Group delay shift covariant quadratic time-frequency representations

We propose classes of quadratic time-frequency representations (QTFRs) that are covariant to group delay shifts (GDSs). The GDS covariance QTFR property is important for analyzing signals propagating through dispersive systems with frequency-dependent characteristics. This is because a QTFR satisfying this property provides a succinct representation whenever the time shift is selected to match the frequency-dependent changes in the signal's group delay that may occur in dispersive systems. We obtain the GDS covariant classes from known QTFR classes (such as Cohen's (1995) class, the affine class, the hyperbolic class, and the power classes) using warping transformations that depend on the relevant group delay change. We provide the formulation of the GDS covariant classes using two-dimensional (2-D) kernel functions, and we list desirable QTFR properties and kernel constraints, as well as specific class members. We present known examples of the GDS covariant classes, and we provide a new class: the power exponential QTFR class. We study the localized-kernel subclasses of the GDS covariant classes that simplify the theoretical development as the kernels reduce from 2-D to one-dimensional (1-D) functions, and we develop various intersections between the QTFR classes. Finally, we present simulation results to demonstrate the advantage of using QTFRs that are matched to changes in the group delay of a signal.