Complex number arithmetic occurs frequently in Digital Signal Processing. A Butterfly usually constitutes at least of one complex product and one complex sum. The standard binary implementation of a product requires at best three real multiplations and five real additions. Besides, the two components of a complex number must be tracked down at every stage. In the past, many authors have proposed different digital representations, in order to cirmcumvent these problems. The choice of any particular one must be based on three factors: completeness, complexity of implementation, and ease of conversion from and to binary. In this paper, the Knuth's representation or "Qua-ter-Imaginary" is revisited and its practical merits evaluated. We then propose the design of some arithmetics and describe in details their hardware implementation. The additional conversion arithmetic is also described. The existence of this overhead is highly justifiable in some systems.

Byte-organized memory requires an error control scheme which can handle errors involving one or several entire bytes. A special parallel nonbinary single error correction and double error detection SEC-DED block code is constructed by dynamic programming. This code is optimum in the sense that it lends itself to a simple and high-speed hardware implementation either in binary or in quaternary logic. A double extension field GF(22m) of the subfield GF(2m) is then introduced. As a design example, a(80,64) SED-DED code in GF(24) is constructed.

Digital signal processing such as discrete Fourier transform requires complex number arithmetic. Number-theoretical properties favor odd-radix systems for simple implementation of complex number operations. An array ternary complex multiplier is presented and compared to other complex multipliers. As an illustration, a perfectly balanced ternary system is implemented by means of current-mode threshold logic gates.

A family of circuits for multivalued, in particular quaternary, integrated injection logic is described. The basic elements are the I2L current mirror and I2L threshold gates.