Foundations of the functional approach for signal analysis

摘要: In this paper, the mathematical foundation of functional (or nonstochastic) approach for signal analysis is established. The considered alternative to classical one that models signals as realizations stochastic processes. work follows fraction-of-time probability introduced by Gardner. By applying concept relative measure used Bochner, Bohr, Haviland, Jessen, Wiener, and Wintner Kac Steinhaus, a probabilistic--but nonstochastic--model built starting from single function time (the at hand). Therefore, are modeled without resorting an underlying ensemble realizations, i.e., process model. Several existing results put in common, rigorous, measure-theory based setup. It shown using concept, distribution function, expectation operator, all familiar probabilistic parameters can be constructed time. new joint measurability two or more functions paper which necessary characterization signals. Moreover, such independence defined. property then prove nonstochastic counterparts several useful theorems analysis. convergence parameter estimators requires (analytical) assumptions on much easier verify than ergodicity As example application, nonrelatively measurable design secure information transmission systems.