The spectral correlation theory of cyclostationary time-series
Signal Processing
Statistical spectral analysis: a nonprobabilistic theory
Statistical spectral analysis: a nonprobabilistic theory
A representation theorem for local LSI operators on two-sided sequences
Signal Processing
Bounded Power Signal Spaces for Robust Control and Modeling
SIAM Journal on Control and Optimization
Time series: data analysis and theory
Time series: data analysis and theory
Prediction for time series in the fraction-of-time probability framework
Signal Processing - Image and Video Coding beyond Standards
Classification of co-channel communication signals using cyclic cumulants
ASILOMAR '95 Proceedings of the 29th Asilomar Conference on Signals, Systems and Computers (2-Volume Set)
The Wold isomorphism for cyclostationary sequences
Signal Processing
IEEE Transactions on Signal Processing
The higher order theory of generalized almost-cyclostationary timeseries
IEEE Transactions on Signal Processing
Least-squares LTI approximation of nonlinear systems and quasistationarity analysis
Automatica (Journal of IFAC)
Some stochastic properties of memoryless individual sequences
IEEE Transactions on Information Theory
IEEE Journal on Selected Areas in Communications
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In this paper, the mathematical foundation of the functional (or nonstochastic) approach for signal analysis is established. The considered approach is alternative to the classical one that models signals as realizations of stochastic processes. The work follows the fraction-of-time probability approach introduced by Gardner. By applying the concept of relative measure used by Bochner, Bohr, Haviland, Jessen, Wiener, and Wintner and by Kac and Steinhaus, a probabilistic--but nonstochastic--model is built starting from a single function of time (the signal at hand). Therefore, signals are modeled without resorting to an underlying ensemble of realizations, i.e., the stochastic process model. Several existing results are put in a common, rigorous, measure-theory based setup. It is shown that by using the relative measure concept, a distribution function, the expectation operator, and all the familiar probabilistic parameters can be constructed starting from a single function of time. The new concept of joint relative measurability of two or more functions is introduced in this paper which is shown to be necessary for the joint characterization of signals. Moreover, by using such a concept, the independence of signals is defined. The joint relative measurability property is then used to prove the nonstochastic counterparts of several useful theorems for signal analysis. It is shown that the convergence of parameter estimators requires (analytical) assumptions on the single function of time that are much easier to verify than the classical ergodicity assumptions on stochastic processes. As an example of application, nonrelatively measurable functions are shown to be useful in the design of secure information transmission systems.