Generating covariance sequences and the calculation of quantization and rounding error variances in digital filters

A linear algorithm is given for the generation of covariance sequences for rational digital filters using numerator and denominator coefficients directly. There is no need to solve a Lyapunov equation or to solve for the residues of a spectrum, as in other methods. By appealing to certain results from the theory of inners, we show that the algorithm provides a unique solution, provided only that the filter is stable. Our results may be used to compute error variances due to product rounding and signal quantization, and to generate covariance strings
{r_{k}}min{0}\max{K}
used in other studies involving second-order properties of digital filters.