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From: Kamran Iranpour on 12 Apr 2010 16:37
Hi,
according to Wkipedia, the Mexican hat wavelet is the negative
normalized second derivative of a Gaussian function. I can verfiy
this analytically and numerically. But I don't understand how to
relate the resulting pulse's frequency to the parameters of the
Gaussian function. Could someone enlighten me?

Kamran
From: Clay on 12 Apr 2010 17:51
On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...(a)gmail.com> wrote:
> Hi,
> according to Wkipedia, the Mexican hat wavelet is the negative
> normalized second derivative  of a Gaussian function. I can verfiy
> this analytically and numerically. But I don't understand how to
> relate the resulting pulse's frequency to the parameters of the
> Gaussian function. Could someone enlighten me?
>
> Kamran

This is not surprising - a gaussian function and its derivatives
consists of an infinite number of frequencies. But if you want the
frequency content of the mexican hat function, just find its fourier
transform. You will have to do integration by parts twice and then
know that the transform of a gaussian is also a gaussian.

But what you are likely wanting is how to compare a windowed fourier
transform kernal to the mexican hat function. This is described in
Daubechies (10 lectures on wavelets).


Clay
From: Tim Wescott on 12 Apr 2010 18:30
Clay wrote:
> On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...(a)gmail.com> wrote:
>> Hi,
>> according to Wkipedia, the Mexican hat wavelet is the negative
>> normalized second derivative of a Gaussian function. I can verfiy
>> this analytically and numerically. But I don't understand how to
>> relate the resulting pulse's frequency to the parameters of the
>> Gaussian function. Could someone enlighten me?
>>
>> Kamran
>
> This is not surprising - a gaussian function and its derivatives
> consists of an infinite number of frequencies. But if you want the
> frequency content of the mexican hat function, just find its fourier
> transform. You will have to do integration by parts twice and then
> know that the transform of a gaussian is also a gaussian.

Or look up the Fourier transform of the Gaussian in a Fourier transform
table, then observe that given a time-domain signal x and its Fourier
transform F{x}, the Fourier transform of dx/dt is j*omega*F{x}. Apply
that twice, and there you are.

> But what you are likely wanting is how to compare a windowed fourier
> transform kernal to the mexican hat function. This is described in
> Daubechies (10 lectures on wavelets).


--
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com
From: glen herrmannsfeldt on 12 Apr 2010 18:35
Tim Wescott <tim(a)seemywebsite.now> wrote:
(snip)

> Or look up the Fourier transform of the Gaussian in a Fourier transform
> table, then observe that given a time-domain signal x and its Fourier
> transform F{x}, the Fourier transform of dx/dt is j*omega*F{x}. Apply
> that twice, and there you are.

The latter should also be in a Fourier transform table, likely
earlier than the transform of a Gaussian.

>> But what you are likely wanting is how to compare a windowed fourier
>> transform kernal to the mexican hat function. This is described in
>> Daubechies (10 lectures on wavelets).

-- glen
From: Kamran Iranpour on 13 Apr 2010 12:15
On Apr 13, 12:30 am, Tim Wescott <t...(a)seemywebsite.now> wrote:
> Clay wrote:
> > On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...(a)gmail.com> wrote:
> >> Hi,
> >> according to Wkipedia, the Mexican hat wavelet is the negative
> >> normalized second derivative  of a Gaussian function. I can verfiy
> >> this analytically and numerically. But I don't understand how to
> >> relate the resulting pulse's frequency to the parameters of the
> >> Gaussian function. Could someone enlighten me?
>
> >> Kamran
>
> > This is not surprising - a gaussian function and its derivatives
> > consists of an infinite number of frequencies. But if you want the
> > frequency content of the mexican hat function, just find its fourier
> > transform. You will have to do integration by parts twice and then
> > know that the transform of a gaussian is also a gaussian.
>
> Or look up the Fourier transform of the Gaussian in a Fourier transform
> table, then observe that given a time-domain signal x and its Fourier
> transform F{x}, the Fourier transform of dx/dt is j*omega*F{x}.  Apply
> that twice, and there you are.
>
> > But what you are likely wanting is how to compare a windowed fourier
> > transform kernal to the mexican hat function. This is described in
> > Daubechies (10 lectures on wavelets).
>
> --
> Tim Wescott
> Control system and signal processing consultingwww.wescottdesign.com

I am a bit slow here. Suppose you wanted to have a pulse with a
certain center frequency and a certain length. How would one then go
from this to deciding what are the appropriate parameter values of
'a', 'b' and 'c' in the Gaussian a*exp^(-(x-b)^2/(2*c^2)) where a, b
and c are some positive numbers ? Or given its statistical equivalent
b would be the mean and c^2 the varaiance.
Sorry about taking your time.

Kamran
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