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

Abstract

For pt.I see ibid., vol.41, p.3425-444 (1993). Part I introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFRs). The present paper defines and studies four subclasses of the HC: (1) The focalized-kernel subclass of the HC is related to a time-frequency concentration property of QTFRs. It is analogous to the localized-kernel subclass of the affine QTFR class. (2) The affine subclass of the HC (affine HC) consists of all HC QTFRs that satisfy the conventional covariance property. It forms the intersection of the HC with the affine QTFR class. (3) The subclasses of the HC consist of all HC QTFRs that satisfy a power time-shift covariance property. They form the intersection of the HC with the recently introduced classes. (4) The power-warp subclass of the HC consists of all HC QTFRs 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 1D kernel functions. The affine HC is contained in both the localized kernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary Po-distribution by a convolution. We furthermore consider the properties of regularity and unitarity 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 Altes-Marinovich Q-distribution.