Autonomous response of a third-order digital filter with two's complement arithmetic realized in cascade form

This paper investigates the output and state trajectories of a third-order digital filter with two’s complement arithmetic realized in parallel form. Although the output of the thirdorder digital filter seems to behave randomly, some regular patterns can be displayed on the plot of y (k − 1) versus y (k), where those regular patterns are similar to the second-order case. When the first-order subsystem is operated at the marginally stable points, the output of the third-order system is still mainly dependent on the behaviors of the corresponding second-order digital filter, even though overflow occurs. Explicit equations relating the trajectories of the system to the filter parameters and the initial conditions provide further insights into the behaviors of the system.