In this article, the second order digital filter with two’s complement arithmetic in  is considered. Necessary conditions for the symbolic sequences to be periodic after a number of iterations are given when the filter parameters are at b=a+1 and b=-a+1. Furthermore, for some particular values of a, even when one of the eigenvalues is outside the unit circle, the system may behave as a linear system after a number of iterations and the state vector may toggle between two states or converge to a fixed point at the steady state. The necessary and sufficient conditions for these phenomena are given in this article
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