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# 6.3 Almost-Periodically Time-Variant Model

In the probabilistic framework built by distribution functions like that in (6.10) and (6.14), the infinite-time average plays the role of the expectation operator. Consequently, statistical functions defined in terms of this expectation operator do not depend on t (see e.g., the nth-order moment (6.11) and the cross-correlation function (6.12a)). That is, such a model corresponds to a stationary description of the signal.

## 6.3.1 Almost-Periodic Component Extraction Operator

A time-variant probabilistic model which is based on a single time series is proposed in (Gardner 1987d), (Gardner and Brown 1991), (Gardner 1994), and referred to as fraction-of-time probability for time series that exhibit cyclostationarity. It is developed in (Gardner and Spooner 1994), (Spooner and Gardner 1994), (Izzo and Napolitano 1998a, 2002a). Such an approach is based on the decomposition of time series, or functions of time series, into a (possibly zero) almost-periodic component and a residual term. Several kinds of decompositions are possible according to the results presented in Sections 1.2.3, 1.2.4, and 1.2.5. In this section, the decomposition of Hartman and Ryll-Nardzewski (Section 1.2.5) will be considered. In fact, this is the only decomposition which is compatible with a finite-power residual term.

Let , p > 1, where is the set of AP functions in ...

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