Large amplitude, self-sustained oscillations in difference equations which describe digital filter sections using saturation arithmetic

The possibility of oscillations due to adder overflow in digital filter sections of order exceeding two is investigated. Specifically, the following problem is posed: is the property that large amplitude limit cycles cannot be self-sustaining in saturation arithmetic special to second-order sections, or is it also true for some or all higher order sections? This issue is resolved and, beyond that, some new and rather interesting results are presented on the stability implications of approximating a nonlinear element in standard filter structures by a range of linear gains. For every order of the section beyond two we are able to identify a set of coefficients corresponding to an absolutely stable underlying linear system, and a set of initial conditions for which the solution (to the nonlinear recursion) is a limit cycle. As an aid to understanding this behavior, we show that a particular natural class of associated linear recursions is always stable if and only if their order does not exceed two. Finally, motivated by the above result and by the viewpoint of describing functions, we make a conjecture according to which these oscillations do not exist if every member of the above stated class of linear systems is absolutely stable. We show that even this conjecture is false for recursions of order three. Thus, to ensure that overflow oscillations do not occur in high-order sections in which saturation arithmetic is used particular care has to be exercised on a case by case basis.