Quantization noise, fixed-point multiplicative roundoff noise, and dithering

The author considers the characteristics of the error resulting when a continuous amplitude signal x_n is quantized and then multiplied by a constant multiplier a under fixed-point roundoff arithmetic. It is shown that the overall error of such an operation can be decomposed into two terms: one being a scaled version of the error due to the quantization of x_n and the other due to rounding off the product aQ(x_n). Exact first- and second-order moments are derived for the quantization error, the roundoff error, and the overall error as a function of the multiplier a and the distribution of x_n. Sufficient conditions are given for the quantization error and the roundoff error to be individually uniformly distributed and white up to the first- and second-order moments, and also for them to be mutually uncorrelated. It is also shown that regardless of the probability distribution of the input signal x_n, it is always possible to add a suitable dither signal to the input of the system so that both the quantization error and the roundoff error are uniformly distributed, white, and mutually uncorrelated. For Gaussian inputs, the sufficient conditions given are not satisfied