THE HYPERBOLIC CLASS OF QUADRATIC TIME-FREQUENCY REPRESENTATIONS .1. 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 TP 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 P-0 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.