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A time-frequency calculus for time-varying systems and non-stationary processes with applications

This thesis introduces an approximate time-frequency calculus for underspread linear time-varying systems (i.e., time-varying systems that effect only small time-frequency shifts of the input signal) and underspread nonstationary random processes (i.e., nonstationary processes that feature only small time-frequency correlations). After briefly describing the major difficulties encountered with time-varying systems and nonstationary processes, we introduce an extended definition of underspread systems. Our extended underspread concept is based on weighted integrals and moments of the system’s generalized spreading function. Subsequently, numerous approximations are presented which show that in the case of underspread systems the generalized Weyl symbol constitutes an approximate time-frequency transfer function. As a mathematical underpinning of our transfer function approximations, we provide bounds on the associated approximation errors that involve the previously defined weighted integrals and moments of the generalized spreading function. We then consider nonstationary random processes and provide an extended definition of underspread processes. This extended underspread concept is based on weighted integrals and moments of the generalized expected ambiguity function of the process. Subsequently, two fundamental classes of time-varying power spectra are introduced and analyzed: “type I” spectra that extend the generalized Wigner-Ville spectrum and “type II” spectra that extend the generalized evolutionary spectrum. We show that in the case of underspread processes, the various members of these two classes of spectra are approximately equivalent to each other and (at least) approximately satisfy several desirable properties. Our approximations are again supported by bounds on the associated approximation errors. These bounds are formulated in terms of the previously defined weighted integrals and moments of the generalized expected ambiguity function. The definition and analysis of time-frequency coherence functions concludes our discussion of time-varying power spectra. Finally, we illustrate the practical relevance of our theoretical findings by considering several applications in the areas of statistical signal processing and wireless communications. These applications include nonstationary signal estimation and detection, the sounding of mobile radio channels, multicarrier communications over time-varying channels, and the analysis of car engine signals.