### Abstract

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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 3425-3444 |

Number of pages | 20 |

Journal | IEEE Transactions on Signal Processing |

Volume | 41 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1993 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Signal Processing*,

*41*(12), 3425-3444. https://doi.org/10.1109/78.258084

**Hyperbolic class of quadratic time-frequency representations Part I : Constant-Q warping, the hyperbolic paradigm, properties, and members.** / Papandreou-Suppappola, Antonia; Hlawatsch, Franz; Boudreaux-Bartels, G. Faye.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 41, no. 12, pp. 3425-3444. https://doi.org/10.1109/78.258084

}

TY - JOUR

T1 - Hyperbolic class of quadratic time-frequency representations Part I

T2 - Constant-Q warping, the hyperbolic paradigm, properties, and members

AU - Papandreou-Suppappola, Antonia

AU - Hlawatsch, Franz

AU - Boudreaux-Bartels, G. Faye

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.

UR - http://www.scopus.com/inward/record.url?scp=0027798109&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0027798109&partnerID=8YFLogxK

U2 - 10.1109/78.258084

DO - 10.1109/78.258084

M3 - Article

AN - SCOPUS:0027798109

VL - 41

SP - 3425

EP - 3444

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 12

ER -