Quadratic Time-Frequency Representations with Scale Covariance and Generalized Time-Shift Covariance: A Unified Framework for the Affine, Hyperbolic, and Power Classes

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

“…The logarithmic form of leads to the hyperbolic class considered in [14]- [17]. The power function , corresponding to the powerlaw group delay function with , leads to the power classes studied in Sections III through VIII.…”

“…An alternative framework for constant-Q time-frequency analysis is the hyperbolic class [14]- [17], which consists of QTFR's that are covariant to time-frequency scalings and hyperbolic time shifts.…”

“…It will be shown that powerfunction group delays play a particularly important role for this type of QTFR's since they are the only functions that lead to maximally wide QTFR classes, i.e., QTFR classes that are parameterized by an arbitrary two-dimensional (2-D) kernel function (see Section II). This will motivate the definition of the power classes (PC's) of QTFR's, for which the group delay is described by a power function 1 [16], [17], [23]- [25]. Subsequently, the PC's will be studied in some detail.…”

“…• The hyperbolic QTFR class [14]- [17] is obtained with the logarithmic phase function and the hyperbolic group delay function . In this 1 At this point, we note that alternative methods for a time-frequency analysis of signals localized along a power-law (or, more generally, polynomial) group delay are the time-frequency representations proposed in [26]- [28].…”

“…In Section III, we develop the basic theory of the PC's. We show that each PC can be derived from the affine class by a unitary "power warping" mapping similar conceptually to the warping method used to derive the hyperbolic class from Cohen's class [14], [16], [17], [29]- [31]. We discuss the relation of the PC's to a signal expansion into "power impulses," which generalizes the conventional Fourier transform.…”