THE HYPERBOLIC CLASS OF QUADRATIC TIME-FREQUENCY REPRESENTATIONS .2. 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 f or 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 time-frequency concentration prope rty 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 cla ss. The power subclasses of the HC consist of all HC QTFR's that satis fy a ''power time-shift'' covariance property. They form the intersect ion of the HC with the recently introduced power classes. The power-wa rp subclass of the HC consists of all HC QTFR's that satisfy a covaria nce to power-law frequency warpings. It is the HC counterpart of the s hift-scale covariant subclass of Cohen's class. All of these subclasse s are characterized by 1-D kernel functions. It is shown that the affi ne HC is contained in both the localized-kernel hyperbolic subclass an d the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary P-0-distribution by a convolutio n. We furthermore consider the properties of regularity (invertibility of a QTFR) and unitarity (preservation of inner products, Moyal's for mula) in the HC. The calculus of inverse kernels is developed, and imp ortant implications of regularity and unitarity are summarized. The re sults comprise a general method for least-squares signal synthesis and new relations for the Altes-Marinovich Q-distribution.