6.1 Introduction

An approach for signal analysis which is alternative to classical one based on stochastic processes is the fraction-of-time (FOT) probability framework or functional approach (Gardner 1987d, 1991c), (Gardner and Brown 1991), (Gardner 1994), (Lekow and Napolitano 2006). In such an approach, signals are modeled as single functions of time (time series) rather than sample paths of stochastic processes. This is a more appropriate model when an ensemble of realizations does not exist and, consequently, the stochastic process turns out to be artificially introduced just to create a mathematical model. Common pitfalls that can arise from the adoption of a non appropriate stochastic process model are described in (Gardner 1987d, 1991c), (Gardner and Brown 1991), (Gardner 1994), (Izzo and Napolitano 2002a).

In the FOT approach, starting from the concept of relative measurability of sets and functions (Kac and Steinhaus 1938), (Lekow and Napolitano 2006), for a single function of time (or time series) a distribution function is constructed and its corresponding expected value is shown to be the infinite-time average. All familiar probabilistic parameters and functions, such as variance, moments, and cumulants, are built starting from the single time series at hand in terms of infinite-time averages. Time series for which this model does not lead to trivial results should be persistent or finite power, that is, the time-averaged power

(6.1)

must exist and be finite. Such a necessary, but not sufficient, condition, with lim replaced by lim sup is also required in Wiener work on generalized harmonic analysis (Wiener 1930), where autocorrelation function and power spectrum are defined for single functions of time.

In the functional approach, the relative measure plays the role played by the probability measure in the stochastic approach. The relative measure, however, does not posses the sigma-additivity property and is not continuous. Such a result constitutes a strong motivation to adopt the functional approach since it enlightens a deep difference between properties of stochastic processes and properties of functions. In other words, the stochastic process model for a single realization at hand should be used carefully, since properties of the stochastic process could not correspond to analogous properties of the function of time at hand.

A time-variant probabilistic model which is based on a single time series is introduced in (Gardner 1987d), (Gardner and Brown 1991), (Gardner 1994), by showing that the almost-periodic component extraction operator is an expectation operator. Starting from this expectation operator, almost-periodically time-variant distributions, moments, and cumulants are defined. Moreover, concepts such as statistical independence, stationarity, and nonstationarity can be introduced. Therefore, this model can be used to statistically characterize ACS and GACS signals. In contrast, at the moment, no single-function-based probabilistic model exists for SC signals.

A rigorous link between the time-average-based and the stochastic-process frameworks in the stationary case is established in (Wold 1948), where an isometric isomorphism (Wold isomorphism) between a stationary ergodic stochastic process and the Hilbert space generated by a single sample path is singled out. The Wold isomorphism is extended to cyclostationary signals in (Gardner 1987d), (Gardner and Brown 1991), (Hurd and Koski 2004).