## Optimum Time-Frequency Representations for the Classification and Detection of Signals (1995)

Venue: | Applied Signal Processing |

Citations: | 9 - 2 self |

### BibTeX

@INPROCEEDINGS{Heitz95optimumtime-frequency,

author = {Christoph Heitz and Fakultat Fur Physik},

title = {Optimum Time-Frequency Representations for the Classification and Detection of Signals},

booktitle = {Applied Signal Processing},

year = {1995},

pages = {124--143}

}

### OpenURL

### Abstract

Time-frequency representations (TFRs) are powerful tools for signal analysis and thus widely used in signal processing. It is well known, however, that there is no single TFR which is "the best" for all problems. It is still an unsolved problem how to determine the optimum TFR for a given signal class and analysis task. In this article we develop a new theory of optimum TFRs for classification and detection problems, where two (or more) classes are to be discriminated from one another. Usually this is performed by mapping the signals into some representation space (e.g. a feature space) where a distinction is easily possible. If we regard TFRs as representations in time-frequency space we have to look for TFRs where all signals of one class are similar to each other but dissimilar to all signals of other classes. After introducing a quantitative measure for similarity or, in contrast, for the distance, a measure for the quality of a given TFR for a given classification or detection pro...

### Citations

4142 |
Pattern Classification and Scene Analysis
- Duda, Hart
(Show Context)
Citation Context ...ean distance between the classes is 20 times larger than the mean inner class distance 1=2 [d(R 1 ; R 1 ) + d(R 2 ; R 2 )]. The quantity m is similar to the Fisher criterion in discriminance analysis =-=[6, 8]-=- for the case of functions on the unit sphere. If, furthermore, the inner class distances d(R i ; R i ) are equal, m resembles the test quantity of the t-test [14]. There, a large test quantity means ... |

2889 |
Introduction to Statistical Pattern Recognition, second edition
- Fukunaga
- 1990
(Show Context)
Citation Context ... on the evaluation of some sort of distance between the signal and representatives of the different classes in the feature space. The classificators differ by the definition of their distance measure =-=[8]-=-. It is a fundamental problem that in order to find a good feature set one has to know in advance which properties of the signals are likely to be important for the classification. In this article, in... |

834 |
Numerical Recipes
- Press, Flannery, et al.
- 1986
(Show Context)
Citation Context ...riterion in discriminance analysis [6, 8] for the case of functions on the unit sphere. If, furthermore, the inner class distances d(R i ; R i ) are equal, m resembles the test quantity of the t-test =-=[14]-=-. There, a large test quantity means that the classes are well separated compared to their variances. Regarding the TFRs of the functions, instead of the two regions R 1 and R 2 we consider now the co... |

597 |
Spectral Analysis and Time Series
- Priestley
- 1981
(Show Context)
Citation Context ...h(t) sin(!t + ') + affl(t) The noise parameters a has been set to 0:5 in this example. For h(t) j 1 this is a standard problem of detection theory for which a lot of solutions already exist (see e.g. =-=[15]). In our ca-=-se, for detection we have to maximize ~ m \Phi as defined in section 4. The algorithm renders ~ m = 63:4 with an optimum kernel given by �� 0 = �� u ; �� 0 = 5:3u. Once again with this ker... |

85 |
Improved time-frequency representation of multicomponent signals using exponential kernels
- Choi, Williams
- 1989
(Show Context)
Citation Context ...ajor problem which is still unsolved for the general case: given a class of signals and a problem to be solved, which is the best kernel to be used? There have been some answers in the past (see e.g. =-=[3, 17, 13, 12]-=-), but each of them is restricted either to some very special signal classes (e.g. chirps) or to special problems. Since it is impossible to find a TFR which is optimum for all signals and problems, i... |

54 | A signal dependent time-frequency representation optimal kernel design
- Baraniuk, Jones
- 1993
(Show Context)
Citation Context ...ses (e.g. chirps) or to special problems. Since it is impossible to find a TFR which is optimum for all signals and problems, it has been argued that one should use data adaptive locally optimum TFRs =-=[11, 1, 2]-=-. There, for each point of the time-frequency plane another TFR is used. In this article we focus on the problem of globally optimum TFRs. We formulate this problem in the very general context of clas... |

44 | The Scale Representation
- Cohen
- 1993
(Show Context)
Citation Context ... no weighting of some time, frequency or time-frequency region. 2. The ideas of this article can be generalized for other quadratic signal representations, such as , e.g. , time-scale representations =-=[5, 16]-=-. The use of TFRs implies (see section 2) that the representation properties are independent of time and frequency shifts. That means that if all signals are shifted in time and/or frequency by the sa... |

38 |
Time-scale energy distributions: A general class extending wavelet transforms
- Rioul, Flandrin
- 1992
(Show Context)
Citation Context ... no weighting of some time, frequency or time-frequency region. 2. The ideas of this article can be generalized for other quadratic signal representations, such as , e.g. , time-scale representations =-=[5, 16]-=-. The use of TFRs implies (see section 2) that the representation properties are independent of time and frequency shifts. That means that if all signals are shifted in time and/or frequency by the sa... |

34 |
A high resolution data-adaptive time-frequency representation
- Jones, Parks
- 1990
(Show Context)
Citation Context ...ses (e.g. chirps) or to special problems. Since it is impossible to find a TFR which is optimum for all signals and problems, it has been argued that one should use data adaptive locally optimum TFRs =-=[11, 1, 2]-=-. There, for each point of the time-frequency plane another TFR is used. In this article we focus on the problem of globally optimum TFRs. We formulate this problem in the very general context of clas... |

34 |
The use of cone-shaped kernels for generalized time-frequency representations of nonstationary signals
- Zhao, Atlas, et al.
- 1990
(Show Context)
Citation Context ...ajor problem which is still unsolved for the general case: given a class of signals and a problem to be solved, which is the best kernel to be used? There have been some answers in the past (see e.g. =-=[3, 17, 13, 12]-=-), but each of them is restricted either to some very special signal classes (e.g. chirps) or to special problems. Since it is impossible to find a TFR which is optimum for all signals and problems, i... |

9 |
Boudreaux-Bartels, “On the optimality of the Wigner distribution for detection
- Kay, F
- 1985
(Show Context)
Citation Context ...ajor problem which is still unsolved for the general case: given a class of signals and a problem to be solved, which is the best kernel to be used? There have been some answers in the past (see e.g. =-=[3, 17, 13, 12]-=-), but each of them is restricted either to some very special signal classes (e.g. chirps) or to special problems. Since it is impossible to find a TFR which is optimum for all signals and problems, i... |

3 | Time-frequency--distributions -- a review
- Cohen
- 1989
(Show Context)
Citation Context ...on 1 . In the literature one finds many derivations of kernels which are optimized to extract special predefined properties of the signal, e.g. the chirp rate or the instantaneous frequency (see e.g. =-=[4]-=- for a review). However, these discussions are restricted to those very special cases where the property to be examined is known in advance and can be formulated in a simple way. Furthermore the class... |

2 |
inversion and conversion of bilinear signal representations
- Transformation
- 1985
(Show Context)
Citation Context ...a real TFR. This is the case if \Phi(��; ��) = \Phi (\Gamma��; \Gamma�� ): (1) ffl Non-uniqueness: In general the signal cannot be recovered out of its TFR, although this is possible f=-=or some kernels [10]-=-. Usually this fact is regarded as a disadvantage, because information is lost. However, in our context it may be advantageous, for if the dropped information is irrelevant to the problem, its suppres... |

2 |
Boudreaux-Bartels. Generalization of the Choi-Williams distribution and the Butterworth distribution for timefrequency analysis
- Papandreou, Faye
- 1993
(Show Context)
Citation Context |

1 | Using optimized time-frequency representations for acoustic quality tests
- Heitz
(Show Context)
Citation Context ...tor was faulty or not. In this section it is shortly described how, for one class of faults, this task could be performed by finding an optimum TFR. For additional information and details we refer to =-=[9]. One of the faults -=-to be detected was "beating". In regular time intervals of about 0:3s a short "tick" could be heard in the motor's noise. The origin of these beats can be explained as follows: The... |