Abstract
The stability of a particular class of shaping filters, namely, the periodic shaping filters, is considered in this paper. The main objective of the present paper is to show that by considering the output covariance function of this periodic shaping filter it is possible to determine the stability of this class of shaping filters in the bounded input-bounded output sense. Furthermore, it is demonstrated that if γ(t, τ) is separable in t and τ, and the components of the separating vectors that result in the scalar covariant function are jointly linearly independent over one period of the time interval, then the resulting shaping filter is uniformly asymptotically stable. The results of this paper are presented in a series of theorems that, in effect, enable the determination of stability properties of such systems by using their structural properties only. The approach taken in this paper is thus parallel to that of Lyapunov's second method in stability analysis of linear time-varying systems.
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