Chaotic behaviours of stable second-order digital filters with two's complement arithmetic

In this paper, the behaviors of stable second-order digital filters with two’s complement arithmetic are investigated. It is found that even though the poles are inside the unit circle and the trajectory converges to a fixed point on the phase plane, that fixed point is not necessarily the origin. That fixed point is found and the set of initial conditions corresponding to such trajectories is determined. This set of initial conditions is a set of polygons inside the unit square, whereas it is an ellipse for the marginally stable case. Also, it is found that the occurrence of limit cycles and chaotic fractal pattern on the phase plane can be characterized by the periodic and aperiodic behaviors of the symbolic sequences, respectively. The fractal pattern is polygonal, whereas it is elliptical for the marginally stable case.