Lecture 1
Introduction to Joint
Time-Frequency Analysis
Time-frequency analysis, adaptive
filtering and source separation
José Biurrun Manresa
08.02.2011
Lecture 1 – Introduction to JTFA
Overview of the course
• Joint Time-Frequency Analysis (JBM)
• Short Time Fourier Transform
• Wigner-Ville distribution
• Kernel properties and design in Cohen’s class time-frequency
distributions
• Wavelet analysis (ENK)
• Continuous wavelet transform
• Discrete wavelet transform
• Wavelet packets
2
Lecture 1 – Introduction to JTFA
Overview of the course
• Biomedical Applications of JTFA
• Applications of JTFA to biomedical signals (JBM)
• Applications of wavelet analysis to biomedical signals (ENK)
• Adaptive filtering
• Source separation
3
Lecture 1 – Introduction to JTFA
Overview of the course
• Exam
• Mini-project with oral defense
• Students will form groups of maximum 4 people each
• Each group will be provided with previously recorded
biological signals with a clearly stated research question that
the group has to solve
• Each group has to deliver a small report
• The mini project will be defended individually in an oral
session in order to assess the theoretical level of each student
4
Lecture 1 – Introduction to JTFA
Classical spectral analysis
• In classical spectral analysis, we isolate some components of a
complex system in order to understand its nature
• In particular, Fourier analysis uses a family of sine functions at
harmonically related frequencies
• Sinusoidal functions contain energy at only one specific frequency
5
Lecture 1 – Introduction to JTFA
What the Fourier transform misses...
6
Lecture 1 – Introduction to JTFA
Timing is also important!
• Classical spectral analysis provides a good description of the
frequencies in a waveform, but not the timing
• The Fourier transform of a musical passage tells us which notes are
played, but it is extremely difficult to figure out when they are played
• The timing information must be somewhere, because the
transformation is bilateral
• Timing is encoded in the phase!
7
Lecture 1 – Introduction to JTFA
Timing is also important!
• For many signals, it is not enough to know the global frequecy
content
• We also need to know the timing in which these changes in
frequency occurr, in order to follow the dynamics of the signal
• Which signals are these?
• Non-stationary, transient, whose parameters change with time
(derived from a non-LTI system)
8
Lecture 1 – Introduction to JTFA
Some toy examples...
9
Lecture 1 – Introduction to JTFA
Some toy examples...
10
Lecture 1 – Introduction to JTFA
Some toy examples...
11
Lecture 1 – Introduction to JTFA
... And some real examples
12
Lecture 1 – Introduction to JTFA
... And some real examples
13
Lecture 1 – Introduction to JTFA
... And some real examples
14
Lecture 1 – Introduction to JTFA
Analyze by segments using the FT
15
Lecture 1 – Introduction to JTFA
Short-Time Fourier Transform (STFT)
• Basic approach: slicing the wavefor of interest into a number
of short segments and performing the Fourier transform on
each one of them
• A window function is applied to a segment of the signal, thus
isolating it from the overall waveform
16
Lecture 1 – Introduction to JTFA
Short-Time Fourier Transform (STFT)
• The windows function is chosen to leave the signal more or
less unaltered around time but to supress the signal for
times distant from the time of interest
fornear
0forfarawayfrom
• Common windows: Hamming, Hanning, Bartlett, BlackmanHarris, Kaiser, Gabor, etc.
• What about a square window?
17
Lecture 1 – Introduction to JTFA
Short-Time Fourier Transform (STFT)
• Since the modified signal emphasizes the original signal around
the time , the Fourier transform will reflect the distribution
of frequencies around that time
1
2
1
2
!
!
18
Lecture 1 – Introduction to JTFA
Short-Time Fourier Transform (STFT)
19
Lecture 1 – Introduction to JTFA
Short-Time Fourier Transform (STFT)
• The energy density spectrum at time is
"#$ , &
1
2
!
&
• For each different time we get a different spectrum, and the
totality of these spectra is the time-frequency distribution "#$ .
• The most common name for this distribution is spectrogram
20
Lecture 1 – Introduction to JTFA
Spectrogram
21
Lecture 1 – Introduction to JTFA
Spectrogram
22
Lecture 1 – Introduction to JTFA
Spectrogram
• If we want to estimate frequency properties for a particular
time, we take narrow windows (wideband spectrograms)
• However, if we want to estimate time properties for a
particular time, we take long windows (narrowband
spectrograms)
• In the latter case, the Short-Time Fourier Transform may be
appropriately called the Long-Time Fourier Transform or the
Short-Frequency Time Transform
23
Lecture 1 – Introduction to JTFA
Spectrogram
• Another way to look at it is as a change of the function basis
Fourier
Transform
STFT
Narrowband
STFT
Wideband
24
Lecture 1 – Introduction to JTFA
Short-Frequency Time Transform
• It is defined by
1
2
*
' ( ′ !′
• The window in time is related with the window ( in
frequency by
( 1
2
!
25
Lecture 1 – Introduction to JTFA
Short-Frequency Time Transform
• Then
• Since the distribution is the absolute square, the phase factor
does not enter into it and the joint distribution can be
defined as
"#$ , &
&
26
Lecture 1 – Introduction to JTFA
General properties of the spectrogram
• Total energy
+#$ , "#$ , !! & !
- & !
• If the energy of the windows is selected to be one, the energy
of the spectrogram is equal to the energy of the signal
27
Lecture 1 – Introduction to JTFA
General properties of the spectrogram
• Mean time and mean frequency
#$ , #$ , & !!
&
. /
!! . 0 /
• If the window is selected so that its mean time and frequency
are zero (e.g. a window symetrical in time whose spectrum is
symmetrical in frequency), then the mean time and frequency
of the spectrogram will be equal to those of the signal
28
Lecture 1 – Introduction to JTFA
General properties of the spectrogram
• Time and frequency variances
1& , #$
&
1& , #$
&
& !
&
!
• Time and frequency std. devs. 1 and 1 have to satisfy
1 1 2 132
29
Lecture 1 – Introduction to JTFA
Uncertainty principle
30
Lecture 1 – Introduction to JTFA
Uncertainty principle
• We cannot simultaneously know time and frequency aspects
of a signal at an arbitrary resolution
• For each value of and , there is a rectangle whose sides are
determined by 1 and 1 , and whose area is at least 4⁄&
• When the window is selected, the inequality becomes an
equality
• Since the window is always equal and just shifts in time,
the STFT has an uniform resolution both in time and
frequency
31
Lecture 1 – Introduction to JTFA
Uncertainty principle
32
Lecture 1 – Introduction to JTFA
Frequency
Uncertainty principle
Time
33
Lecture 1 – Introduction to JTFA
References and further reading
• Time Frequency Analysis: Theory and Applications by Leon
Cohen. Prentice Hall; 1994. Chapters 6 and 7
• Biosignal and Medical Image Processing, Second Edition by
John L. Semmlow. CRC press; 2009. chapter 6 pp. 139-146
• The Time Frequency Toolbox tutorial
(http://tftb.nongnu.org/tutorial.pdf)
• Material from Signals and System course, FI-UNER
34