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 used in other studies involving second-order properties of digital filters.