The zero-input response of digital filters using saturation arithmetic

It is known that second-order digital filters realized in direct form using saturation arithmetic have the property that the amplitudes of all self-sustained limit cycles can be made arbitrarily small by increasing the numbers of bits associated with the representation of the data samples. In this paper we observe that a recent extension to certain higher order cases of an idealized version of that result (quantization is ignored in the Idealized model) is a direct corollary of a theorem in the theory of feedback systems, and that the extension (and related more general propositions) can be proved without the earlier theorem by using simple operator-theoretic methods. The main result given in the paper is an extension of the result for nonidealized second-order sections to a class of filters of arbitrary order.