### Abstract

Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFR's) as a new framework for constant-Q time-frequency analysis. The present Part II defines and studies the following four subclasses of the HC: The localized-kernel subclass of the HC is related to a timefrequency concentration property of QTFR's. It is analogous to the localized-kernel subclass of the affine QTFR class. The affine subclass of the HC (affine HC) consists of all HC QTFR's that satisfy the conventional time-shift covariance property. It forms the intersection of the HC with the affine QTFR class. The power subclasses of the HC consist of all HC QTFR's that satisfy a "power time-shift" covariance property. They form the intersection of the HC with the recently introduced power classes. The power-warp subclass of the HC consists of all HC QTFR's that satisfy a covariance to power-law frequency warpings. It is the HC counterpart of the shift-scale covariant subclass of Cohen's class. All of these subclasses are characterized by 1-D kernel functions. It is shown that the affine HC is contained in both the localizedkernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary P0 -distribution by a convolution. We furthermore consider the properties of regularity (invert ibility of a QTFR) and unitarity (preservation of inner products, Moyal's formula) in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for least-squares signal synthesis and new relations for the AltesMarinovich Q-distribution.

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

Pages (from-to) | 303-315 |

Number of pages | 13 |

Journal | IEEE Transactions on Signal Processing |

Volume | 45 |

Issue number | 2 |

State | Published - 1997 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Signal Processing

### Cite this

*IEEE Transactions on Signal Processing*,

*45*(2), 303-315.

**The hyperbolic class of quadratic time-frequency representations-part ii : subclasses, intersection with the affine and power classes, regularity, and unitarity.** / Hlawatsch, Franz; Papandreou-Suppappola, Antonia; Paye Boudreaux-Bartels, G.; Hlawatsch, F.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 45, no. 2, pp. 303-315.

}

TY - JOUR

T1 - The hyperbolic class of quadratic time-frequency representations-part ii

T2 - subclasses, intersection with the affine and power classes, regularity, and unitarity

AU - Hlawatsch, Franz

AU - Papandreou-Suppappola, Antonia

AU - Paye Boudreaux-Bartels, G.

AU - Hlawatsch, F.

PY - 1997

Y1 - 1997

N2 - Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFR's) as a new framework for constant-Q time-frequency analysis. The present Part II defines and studies the following four subclasses of the HC: The localized-kernel subclass of the HC is related to a timefrequency concentration property of QTFR's. It is analogous to the localized-kernel subclass of the affine QTFR class. The affine subclass of the HC (affine HC) consists of all HC QTFR's that satisfy the conventional time-shift covariance property. It forms the intersection of the HC with the affine QTFR class. The power subclasses of the HC consist of all HC QTFR's that satisfy a "power time-shift" covariance property. They form the intersection of the HC with the recently introduced power classes. The power-warp subclass of the HC consists of all HC QTFR's that satisfy a covariance to power-law frequency warpings. It is the HC counterpart of the shift-scale covariant subclass of Cohen's class. All of these subclasses are characterized by 1-D kernel functions. It is shown that the affine HC is contained in both the localizedkernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary P0 -distribution by a convolution. We furthermore consider the properties of regularity (invert ibility of a QTFR) and unitarity (preservation of inner products, Moyal's formula) in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for least-squares signal synthesis and new relations for the AltesMarinovich Q-distribution.

AB - Part I of this paper introduced the hyperbolic class (HC) of quadratic/bilinear time-frequency representations (QTFR's) as a new framework for constant-Q time-frequency analysis. The present Part II defines and studies the following four subclasses of the HC: The localized-kernel subclass of the HC is related to a timefrequency concentration property of QTFR's. It is analogous to the localized-kernel subclass of the affine QTFR class. The affine subclass of the HC (affine HC) consists of all HC QTFR's that satisfy the conventional time-shift covariance property. It forms the intersection of the HC with the affine QTFR class. The power subclasses of the HC consist of all HC QTFR's that satisfy a "power time-shift" covariance property. They form the intersection of the HC with the recently introduced power classes. The power-warp subclass of the HC consists of all HC QTFR's that satisfy a covariance to power-law frequency warpings. It is the HC counterpart of the shift-scale covariant subclass of Cohen's class. All of these subclasses are characterized by 1-D kernel functions. It is shown that the affine HC is contained in both the localizedkernel hyperbolic subclass and the localized-kernel affine subclass and that any affine HC QTFR can be derived from the Bertrand unitary P0 -distribution by a convolution. We furthermore consider the properties of regularity (invert ibility of a QTFR) and unitarity (preservation of inner products, Moyal's formula) in the HC. The calculus of inverse kernels is developed, and important implications of regularity and unitarity are summarized. The results comprise a general method for least-squares signal synthesis and new relations for the AltesMarinovich Q-distribution.

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M3 - Article

VL - 45

SP - 303

EP - 315

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 2

ER -