Søren Andersen andersen at rz.uni-leipzig.de
Fri Sep 29 07:40:22 PDT 2006

```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.

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.
So the product of both standard deviations will be equal to 1.

The result of the Fourier-transformation is an
angular frequency ('omega'), which is related to the 'normal' frequency 'f' by
omega=2*pi*f

So the above relation, which can be written as
sigma_t * sigma_omega = 1
becomes
sigma_t * sigma_f = 1 / (2*pi)

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.

The original question concerned the bandwidth 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.

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:

FWHM = 2 * sqrt( ln(2) ) * sigma

Combining this with the uncertainty-formula above we obtain

FWHM_t * FWHM_f = 2 * ln(2) / pi     (=0.44...)

you have any comments or corrections !

best regards,
Søren Andersen

At 11:40 29.09.2006, you wrote:
>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>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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Søren Andersen, Dipl.-Psych.

Universität Leipzig
Institut Psychologie I
Professur Allgemeine Psychologie & Methodenlehre
Seeburgstr. 14-20
D-04103 Leipzig
Tel.: +49 - 341 - 97 39 53 3
Fax: +49 - 341 - 97 35 96 9

E-Mail: <mailto:andersen at rz.uni-leipzig.de>andersen at rz.uni-leipzig.de
http://www.uni-leipzig.de/~psyall2/andersen/index.html
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