A New and Interesting Class of Limit Cycles in Recursive Digital Filters

Oscillations often occur in recursive digital filters as a result of the nonlinear action of quantizing the products in the feedback sections. These oscillations occur in the least significant bits of the data and are called limit cycles.* These limit cycles influence the required internal data word length and hence the cost of the filter. 1 This paper describes a new and important class of limit cycles that exist in second-order recursive digital filters. These limit cycles often occur in filters with high Q poles located near dc or half the sampling * These limit cycles are to be distinguished from the large limit cycles caused by overflow. 379 frequency. They are called rolling-pin limit cycles and derive this name from their characteristic shape when plotted in the successive value plane (Fig. 2a). The second-order section under consideration employs rounding of both feedback products in either sign-magnitude or twoscomplement number format and is shown in Fig. 1. The rolling-pin limit cycles are defined by three integers, K, L, and M, and a simple construction rule. As seen in Fig. 2b, K is the constant step size in the handle of the rolling pin, L is the constant step size in the body of the rolling pin, and M is the number of steps of step size L. For each value of K, L, M, a unique set of limit cycles is completely defined. In this paper, emphasis is on the simplest case of K = 1. This class of rolling-pin limit cycles is important because of its unusually large amplitude.