Impact of Shift Operations on (-1+j)-Base Complex Binary Numbers

Abstract

Complex numbers play a very important role in various applications of electrical and computer engineering. These days, arithmetic operations dealing with these numbers rely on a “divide-and- conquer” technique wherein a complex number is broken into its real and imaginary parts and then, after representing each part in binary number system, operation is carried out on each part as if part of the real arithmetic. Thus, addition of two complex numbers requires two separate additions, and their multiplication requires four individual multiplications, one subtraction and one addition. In an effort to reduce the number of arithmetic operations within the realm of complex arithmetic, binary number system with base (−1+j), called complex binary number system, has been proposed in the literature which allows a complex number to be represented as single-unit instead of two separate units as in the base-2 binary number system. In this paper, the effects of shift operations on complex binary numbers have been examined and mathematical equations describing their behavior have been obtained. Analysis of these equations leads to the conclusion that the impact of shift operations on a complex binary number is, to a large extent, similar to typical multiply-by-2 (for per-bit shift-left) and divide-by-2 (for per-bit shift-right) operations of traditional base-2 binary number.