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

Abstract

Abstract-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-Loève (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 timevarying 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.Index Terms-Evolutionary spectrum, Gabor expansion, instantaneous power spectrum, Karhunen-Loève (KL) expansion, nonstationary random processes, nonstationary spectral analysis, time-frequency (TF) analysis, time-varying systems, Wigner-Ville spectrum.

Time-Frequency ARMA Models and Parameter Estimators for Underspread Nonstationary Random Processes

“…This results in nonstationary processes with small high-lag temporal and spectral correlations or, equivalently, with a temporal correlation length that is much smaller than the duration over which the time-varying second-order statistics are approximately constant. Such underspread processes [28], [29] are encountered in many applications.…”

“…Simulation results demonstrate that our methods outperform existing TVAR, TVMA, and TVARMA parameter estimators with respect to accuracy and/or complexity. For processes that are not underspread (called "overspread" [28], [29]), the models discussed here will not be parsimonious and those estimators that involve an underspread approximation must be expected to exhibit poor performance.…”

“…The expected ambiguity function (EAF) of the process , denoted , is defined as [28], [29] (6) where is the inner product of two signals and .…”

“…The value of at a given TF lag point characterizes the average statistical correlations of any two components of separated by time lag and frequency lag . A nonstationary process is said to be underspread if its TF correlations are effectively zero for larger time and frequency lags [28], [29]. The EAF then is effectively zero outside a small region about the origin of the plane.…”

Localisation of Geometric Anisotropy

“…These facts justify the construction and study of the proposed operators. 1-D localisation operators have already been the focus of considerable study, see for example the references in [1], [4], [5]. Consecutive truncations form a possible method of constructing localisation operators: such procedures treat the space and spatial frequency variables inhomogeneously.…”