Fixed-point multiplication: A probabilistic bit-pattern view

The truncation error of fixed-point multiplication is commonly modelled as white noise with a uniform distribution. This paper shows the inaccuracy of this assumption by using a bitwise approach. A probabilistic bit-pattern analysis of the fixed-point multiplication is presented from which an accurate probability model for the truncation error is derived. Using this PDF model, exact values for the mean and variance of the truncation error in the output of the fixed-point multiplication are formulated for the first time. The proposed PDF model, mean and variance values are compact and usable in word-length optimization algorithms.