## Covariant Time-Frequency Representations Through Unitary Equivalence (1996)

Venue: | IEEE Signal Processing Letters |

Citations: | 9 - 2 self |

### BibTeX

@ARTICLE{Baraniuk96covarianttime-frequency,

author = {Richard G. Baraniuk},

title = {Covariant Time-Frequency Representations Through Unitary Equivalence},

journal = {IEEE Signal Processing Letters},

year = {1996},

volume = {3},

pages = {79--81}

}

### OpenURL

### Abstract

We propose a straightforward characterization of all quadratic time-frequency representations covariant to an important class of unitary signal transforms (namely, those having two continuous-valued parameters and an underlying group structure). Thanks to a fundamental theorem from the theory of Lie groups, we can describe these representations simply in terms of unitary transformations of the well-known Cohen's and affine classes. This work was supported by the National Science Foundation, grant no. MIP--9457438, the Office of Naval Research, grant no. N00014--95--1--0849, and the Texas Advanced Technology Program, grant no. TX--ATP 003604-- 002. I. Introduction Quadratic time-frequency representations (TFRs) have found wide application in problems requiring time-varying spectral analysis [1, 2]. Since the distribution of signal energy jointly over time and frequency coordinates does not have a unique representation, there exist many different TFRs and many different ways to obtai...

### Citations

65 |
Time–Frequency Analysis. Englewood Cliffs
- Cohen
- 1995
(Show Context)
Citation Context ... Technology Program, grant no. TX--ATP 003604-- 002. 1 I. Introduction Quadratic time-frequency representations (TFRs) have found wide application in problems requiring time-varying spectral analysis =-=[1, 2]-=-. Since the distribution of signal energy jointly over time and frequency coordinates does not have a unique representation, there exist many different TFRs and many different ways to obtain them. Up ... |

48 | Unitary equivalence: A new twist on signal processing,” submitted to
- Baraniuk, Jones
- 1993
(Show Context)
Citation Context ...time shifts" and scale changes, while the TFRs of the power classes [6] are covariant to "power time shifts" and scale changes. Unitarily transformed Cohen's and affine classes furnish =-=even more TFRs [7, 8]. While ex-=-tremely simple both in concept and in application, the unitary equivalence or "warping" procedure that generates these TFRs leads at once to an infinite number of new TFR classes covariant t... |

35 | Time-scale energy distributions: A general class extending wavelet transforms - Rioul, Flandrin - 1992 |

21 |
and P.Bertrand, “A class of affine Wigner functions with extended covariance properties
- Bertrand
- 1992
(Show Context)
Citation Context ...class TFRs, because T and F comprise the heart of the unitary representation on L 2 (IR) of the Weyl-Heisenberg group, with (Fm 1 T n 1 )(F m 2 T n 2 ) = e \Gammaj 2m 2 n 1 Fm 1 +m 2 T n 1 +n2 : (See =-=[4, 9, 10]-=- for more details on the role of group theory in time-frequency analysis.) Each TFR in the affine class [2--4] can be expressed as (Qs)(t; f) = ZZ (As)(`; ) /(`=f; f) e \Gammaj 2(`t+f) d` d; with kern... |

19 |
Wideband ambiguity functions and the affine Wigner distributions,” submitted to
- Shenoy, Parks
- 1993
(Show Context)
Citation Context ...l for Cohen's class and affine class TFRs. Physical considerations (invertibility, composition, etc.) dictate that each of these transformations be a unitary group representation with group law "=-=ffl" [4, 9, 10] G (p 1 ;q-=- 1 ) G (p 2 ;q 2 ) = G (p 1 ;q 1 )ffl(p 2 ;q 2 ) : (5) We say a TFR (Ps)(t; f) is covariant to G (p;q) if (PG (p;q) s)(t; f) = (Ps)(t; f) \Pi (p; q); where "\Pi" is the representation of G (... |

14 |
Boudreaux-Bartels, "The hyperbolic class of quadratic time-frequency representations. Part I: Constant-Q warping, the hyperbolic paradigm, properties, and members
- Papandreou, Hlawatsch, et al.
- 1993
(Show Context)
Citation Context ...changes are not the only important signal transformations occurring in nature, several new TFR classes matching different transformations have been proposed recently. The TFRs of the hyperbolic class =-=[5] are covariant to &q-=-uot;hyperbolic time shifts" and scale changes, while the TFRs of the power classes [6] are covariant to "power time shifts" and scale changes. Unitarily transformed Cohen's and affine c... |

11 | Marginals vs. covariance in joint distribution theory
- Baraniuk
- 1995
(Show Context)
Citation Context ...class TFRs, because T and F comprise the heart of the unitary representation on L 2 (IR) of the Weyl-Heisenberg group, with (Fm 1 T n 1 )(F m 2 T n 2 ) = e \Gammaj 2m 2 n 1 Fm 1 +m 2 T n 1 +n2 : (See =-=[4, 9, 10]-=- for more details on the role of group theory in time-frequency analysis.) Each TFR in the affine class [2--4] can be expressed as (Qs)(t; f) = ZZ (As)(`; ) /(`=f; f) e \Gammaj 2(`t+f) d` d; with kern... |

10 | Displacement-covariant time-frequency energy distributions
- Hlawatsch, Bölcskei
- 1995
(Show Context)
Citation Context ...PG (p 1 ;q 1 ) G (p 2 ;q 2 ) s)(t; f) = (Ps)(t; f) \Pi [(p 1 ; q 1 ) ffl (p 2 ; q 2 )]: A similar (and equivalent) approach to covariance has been developed independently by Hlawatsch and Bolcskei in =-=[11,12]. In their-=- terminology, G is a time-frequency displacement operator with "\Pi" the associated displacement function. 5 Examples of displacement operators and functions include: G (m;n) = FmT n and (1)... |

9 | On joint distributions of arbitrary variables
- Baraniuk, Cohen
- 1995
(Show Context)
Citation Context ...time shifts" and scale changes, while the TFRs of the power classes [6] are covariant to "power time shifts" and scale changes. Unitarily transformed Cohen's and affine classes furnish =-=even more TFRs [7, 8]. While ex-=-tremely simple both in concept and in application, the unitary equivalence or "warping" procedure that generates these TFRs leads at once to an infinite number of new TFR classes covariant t... |

9 |
Unified theory of displacementcovariant time-frequency analysis
- Hlawatsch, Bölcskei
- 1994
(Show Context)
Citation Context ...PG (p 1 ;q 1 ) G (p 2 ;q 2 ) s)(t; f) = (Ps)(t; f) \Pi [(p 1 ; q 1 ) ffl (p 2 ; q 2 )]: A similar (and equivalent) approach to covariance has been developed independently by Hlawatsch and Bolcskei in =-=[11,12]. In their-=- terminology, G is a time-frequency displacement operator with "\Pi" the associated displacement function. 5 Examples of displacement operators and functions include: G (m;n) = FmT n and (1)... |

5 |
Boudreaux-Bartels, "The power classes of quadratic time-frequency representations: A generalization of the affine and hyperbolic classes
- Hlawatsch, Papandreou, et al.
- 1993
(Show Context)
Citation Context ...s matching different transformations have been proposed recently. The TFRs of the hyperbolic class [5] are covariant to "hyperbolic time shifts" and scale changes, while the TFRs of the powe=-=r classes [6] are covar-=-iant to "power time shifts" and scale changes. Unitarily transformed Cohen's and affine classes furnish even more TFRs [7, 8]. While extremely simple both in concept and in application, the ... |