Criteria for Determining if a High-Order Digital Filter Using Saturation Arithmetic Is Free of Overflow Oscillations

Abstract

Recently we found that, among recursive digital filters using saturation arithmetic to contend with overflow, a fundamental difference exists between second and higher order filters: the latter may sustain large-amplitude overflow oscillations. In this paper we have derived a new criterion expressly designed for determining when a given high-order recursive system using saturation arithmetic is free of overflow oscillations. The new criterion, which is easy to use, follows from this result: we associate with the given system two trigonometric polynomials in θ of degree equal to the order of the given system; if any linear combination of the polynomials with nonnegative weights is positive for all θ in [0, π] then the system is free of all nontrivial periodic oscillations. We prove that the new criterion subsumes certain well-known criteria, such as Tsypkin's criterion, from the literature on nonlinear systems. To illustrate, three classes of special systems are investigated, and in each case the new criterion gives substantial improvements. Finally, the new test is applied in the synthesis of high order sections for a realistic eighth-order system.