Necessary and sufficient conditions for the absence of overflow phenomena in a second-order recursive digital filter

Due to adder overflow, different stationary output signals can be obtained from a digital filter for the same input signal. This phenomenon is discussed for a second-order recursive digital filter, resulting in a definition of stability of a digital system with respect to a certain input signal. A necessary and sufficient condition for unique steady-state solutions for a filter using a saturation characteristic is given. It is indicated that this condition is also a necessary condition for any other possible overflow characteristic. For a class of overflow characteristics, sufficient conditions for unique solutions are derived. These sufficient conditions are in general not necessary, but the necessity of these conditions is proven for one specific nonlinear characteristic.