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

Abstract

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 P 0-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 Altes-Marinovich Q-distribution.

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

“…• The hyperbolic class [27,[29][30][31][32][33][34] provides an alternative framework for constant-Q analysis. It includes QTFRs that are covariant to hyperbolic time shifts and TF scalings such as the AltesMarinovich Q-distribution and the Bertrand P 0 -distribution.…”

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

“…In a similar manner, application of the localized-kernel principle within the hyperbolic class led to the definition of the hyperbolic localized-kernel subclass [15]. Here, we will introduce and discuss the localized-kernel subclass of PC , denoted LPC , that is conceptually analogous to the affine and hyperbolic localized-kernel subclasses.…”

“…The PC subclass that consists of all PC QTFR's satisfying the hyperbolic time-shift covariance (40) forms the intersection of the PC with the hyperbolic class. It will therefore be called the hyperbolic-power intersection subclass of PC or, briefly, HPC [15]- [17]; its members will be denoted . Since the hyperbolic class is only defined for analytic signals and , the HPC is likewise defined only for analytic signals and .…”

“…In addition, the kernel structure (35) is also important as it is necessary for the following QTFR property (cf. [15], [53]): For a signal with phase function , the QTFR is perfectly concentrated along the corresponding group delay curve , i.e., for all (37) Here, is a given amplitude function, and is a given phase function. The next theorem, whose proof is similar to the proof of an analogous theorem for the hyperbolic localized-kernel subclass [15], states that (37) can only be satisfied by a PC QTFR if its kernel has the localized-kernel structure in (35).…”

Range bias modeling for hyperbolic frequency modulated waveforms in target tracking

“…HFM is a popular waveform for active acoustic systems, and a comprehensive time-frequency analysis on it can be found in [10] and [11]. HFM waveforms have been used in reverberation suppression [12], target phase extraction [5], and Doppler estimation [6] [7]; this paper dedicates itself to target tracking.…”