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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.<br><br>
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.<br><br>
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<br><br>
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)<br><br>
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.<br><br>
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.<br><br>
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:<br><br>
<x-tab> </x-tab>FWHM = 2 *
sqrt( ln(2) ) * sigma<br><br>
Combining this with the uncertainty-formula above we obtain<br><br>
<x-tab> </x-tab>FWHM_t *
FWHM_f = 2 * ln(2) / pi (=0.44...)<br><br>
<br>
I hope this is of any help. Please contact me if you have any comments or
corrections !<br><br>
best regards,<br>
Søren Andersen<br><br>
<br><br>
At 11:40 29.09.2006, you wrote:<br>
<blockquote type=cite class=cite cite="">That's a tough question, that we
were just discussing with Rey Ramirez in our Lab.<br>
<br>
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.<br>
<br>
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).<br><br>
3. However, according to this paper for instance, <br><br>
<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>
<br><br>
for any type of Gaussian wavelet, we have according the Heisenberg's
uncertainty principle<br><br>
<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))<br><br>
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.<br><br>
4. Yes, other references state that <br><br>
<font face="Symbol">D</font>f<font face="Symbol"> D</font>t >=
1
(or without the special characters delta_f * delta_t >= 1)<br><br>
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.<br><br>
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.<br><br>
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.<br><br>
Best,<br><br>
Arno<br><br>
Clemens Brunner wrote: <br>
<blockquote type=cite class=cite cite=""><br>
<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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Søren Andersen, Dipl.-Psych.<br><br>
<i>Universität Leipzig <br>
Institut Psychologie I<br>
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<br>
Fax: +49 - 341 - 97 35 96 9</i> <br><br>
E-Mail:
<a href="mailto:andersen@rz.uni-leipzig.de">andersen@rz.uni-leipzig.de</a>
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