Fraction-of-time probability for time-series that exhibit cyclostationarity

Abstract

Abstract A nonstochastic alternative to the stochastic process framework for conceptualizing, modeling and analyzing time-series encountered in communications, radar and telemetary systems is proposed. Wold's isomorphism between a single time-series and an ergodic stationary stochastic process is generalized to accomodate time-series with periodic structure and corresponding cycloergodic cyclostationary stochastic processes. This reveals the existence of a nonstochastic theory for single time-series with periodic structure that completely parallels the theory of cycloergodic cyclostationary stochastic processes. In particular, the concept of a nonstochastic stationary fraction-of-time probability (temporal-probability) model for a single time-series, which is closely associated with Wold's isomorphism, is generalized to cyclostationary and almost cyclostationary nonstochastic temporal-probability models for time-series with periodic structure corresponding to a single period and to multiple incommensurate periods, respectively. Gaussian time-series are considered as a specific illustrative case. Applications to signal processing are cited.