The hyperbolic class of quadratic time-frequency representations. I. Constant-Q warping, the hyperbolic paradigm, properties, and members

Abstract

The time-frequency (TF) version of the wavelet transform and the "affine" quadraticlbilinear TF representations can be used for a TF analysis with constant-Q characteristic. This paper considers a new approach to constant-Q TF analysis. A specific TF warping transform is applied to Cohen's class of quadratic TF representations, which results in a new class of quadratic TF representations with constant-Q characteristic. The new class is related to a "hyperbolic TF geometry" and is thus called the hyperbolic class (HC). Two prominent TF representations previously considered in the literature, the Bertrand Po distribution and the Altes-Marinovic Q-distribution, are members of the new HC. We show that any hyperbolic TF representation is related to both the wideband ambiguity function and a "hyperbolic ambiguity function." It is also shown that the HC is the class of all quadratic TF representations which are invariant to "hyperbolic time-shifts" and TF scalings, operations which are important in the analysis of Doppler-invariant signals and self-similar random processes. The paper discusses the definition of the HC via constant-Q warping, some signal-theoretic fundamentals of the "hyperbolic TF geometry,'' and the description of the HC by 2-D kernel functions. Several members of the HC are considered, and a list of desirable properties of hyperbolic TF representations is given together with the associated kernel constraints. I. INTRODUCTION UADRATIC time-frequency representations (QTFR's) are useful in the analysis of nonstationary signals [l]. This paper discusses a new class of "constant-Q" QTFR's. By way of introduction, and to establish the notation used, we first give a brief review of two basic QTFR classes. In the following, let x (t) be a signal with Fourier transform X (f) = jYm x(t)e-'2*fr dt.

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.…”

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

“…case, the generalized time-shift covariance property in (2) simplifies to the hyperbolic time-shift covariance property [14]. The paper is organized as follows.…”

“…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.…”

“…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.…”

“…• 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].…”