Roundoff Noise Minimization of State-Space Digital Filters Using Separate and Joint Error Feedback/Coordinate Transformation Optimization

This paper investigates the problem of minimizing roundoff noise under 2-norm dynamic-range scaling constraints in state-space digital filters by means of error feedback as well as joint error feedback/coordinate transformation optimization. First, several techniques for the determination of optimal full-scale, diagonal, and scalar error-feedback matrices for a given state-space digital filter are proposed, where three realization schemes, namely, the general state-space realization, input-balanced realization, and optimal realization in the sense of Hwang–Mullis–Roberts are examined. Furthermore, an iterative approach is developed for jointly optimizing a scalar error-feedback matrix and a coordinate transformation matrix so as to minimize the roundoff noise subject to the 2-norm dynamic-range scaling constraints. The proposed method may be regarded as an alternative, but much simpler and more general, approach to Hwang’s method for synthesizing the optimal filter structure with minimum roundoff noise. A case study is included to illustrate the utility of the proposed techniques.