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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 class="moz-txt-link-freetext" 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:
<blockquote cite="midE5209667-3F0D-4CEB-9A38-5D8EDF5C1552@tugraz.at"
type="cite">
<pre wrap="">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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