ARITHMETIC CO-TRANSFORMATIONS IN THE REAL AND COMPLEX LOGARITHMIC NUMBER-SYSTEMS

The real logarithmic number system, which represents a value with a si gn bit and a quantized logarithm, can be generalized to create the com plex logarithmic number system, which replaces the sign bit with a qua ntized angle in a log/polar coordinate system. Although multiplication and related operations are easy in both real and complex systems, add ition and subtraction are hard, especially when interpolation is used to implement the system. Both real and complex logarithmic arithmetic benefit from the use of co-transformation, which converts an addition or subtraction from a region where interpolation is expensive to a reg ion where it is easier. Two co-transformations that accomplish this go al are introduced. The first is an approximation based on real analysi s of the subtraction logarithm. The second is based on simple algebra that applies for both real and complex values and that works for both addition and subtraction.