System on Chip Seminar

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In pursuit of equal opportunity representation for our complex neighbors...

Dr Tariq Jamil

Complex numbers play a unique and important role in engineering applications such as digital signal processing and image processing. These days, arithmetic operations involving complex numbers are usually carried out by the application of "divide-and-conquer" technique, whereby a complex number is broken-up into its real and imaginary parts and then operations are carried out on each part as if it was a part of the real arithmetic. Finally, the overall result of the complex operation is obtained by accumulation of the individual results. For instance, addition of two complex numbers (a+jb) and (c+jd) requires two separate additions (one for the real part and one for the imaginary part) while multiplication of the same two complex numbers requires four multiplications (ac, ad, bc, bd ), one subtraction (j2bd = -bd ), and one overall addition. This can be effectively reduced to just one complex addition or only one multiplication and addition respectively for the given cases if each complex number is represented as one unit instead of two individual units.

In an effort to provide equal opportunity representation to complex numbers, both mathematicians and engineers have tried to define binary numbers with bases other than 2. This includes work by Donald E. Knuth in 1960, Walter Penney in 1964, and V. Stepanenko in 1996. In this presentation, we'll endeavor to define a (-1+j)-base binary number system for complex numbers which provides them opportunity to be represented as a single unit like their counterparts in the "real"-world. We'll outline procedures for addition, subtraction, multiplication, and division of two such complex binary numbers and, finally, hardware designs of nibble-size complex binary-adder units based on the traditional minimum-delay and ripple-carry principles will be presented.

nurmi@cs.tut.fi