TY - JOUR

T1 - The Hyperbolic Class of Quadratic Time-Frequency Representations Part I

T2 - Constant-Q Warping, the Hyperbolic Paradigm, Properties, and Members

AU - Papandreou, Antonia

AU - Boudreaux-Bartels, G. Faye

AU - Hlawatsch, Franz

N1 - Funding Information:
Manuscript received September 8, 1992; revised May 5, 1993. The Guest Editor coordinating the review of this paper and approving it for publication was Dr. Patrick Flandrin. This work was supported in part by Office of Naval Research under Grants N00014-89-J-1812 and N00014-92-J-1499 and in part by the Fonds zur Forderung der wisaenschaftlichen Forschung under Grants P7354-PHY and J0530-TEC.

PY - 1993/12

Y1 - 1993/12

N2 - 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.

AB - 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.

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U2 - 10.1109/78.258084

DO - 10.1109/78.258084

M3 - Article

AN - SCOPUS:0027798109

SN - 1053-587X

VL - 41

SP - 3425

EP - 3444

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

IS - 12

ER -