F. Hlawatsch et al., The power classes - Quadratic time-frequency representations with scale covariance and dispersive time-shift covariance, IEEE SIGNAL, 47(11), 1999, pp. 3067-3083
We consider scale-covariant quadratic time-frequency representations (QTFR'
s) specifically suited for the analysis of signals passing through dispersi
ve systems. These QTFR's satisfy a scale covariance property that is equal
to the scale covariance property satisfied by the continuous wavelet transf
orm and a covariance property with respect to generalized time shifts. We d
erive an existence/representation theorem that shows the exceptional role o
f time shifts corresponding to group delay functions that are proportional
to powers of frequency. This motivates the definition of the power classes
(PC's) of QTFR's. The PC's contain the affine QTFR class as a special case,
and thus, they extend the affine class. We show that the PC's can be defin
ed axiomatically by the two covariance properties they satisfy, or they can
be obtained from the affine class through a warping transformation. We dis
cuss signal transformations related to the PC's, the description of the PC'
s by kernel functions, desirable properties and kernel constraints, and spe
cific PC members. Furthermore, we consider three important PC subclasses, o
ne of which contains the Bertrand P-k distributions. Finally, we comment on
the discrete-time implementation of PC QTFR's, and we present simulation r
esults that demonstrate the potential advantage of PC QTFR's.