The Hyperbolic Class of Quadratic Time-Frequency Representations Part I: Constant-Q Warping, the Hyperbolic Paradigm, Properties, and Members

The time-frequency (TF) version of the wavelet transform and the “affine” quadratic/bilinear 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 P₀ 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.