### 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) |
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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 |

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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