Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

6.4 Nonstationarity Classification in the Functional Approach

In the classical approach for statistical signal analysis, signals are modeled as stochastic processes, that is, ensemble of sample paths or realizations. In such a framework, nonstationarity is the property that statistical functions defined by ensemble averages depend on the time parameter t.

In the functional approach, once the almost-periodically time-variant model (characterized at second-order by (6.24)–(6.27b)) is adopted, the classification of the kind of second-order nonstationarity in the wide sense for a time series can be made on the basis of the elements contained in the set A_τ in (6.26). If A_τ contains incommensurate cycle frequencies α, then the time series x(t) is said to be wide-sense generalized almost cyclostationary. If is countable, then the time series is said to be wide-sense almost cyclostationary. In the special case where , the time series is said to be wide-sense cyclostationary. If the set A contains only the element α = 0, then the time series is said to be wide-sense stationary. In this classification, WSS time series are a subclass of the cyclostationary time series which are a subclass of the ACS time series which in turn are a subclass of the GACS time series. A similar classification is ...