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 made in the strict sense in Section 6.3.2 with reference to the sets Γ_τ and Γ.

In the functional approach a different classification can be made on the basis of the existence or not of the stationary and almost-cyclostationary probabilistic models. Specifically, time series for which a probabilistic model based on almost-periodic distributions like (6.24) exists are a subclass of time series for which a probabilistic model based on stationary distributions like (6.13) (with y ≡ x) exists. In fact, the existence of (6.24) requires more stringent conditions on the time series x(t) with respect to those for the existence of (6.13) (with y ≡ x) (Section 6.3.2), (Lekow and Napolitano 2006, Ex. 6.1). Therefore, from this point of view, almost-cyclostationary time series are a subclass of the stationary time series.

The existence of two possible classifications of nonstationarity of time series in the functional approach constitutes a significative difference with respect to the classical stochastic approach. This difference is consequence of the difference existing between the expectation operators in the two approaches. In the stochastic approach the expectation operator makes an average on the variable ranging in the sample space and produces statistical functions of a stochastic process (possibly) depending on the time variable t (see (1.1)–(1.5)). The kind of time variability with respect to t characterizes the kind of nonstationarity of the process. In contrast, in the functional approach, the expectation operator (the infinite-time average or the almost-periodic component extraction operator) acts on the time variable t and is a priori chosen with the probabilistic model (see (6.3)–(6.13) for the stationary model and (6.24)–(6.26) for the (generalized) almost-cyclostationary model). The choice of the model implicitly imposes conditions on the admissible time series x(t), i.e., the conditions for which the statistical functions of the considered model exist.

Nonstationary signal analysis in the functional approach should be carefully handled. In (Gardner 1994), possible pitfalls associated with the possibility of choosing non unique statistical models (e.g., stationary and almost cyclostationary) are discussed. In addition, it should be noted that not every kind of nonstationarity (i.e., time variability of the statistical functions) that can be modeled in the stochastic approach can also be modeled in the functional approach. This is consequence of the fact that statistical functions, and in particular autocorrelation functions, of finite-power nonstationary stochastic processes can be reliably estimated starting from a single realization only if the kind of nonstationarity is of known form or of almost-periodic type (Gardner 1988a). For example, as shown in Chapter 4, a consistent estimator of the spectral correlation density of SC processes is obtained only when the location in the bifrequency plane of the support curves of the Loève bifrequency spectrum is known (Napolitano 2003), (Napolitano 2007c).