Stability Analysis of Digital Filters under Finite Word Length Effects via Normal-Form Transformation

In this paper, a novel approach is proposed to analyze and minimize fixed-point arithmetic errors for digital filter implementations based on eigenvalue sensitivity measures. First, uncertainties of the filter-parameter caused by roundoff and computational errors are expressed in the function of register length for fixed-point implementations. Sequentially, based on a fixed-point statistical model and a normal-form transformation, a stability criterion of the dynamical filter model is derived to form a similarity transformation with a sufficient and necessary condition of the stability. Thus, the eigenvalue sensitivity measure with respect to filter parameters is constructed in the sense of induced 2-norm. This measure is minimized by an optimal similarity transformation (normal-form transformation) obtained from an analytically algebraic method. Based upon this transformation as well as the stability criterion, a minimum register length can be obtained. Finally, an example is performed to illustrate the effectiveness of the proposed scheme.