Generalizations of Cyclostationary Signal Processing: Spectral Analysis and Applications

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 ...