Overflow oscillations in state-space digital filters

A sufficient condition is established for the stability of general state-space digital filters implemented with two's complement arithmetic for the addition operation. A necessary and sufficient condition is established for the existence of period-T limit cycles in second-order direct-form digital filters. This theorem can be used to determine existence regions in the parameter plane for limit cycles of arbitrary period. Given the filter coefficients, this theorem can be used to find initial conditions that yield period-T limit cycles for any T, and all possible limit cycle vector sequences.>