The delay operator z-1 - inappropriate for use in recursive digital filters?

Abstract

The paper questions the appropriateness of using the delay operator z-¹ for recursive digital filters, particularly when applied to the realisation of digital control. The well known problems of coefficient sensitivity at high sample frequencies are reviewed, and the fundamental reason for this lack of robustness is identified. An analogy is used to argue that the approach whereby digital filters are designed using discrete transfer functions in z is, in fact, mathematically dubious, perhaps unperceived by their users, and that the sensitivity problems are inevitable when z is used. The analogy is extended to argue strongly for using an alternative, the δ operator, which inherently avoids the problem. Some of the implications of using the new operator are assessed, but the main purpose of the paper is to challenge the established practice of using the z operator.