Nonstationary spectral analysis based on time-frequency operator symbols and underspread approximations

We present a unified framework for time-varying or time-frequency (TF) spectra of nonstationary random processes in terms of TF operator symbols. We provide axiomatic definitions and TF operator symbol formulations for two broad classes of TF spectra, one of which is new. These classes contain all major existing TF spectra such as the Wigner-Ville, evolutionary, instantaneous power, and physical spectrum. Our subsequent analysis focuses on the practically important case of nonstationary processes with negligible high-lag TF correlations (so-called underspread processes). We demonstrate that for underspread processes all TF spectra yield effectively identical results and satisfy several desirable properties at least approximately. We also show that Gabor frames provide approximate Karhunen-Loeve (KL) functions of underspread processes and TF spectra provide a corresponding approximate KL spectrum. Finally, we formulate simple approximate input-output relations for the TF spectra of underspread processes that are passed through underspread linear time-varying systems. All approximations are substantiated mathematically by upper bounds on the associated approximation errors. Our results establish a TF calculus for the second-order analysis and time-varying filtering of underspread processes that is as simple as the conventional spectral calculus for stationary processes