ith 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 Scientique). It
results from a research eort 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 dierent phases of the development. 4 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
Denition . . . . . . . . . . . . . . . . . . . . . . . . . 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
Denitions and interpretation . . . . . . . . . . . . . . 41
3.2.2
Properties . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Sampling considerations . . . . . . . . . . . . . . . . . . . . . 43
5 6
CONTENTS
3.3.1
The discrete STFT . . . . . . . . . . . . . . . . . . . . 43
3.3.2
The Gabor Representation . . . . . . . . . . . . . . . . 44
3.3.3
The discrete wavelet transform . . . . . . . . . . . . . 46
3.4 From atomic decompositions to energy
distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4.1
The spectrogram . . . . . . . . . . . . . . . . . . . . . 48
3.4.2
The scalogram . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.3
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 54
4 Second class of solutions : the energy distributions
57
4.1 The Cohens class . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.1
The Wigner-Ville distribution . . . . . . . . . . . . . . 58
4.1.2
The Cohens class . . . . . . . . . . . . . . . . . . . . . 67
4.1.3
Link with the narrow-band ambiguity function . . . . . 72
4.1.4
Other important energy distributions . . . . . . . . . . 76
4.1.5
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 The ane class . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1
Axiomatic denition . . . . . . . . . . . . . . . . . . . 83
4.2.2
Some examples . . . . . . . . . . . . . . . . . . . . . . 86
4.2.3
Relation with the ambiguity domain . . . . . . . . . . 95
4.2.4
The ane Wigner distributions . . . . . . . . . . . . . 98
4.2.5
The pseudo ane Wigner distributions . . . . . . . . . 102
4.2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.3 The reassignment method . . . . . . . . . . . . . . . . . . . . 108
4.3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3.2
The reassignment of the spectrogram . . . . . . . . . . 109
4.3.3
Reassignment of the Cohens class representations . . . 111
4.3.4
Reassignment of the ane class representations . . . . 113
4.3.5
Numerical examples . . . . . . . . . . . . . . . . . . . 113
4.3.6
Connected approaches . . . . . . . . . . . . . . . . . . 115
4.3.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 116
5 Extraction of information from a time-frequency image
123
5.1 Moments and marginals . . . . . . . . . . . . . . . . . . . . . 123
5.1.1
Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.2
Marginals . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 More on interferences : information on phase . . . . . . . . . . 124
5.3 Renyi information . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 Time-frequency analysis : help to decision . . . . . . . . . . . 128
5.4.1
General considerations . . . . . . . . . . . . . . . . . . 128 CONTENTS
7
5.4.2
An example : detection and estimation of linear FM
signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.5 Analysis of local singularities . . . . . . . . . . . . . . . . . . . 132
GNU Free Documentation License
138
1. Applicability and denitions . . . . . . . . . . . . . . . . . . . . 139
2. Verbatim copying . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3. Copying in quantity . . . . . . . . . . . . . . . . . . . . . . . . . 142
4. Modications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5. Combining documents . . . . . . . . . . . . . . . . . . . . . . . . 144
6. Collections of documents . . . . . . . . . . . . . . . . . . . . . . 145
7. Aggregation with independent works . . . . . . . . . . . . . . . . 145
8. Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
9. Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
10. Future revisions of this license . . . . . . . . . . . . . . . . . . . 146
ADDENDUM: How to use this License for your documents . . . . . 147
Bibliography
148 8
CONTENTS Chapter 1
Introduction
1.1
Presentation
The Time-Frequency Toolbox is a collection of M-les developed for the
analysis of non-stationary signals using time-frequency distributions. This
toolbox includes two groups of les :
the signal generation les, which allow the synthesis of numerous kinds
of non-stationary signals ;
the processing les, including the time-frequency distributions and other
related processing functions.
As usual under MATLAB, each function of the toolbox has a help entry
that you can refer to by typing
>> help name_of_the_file
at the prompt of the matlab command window. In almost every case, a
simple example is given, which facilitates the use of the function.
Seven demonstration M-les are also available, which provide sequences
of examples illustrating the possibilities of the Time-Frequency Toolbox, and
following closely the plan of this tutorial. These les are :
tfdemo
Main menu of the demonstration
tfdemo1
Introduction
tfdemo2
Non-stationary signals
tfdemo3
Linear time-frequency representations
tfdemo4 Cohens class time-frequency distributions
tfdemo5
Ane class time-frequency distributions
tfdemo6
Reassigned time-frequency distributions
tfdemo7
Extraction of information
9 The aim of this Tutorial is to present the way to use the Time-Frequency
Toolbox, and also to introduce the reader in an illustrative and friendly
way to the theory of time-frequency analysis. We advise the reader, when
looking at a chapter of this tutorial, to run simultaneously the corresponding
demonstration le. In this way, he will have a good understanding of the
Toolbox.
1.2
Background, system requirements and
installation
This Toolbox is primarily intended for researchers and engineers with
some knowledge on signal processing theory. In particular, the concepts
of Fourier transform, Shannon sampling and stationarity are important to
understand the following features.
The Time-Frequency Toolbox assumes that MATLAB v.4.2c (or a later
version) is present on your system, as well as the Signal Processing Toolbox
v.3.0 (or a later version).
Instructions for installing this toolbox on a workstation or a large machine
are found in the MATLAB Installation Guide. Instructions for installing on
micro computers are found in the MATLAB Users Guide.
1.3
Introductory examples
1.3.1
Example 1
Let us consider rst a signal with constant amplitude, and with a linear
frequency modulation varying from 0 to 0.5 in normalized frequency (ratio of
the frequency in Hertz to the sampling frequency, with respect to the Shannon
sampling theorem). This signal is called a chirp, and as its frequency content
is varying with time, it is a non-stationary signal. To obtain such a signal, we
can use the M-le fmlin.m, which generates a linear frequency modulation
(see g. 1.1) :
>> sig1=fmlin(128,0,0.5);
>> plot(real(sig1));
From this time-domain representation, it is dicult (except for experienced
specialists) to say what kind of modulation is contained in this signal :
what are the initial and nal frequencies, is it a linear, parabolic, hyper-
bolic. . . frequency modulation ?
10 F