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

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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. The 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 TF geometry and is thus called the hyperbolic class (HC). Two prominent TF representations previously considered in the literature, the Bertrand P/sub 0/ distribution and the Altes-Marinovic Q-distribution, are members of the new HC. The authors show that any hyperbolic TF representation is related to both the wideband ambiguity function and a ambiguity function. It is also shown that the HC is the class of all quadratic TF representations which are invariant to 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 TF geometry, and the description of the HC by 2D 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. >