Knuth´s complex number arithmetic revisited

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.