Time-Frequency
Toolbox
For Use with MATLAB
François Auger *
Patrick Flandrin *
Paulo Gonçalvès °
Olivier Lemoine *
* CNRS (France)
° Rice University (USA)
1995-1996
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Copyright (C) 1996 CNRS (France) and Rice University (USA).
Permission is granted to copy, distribute and/or modify this document under
the terms of the GNU Free Documentation License, Version 1.2 or any later
version published by the Free Software Foundation; with no Invariant Sec-
tions, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license
is included in the section entitled ”GNU Free Documentation License”.
The Time-Frequency Toolbox has been mainly developed under the aus-
pices of the French CNRS (Centre National de la Recherche Scientifique). It
results from a research effort conducted within its Groupements de Recherche
”Traitement du Signal et Images” (O. Macchi) and ”Information, Signal et
Images” (J.-M. Chassery). Parts of the Toolbox have also been developed at
Rice University, when one of the authors (PG) was visiting the Department
of Electrical and Computer Engineering, supported by NSF. Supporting in-
stitutions are gratefully acknowledged, as well as M. Guglielmi, M. Najim,
R. Settineri, R.G. Baraniuk, M. Chausse, D. Roche, E. Chassande-Mottin,
O. Michel and P. Abry for their help at different phases of the development.
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Contents
1 Introduction 9
1.1 Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2 Background, system requirements and
installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Introductory examples . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Non stationary signals 19
2.1 Time representation and frequency representation . . . . . . . 19
2.2 Localization and the Heisenberg-Gabor
principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Instantaneous frequency . . . . . . . . . . . . . . . . . . . . . 22
2.4 Group delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 About stationarity . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 How to synthesize a mono-component non-stationary signal . . 26
2.7 What about multi-component non-stationary signals ? . . . . 29
3 First class of solutions : the atomic decompositions 33
3.1 The Short-Time Fourier Transform . . . . . . . . . . . . . . . 33
3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.2 An example . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.3 Some properties . . . . . . . . . . . . . . . . . . . . . . 36
3.1.4 Time-frequency resolution . . . . . . . . . . . . . . . . 37
3.2 Time-scale analysis and the wavelet transform . . . . . . . . . 40
3.2.1 Definitions and interpretation . . . . . . . . . . . . . . 41
3.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Sampling considerations . . . . . . . . . . . . . . . . . . . . . 43
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