Numerical solution of the Lyapunov equation for narrow-band digital filters

The quantization noise generated by a digital filter can be analyzed via the solution of the Lyapunov equation
K = AKA^{T} + bb^{T}
corresponding to the filter's state variable description. A method given by Mullis and Roberts [1] for the solution of the Lyapunov equation is noted for its speed of implementation, but its applicability is limited to filters with wide bandwidths because of numerical accuracy. A method of extending this algorithm to narrowband filters is presented. It uses a transformation which was also presented by Mullis and Roberts. The transformation distorts the frequency response of the filters and improves the numerical accuracy of the solution of the Lyapunov equation. The development given here is restricted to low-pass or high-pass filters, but it can be extended to bandpass filters.