Digital fixed-point multiplication error structure and some consequences

This paper examines the properties of fixed-point multiplication errors for sign-magnitude (SM), two´s-complement (TC), and one´s-complement (OC) binary number representations. The mappings from the multiplier input values onto the corresponding roundoff and chopping error values are derived. Periodic error patterns are shown to exist as the multiplier input is stepped sequentially through all possible positive and all possible negative values. These patterns are dependent on the multiplication coefficient value as well as the polarity of the multiplier input. The coefficient dependent parameter ν is defined and the relation to the patterns is shown to be the following. As the multiplier input ranges over all possible positive values, a set S+ of 2^νdistinct error values results. A set S- of 2^νdistinct error values also results as the input ranges over all possible negative values. However, S+ overlaps completely with S- only for the TC case. Partial overlap occurs in all the other cases. Previous work is referenced to show that, under certain conditions, the probability of occurrence of each error value within the set S^αapproaches the value 2^-νP^αwhere P^αis the probability of occurrence of the set S^αand α = ±. The statistical error models can be represented in two ways: first as uniformly distributed white noise plus a constant for TC; second as uniformly distributed white noise plus a constant plus a hard-clipper output for the other number representations. The hard-clipper input is connected to the multiplier input. In all cases the white noise is zero-mean with variance equal to (1-2^-2ν)2^-2M/12 where M is the multiplier output word length.