Sinusoidal response of a second-order digital filter with twos complement arithmetic

In this brief, results of the sinusoidal response case are presented. It is found that the visual appearance of the trajectory of the sinusoidal response case is much richer than that of the autonomous and step-response cases. Based on the state-space technique, the state vectors to be periodic are investigated. The set of initial conditions and the necessary conditions on the filter parameters are also derived. When overflow occurs, the system is nonlinear. If the corresponding symbolic sequences are periodic, some trajectory patterns are simulated. Since the state-space technique is not sufficient to efficiently derive the sets of initial conditions and the necessary conditions on the filter parameters, a frequency-domain technique is employed to figure out the set of initial conditions. When the symbolic sequences are aperiodic, an elliptical fractal pattern or random-like chaotic pattern is found.