. An RNS implementation of a 64th-order dual bandpass filter is compared with several alternative filter structures to illustrate tradeoffs between speed and hardware complexity.

The use of residue number systems in the design of finite impulse response digital filters

Interval algebras are introduced as media in which to simulate digital filters for the purpose of analyzing quantization effects due to finite wordlength. The theory presented forms the basis of a computer-aided analysis scheme which generates confidence interval error bounds on the time domain response of a digital filter. The techniques are applicable to filters of arbitrary order and configuration implemented in either fixed or floating-point arithmetic. Experimental data is presented for a fourth-order low-pass Butterworth filter realized with fixed-point arithmetic in the direct, canonical direct, parallel, cascade, and ladder configurations. An example of Parker and Hess is used to illustrate the bounding limit cycles with interval techniques.

B. Leon

Papers

The use of residue number systems in the design of finite impulse response digital filters

An analysis of quantization error in digital filters based on interval algebras