A condition for the overflow stability of second-order digital filters that is satisfied by all scaled state-space structures using saturation

A set of conditions is derived that ensures overflow stability of second-order digital filters for different classes of overflow arithmetics, involving only the elements of the state-transition matrix. The well-known arithmetic saturation, zeroing, and two's-complement lead to different stability conditions, the condition for saturation being the least restrictive. As a result, all properly scaled second-order state-space structures are zero-input overflow stable if saturation is used for overflow correction. Conditions are derived for stable second-order digital filters in a nonzero input situation by introducing a weaker form of stability of the forced response. The analysis is based on determining the set of Lyapunov functions for a general second-order state-transition matrix, given a certain overflow arithmetic.>