A unified approach to the optimal synthesis of fixed-point state-space digital filters

This paper proposes two expressions of the output error variance due to coefficient quantization of fixed-point state-space digital filters in the time domain. One is the deterministic approach, which gives precisely the output error variance of state-space digital filters. The other is the statistical approach, where the errors of coefficient quantization are assumed to be independent random variables. The statistical approach gives a simple and easy way to analyze the output error variance due to coefficient quantization of state-space digital filters. The statistical coefficient sensitivity introduced by the statistical approach is shown to be equivalent to the round off noise power gain. Thus, state-space digital filters which are optimal with respect to both coefficient sensitivity and roundoff noise can be synthesized by the method of minimization of the roundoff noise. Such optimal state-space digital filters which could be of any order are proved to be free of autonomous overflow limit cycles. A numerical example is given to illustrate the effectiveness of the analysis of the output error variance due to coefficient quantization and the synthesis method proposed here.