COMPLEX BEHAVIOR IN DIGITAL FILTERS

Abstract

This paper presents recent results concerning possibilities of complex behavior in extremely simple second-order digital filter structures. Two types of such behavior have been confirmed by simulation experiments and rigorous mathematical analysis: 1) chaotic behavior defined here as the existence of aperiodic, bounded trajectories displaying sensitive dependence to the initial states. Such trajectories often form very complex, self-similar patterns in the state-space. 2) an abundance of oscillatory solutions and final state sensitivity with respect to system parameters. This type of behavior can be identified by an extremely complicated structure of the parameter space (existence of Arnold tongues) and fine structure of changes of dynamic behavior when varying filter parameters (devil’s staircase). The first type of behavior has been discovered in filter sections employing two’s complement adder overflow characteristics and confirmed via symbolic dynamics technique. Properties of the second type have been found in extensive numerical experiments carried out for filter sections employing saturation arithmetic and in which a full confirmation via the mathematical analysis of an associated one-dimensional model of the system has been made.