In this dissertation, we review linear time-frequency representations, such as the wavelet transform, and quadratic time-frequency representations (QTFRs), such as the Wigner distribution (WD). We summarize QTFR classifications, including Cohen's class, the affine class, the hyperbolic class, and other warped Cohen's classes. We provide alternative 'Normal formulations', kernels of QTFR members, and desirable properties in terms of kernel constraints.
We review Fourier transform based higher order moment (and cumulant) functions and higher order moment (and cumulant) spectra. We also review Mellin transform based higher order spectra for self-similar (scaling-stationary) random processes.
We unify existing HO-TFR theory by describing both multi-time and multi-frequency classes of higher order time-frequency representations (HO-TFRs) in terms of alternative Normal formulations, kernels of HO-TFR members, and properties in terms of kernel constraints.
We propose the higher order generalized warped WD (HOG-WD) for analyzing signals and systems with dispersive instantaneous frequency characteristics. The HOG-WD generalizes the higher order WD and the higher order Altes-Marinovic Q-distribution and extends to higher order the quadratic generalized warped WD. We derive desirable properties and discuss links to existing HO-TFRs and QTFRs.
We define a class of HO-TFRs based on the HOG-WD preserving generalized frequency shifts and generalized warped time shifts. Special cases include the higher order Cohen's and hyperbolic classes, and the new power warped and exponentially warped Cohen's classes. We define alternative Normal formulations, kernels of HO-TFR members, and additional properties in terms of kernel constraints. We generalize higher order moment functions and their associated generalized higher order spectra for random processes with dispersive phase laws. We apply our new power warped bispectrum to phase coupled sinusoids with a power dispersive phase law.
We extend to higher order the quadratic affine class. Our affine HO-TFRs preserve scale and time-shift covariance. We provide alternative Normal formulations, HO-TFR members, including the new higher order scalogram which is related to the wavelet transform. We list additional desirable properties and derive their kernel constraints. We derive the higher order affine-Cohen intersection of HO-TFRs. We warp our higher order affine class and discuss possible connections to higher order extensions of the exponential and power classes.
