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

It is well known that the autonomous response of a second-order digital filter with twoʹs complement arithmetic may exhibit chaotic behavior. In this paper, results of the step-response case are presented. Despite the presence of the overflow nonlinearity, it is found that the step-response behaviors can be related to some corresponding autonomous-response behaviors by means of an appropriate affine transformation. Based on this method, some differences between the step response and the autonomous response are explored. The effects of the filter parameter and input step size on the trajectory behaviors are presented. Some previous necessary conditions for the trajectory behaviors, initial conditions and symbolic sequences are extended and strengthened to become necessary and sufficient conditions. Based on these necessary and sufficient conditions, some counter-intuitive results are reported. For example, it is found that for some sets of filter parameter values, the system may exhibit the type I trajectory even when a large input step size is applied and overflow occurs. On the other hand, for some sets of filter parameter values, the system will not give the type I trajectory for any small input step size, no matter what the initial conditions are.