Roundoff Noise

Roundoff Noise:

To determine the roundoff noise at the output of a digital filter we will assume that the noise due to a quantization is stationary, white, and uncorrelated with the filter input, output, and internal variables. This assumption is good if the filter input changes from sample to sample in a sufficiently complex manner. It is not valid for zero or constant inputs for which the effects of rounding are analyzed from a limit cycle perspective.

To satisfy the assumption of a sufficiently complex input, roundoff noise in digital filters is often calculated for the case of a zero- mean white noise filter input signal x(n) of variance a ¹. This simplifies calculation of the output roundoff noise because expected values of the form E{x(n)x(n — k)} are zero for k = 0 and give a² when k = 0. 

Another assumption that will be made in calculating roundoff noise is that the product of two quantization errors is zero. To justify this assumption, consider the case of a 16-b fixed-point processor. In this case a quantization error is of the order 2^{—1 5} , while the product of two quantization errors is of the order 2^{—3 0} , which is negligible by comparison.

If a linear system with impulse response g(n) is excited by white noise with mean m_x and variance a² , the output is noise of mean [3, pp.788-790]

Therefore, if g(n) is the impulse response from the point where a roundoff takes place to the filter output, the contribution of that roundoff to the variance (mean-square value) of the output roundoff noise is given by (3.25) with a² replaced with the variance of the roundoff. If there is more than one source of roundoff error in the filter, it is assumed that the errors are uncorrelated so the output noise variance is simply the sum of the contributions from each source.