The hyperbolic class of quadratic time-frequency representations. II. Subclasses, intersection with the affine and power classes, regularity, and unitarity

Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFR’s) as a new framework for constant-Q time-frequency analysis. The present Part II defines and studies the following four subclasses of the HC: • The localized-kernel subclass of the HC is related to a timefrequency concentration property of QTFR’s. It is analogous to the localized-kernel subclass of the affine QTFR class. • The affine subclass of the HC (affine HC) consists of all HC QTFR’s that satisfy the conventional time-shift covariance property. It forms the intersection of the HC with the affine QTFR class. • The power subclasses of the HC consist of all HC QTFR’s that satisfy a “power time-shift” covariance property. They form the intersection of the HC with the recently introduced power classes. • The power-warp subclass of the HC consists of all HC QTFR’s that satisfy a covariance to power-law frequency warpings. It is the HC counterpart of the shift-scale covariant subclass of Cohen’s class. All of these subclasses are characterized by 1-D kernel functions. It is shown that the affine HC is contained in both the localizedkernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary P0-distribution by a convolution. We furthermore consider the properties of regularity (invertibility of a QTFR) and unitarity (preservation of inner products, Moyal’s formula) in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for least-squares signal synthesis and new relations for the AltesMarinovich Q-distribution.