Unified framework for the scale covariant affine, hyperbolic, and power class quadratic time-frequency representations using generalized time shifts

We propose a framework that unifies and extends the affine, hyperbolic, and power classes of quadratic time-frequency representations (QTFRs). These QTFR classes satisfy the scale covariance property, important in multiresolution analysis, and a generalized time-shift covariance property, important in the analysis of signals propagating through dispersive systems. We provide a general class formulation in terms of 2-D kernels, a generalized signal expansion, a list of desirable QTFR properties with kernel constraints, and a 'central QTFR' generalizing the Wigner distribution and the Altes-Marinovich Q-distribution. We also propose two generalized time-shift covariant (not, in general, scale covariant) QTFR classes by applying a generalized warping to Cohen's class and to the affine class.