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Dear Soren,
<p>If Gaussian dilates and translates it gives better resolution in time-frequency
domain which is S-Transform. But, if Gaussian window does not dilate then
it remains STFT/Gabor Transform and suffer from resolution. You may refer
IEEE Xplore for plenty of papers on S-Transform. Please check Stockwell
homepage for S-Transform papers and code.
<br>
<p>Søren Andersen wrote:
<blockquote TYPE=CITE>I have been thinking about problems similar to the
points 3) & 4) mentioned by Arno, recently. My preliminary conclusions
are summarized below. I'd be grateful for any comments or corrections.
<p>The uncertainty relation becomes minimal (i.e. the equal sign applies)
for a Gabor-Filter/Morlet-Wavelet (a windowed FT using a Gaussian window,
see Arno's explanation below). Therefore the uncertainty relation can be
derived by calculating the Fourier-transformation of a Gaussian. The result
is again a Gaussian (i.e. the Gaussian is an Eigenfunction of the Fourier-transformation).
If the standard deviation of the untransformed Gaussian (time-domain) was
equal to sigma, then the Gaussian derived from the transformation (frequency-domain)
will have a standard deviation of 1/sigma.
<br>So the product of both standard deviations will be equal to 1.
<p>The result of the Fourier-transformation is an angular frequency ('omega'),
which is related to the 'normal' frequency 'f' by
<br><x-tab></x-tab> omega=2*pi*f
<p>So the above relation, which can be written as
<br><x-tab></x-tab>sigma_t * sigma_omega = 1
<br>becomes
<br><x-tab></x-tab>sigma_t * sigma_f = 1 / (2*pi)
<p>If we take the standard deviation as a measure of 'uncertainty', i.e.
what is called 'delta' in the formulas cited by Arno, we get the same formula
as under 3), except for a factor of 1/2. Were this is derived from I do
not know.
<p>The original question concerned the <b>bandwidth</b> of a signal. A
bandwidth is commonly defined as the Full-Width-At-Half-Maximum (FWHM)
of a signal or filter (also known as 3db Bandwidth). So to get any further
we need a formula that relates the FWHM of a Gaussian to sigma.
<p>Note that "half-maximum" applies to the power, which is the squared
amplitude. Filters are usually applied to the non-squared signal (i.e.
the amplitude), therefore the FWHM corresponds to the width of the Gaussian
at the 2 points where its value is equal to sqrt(1/2) of the maximum. The
solution is:
<p><x-tab></x-tab>FWHM = 2 * sqrt( ln(2) ) * sigma
<p>Combining this with the uncertainty-formula above we obtain
<p><x-tab></x-tab>FWHM_t * FWHM_f = 2 * ln(2) / pi
(=0.44...)
<br>
<p>I hope this is of any help. Please contact me if you have any comments
or corrections !
<p>best regards,
<br>Søren Andersen
<br>
<br>
<p>At 11:40 29.09.2006, you wrote:
<blockquote type=cite class=cite cite="">That's a tough question, that
we were just discussing with Rey Ramirez in our Lab.
<p>1. Another approach consist in using the standard deviations of normalized
Morlet wavelets in time. Each wavelet is composed of a Gaussian window
multiplied by a sinus (=Gabor). For the time domain, you simply use 2 standard
deviation of the Gaussian taper (there is 95% of the power within 2 standard
deviation in time). Using 2 is a random definition though. For the frequency
domain, you use also 2 standard deviation of the wavelet in frequency domain
(just FFT the real part of the wavelet). According to Rey, for the type
of wavelets used in Tallon-Baudry et Bertrand, Biomag, 1996, this is always
equal to 0.6366 irrespective of the value of the frequency and number of
cycles.
<p>2. Multitaper theory is all about setting a specific time and frequency
resolution. As you increase the number of tapers, you have to sacrifice
both time and frequency resolution (but you gain in SNR). This is not such
a big problem at high frequencies (40Hz) but really does not make sense
at low frequencies (5Hz).
<p>3. However, according to this paper for instance,
<p><a href="http://ieeexplore.ieee.org/iel5/6171/16493/00762269.pdf?arnumber=762269">http://ieeexplore.ieee.org/iel5/6171/16493/00762269.pdf?arnumber=762269</a>
<p>for any type of Gaussian wavelet, we have according the Heisenberg's
uncertainty principle
<p><font face="Symbol">D</font>f<font face="Symbol"> D</font>t >= 1/(4pi)
(or without the special characters delta_f * delta_t >= 1/(4pi))
<p>They come up with the number 4pi by using a Gaussian modulated pulse.
I could not access the reference (in 1946) so if someone could explain
that to us, that would be great.
<p>4. Yes, other references state that
<p><font face="Symbol">D</font>f<font face="Symbol"> D</font>t >= 1
(or without the special characters delta_f * delta_t >= 1)
<p>But this seems to be related to the Heisenberg's uncertainty principle
in quantum mechanics (about the vibration frequency of a particle) so I
am not sure it applies in our case.
<p>5. In timef(),we are still using sinusoidal wavelets which are nearly
indistinguishable from Morlet from a user perspective (the only difference
rely in the taper which is not Gaussian but a hanning window, the reason
being that you do not loose energy of the wavelet on the extremities as
you do with Gaussian). We will update the timef() function to allow Morlet
in the next release and make it a default.
<p>If you know the exact formula between delta_f and delta_t, then it becomes
easy to compute both the time and frequency resolution (because we can
compute the time used at each single frequency). I hope some signal processing
savvy participants to the list can enlighten us further on this topic.
<p>Best,
<p>Arno
<p>Clemens Brunner wrote:
<blockquote type=cite class=cite cite="">
<pre>I'm using timef to calculate wavelet-based time-frequency maps
with
the parameter cycles = [4 0.75]. Now I was wondering if I can find
out the bandwidth of the returned values as only the center
frequencies are given. Is the bandwidth constant over the frequency
range? Or does it change (i.e. grow) with frequency?
Second, if I would be using the FFT-based method (i.e. cycles = 0),
how could I find out the bandwidth of the single bands?
TIA
Clemens
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<x-sigsep>
<p></x-sigsep>Søren Andersen, Dipl.-Psych.
<p><i>Universität Leipzig</i>
<br><i>Institut Psychologie I</i>
<br><i>Professur Allgemeine Psychologie & Methodenlehre</i>
<br><i>Seeburgstr. 14-20</i>
<br><i>D-04103 Leipzig</i>
<br><i>Tel.: +49 - 341 - 97 39 53 3</i>
<br><i>Fax: +49 - 341 - 97 35 96 9</i>
<p>E-Mail:
<a href="mailto:andersen@rz.uni-leipzig.de">andersen@rz.uni-leipzig.de</a>
<br><a href="http://www.uni-leipzig.de/~psyall2/andersen/index.html" eudora="autourl">http://www.uni-leipzig.de/~psyall2/andersen/index.html</a>
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