[Eeglablist] timef wavelet bandwidth

[arno]

That's a tough question, that we were just discussing with Rey Ramirez 
in our Lab.

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.

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).

3. However, according to this paper for instance,

http://ieeexplore.ieee.org/iel5/6171/16493/00762269.pdf?arnumber=762269

for any type of Gaussian wavelet, we have according the Heisenberg's 
uncertainty principle

Df Dt >= 1/(4pi)                (or without the special characters 
delta_f * delta_t >= 1/(4pi))

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.

4. Yes, other references state that

Df Dt >= 1                      (or without the special characters 
delta_f * delta_t >= 1)

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.

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.

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.

Best,

Arno

Clemens Brunner wrote:
> 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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