The power classes-quadratic time-frequency representations with scale covariance and dispersive time-shift covariance

We consider scale-covariant quadratic time-frequency representations (QTFRs) specifically suited for the analysis of signals passing through dispersive systems. These QTFRs satisfy a scale covariance property that is equal to the scale covariance property satisfied by the continuous wavelet transform and a covariance property with respect to generalized time shifts. We derive an existence/representation theorem that shows the exceptional role of time shifts corresponding to group delay functions that are proportional to powers of frequency. This motivates the definition of the power classes (PCs) of QTFR´s. The PCs contain the affine QTFR class as a special case, and thus, they extend the affine class. We show that the PCs can be defined axiomatically by the two covariance properties they satisfy, or they can be obtained from the affine class through a warping transformation. We discuss signal transformations related to the PCs, the description of the PCs by kernel functions, desirable properties and kernel constraints, and specific PC members. Furthermore, we consider three important PC subclasses, one of which contains the Bertrand (1992) P_k distributions. Finally, we comment on the discrete-time implementation of PC QTFRs, and we present simulation results that demonstrate the potential advantage of PC QTFRs