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Mexican hat and Gaussian function

Started by Kamran Iranpour April 12, 2010
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
On Apr 12, 4:37&#2013266080;pm, Kamran Iranpour <kamran.iranp...@gmail.com> wrote:
> Hi, > according to Wkipedia, the Mexican hat wavelet is the negative > normalized second derivative &#2013266080;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
Clay wrote:
> On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...@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
Tim Wescott <tim@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
On Apr 13, 12:30&#2013266080;am, Tim Wescott <t...@seemywebsite.now> wrote:
> Clay wrote: > > On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...@gmail.com> wrote: > >> Hi, > >> according to Wkipedia, the Mexican hat wavelet is the negative > >> normalized second derivative &#2013266080;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}. &#2013266080;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
On Apr 13, 12:15&#2013266080;pm, Kamran Iranpour <kamran.iranp...@gmail.com>
wrote:
> On Apr 13, 12:30&#2013266080;am, Tim Wescott <t...@seemywebsite.now> wrote: > > > > > > > Clay wrote: > > > On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...@gmail.com> wrote: > > >> Hi, > > >> according to Wkipedia, the Mexican hat wavelet is the negative > > >> normalized second derivative &#2013266080;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}. &#2013266080;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- Hide quoted text - > > - Show quoted text -
I don't think you are understanding the problem. The center frequency of the gaussian pulse is determined by "b." But it contains much more than just one frequency. The pulse length is infinite.
On Apr 13, 9:14&#2013266080;pm, Clay <c...@claysturner.com> wrote:
> On Apr 13, 12:15&#2013266080;pm, Kamran Iranpour <kamran.iranp...@gmail.com> > wrote: > > > > > On Apr 13, 12:30&#2013266080;am, Tim Wescott <t...@seemywebsite.now> wrote: > > > > Clay wrote: > > > > On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...@gmail.com> wrote: > > > >> Hi, > > > >> according to Wkipedia, the Mexican hat wavelet is the negative > > > >> normalized second derivative &#2013266080;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}. &#2013266080;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- Hide quoted text - > > > - Show quoted text - > > I don't think you are understanding the problem. The center frequency > of the gaussian pulse is determined by "b." But it contains much more > than just one frequency. &#2013266080;The pulse length is infinite.
I know there are more than one frequencies, thats why I called it center frequency. For me it appears to be decided not only by 'b' but by 'c' too.
On Apr 14, 2:45&#2013266080;am, Kamran Iranpour <kamran.iranp...@gmail.com> wrote:
> On Apr 13, 9:14&#2013266080;pm, Clay <c...@claysturner.com> wrote: > > > > > > > On Apr 13, 12:15&#2013266080;pm, Kamran Iranpour <kamran.iranp...@gmail.com> > > wrote: > > > > On Apr 13, 12:30&#2013266080;am, Tim Wescott <t...@seemywebsite.now> wrote: > > > > > Clay wrote: > > > > > On Apr 12, 4:37 pm, Kamran Iranpour <kamran.iranp...@gmail.com> wrote: > > > > >> Hi, > > > > >> according to Wkipedia, the Mexican hat wavelet is the negative > > > > >> normalized second derivative &#2013266080;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}. &#2013266080;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- Hide quoted text - > > > > - Show quoted text - > > > I don't think you are understanding the problem. The center frequency > > of the gaussian pulse is determined by "b." But it contains much more > > than just one frequency. &#2013266080;The pulse length is infinite. > > I know there are more than one frequencies, thats why I called it > center frequency. For me it appears to be decided not only by 'b' but > by 'c' too.- Hide quoted text - > > - Show quoted text -
Your "a" is linked to "c" via normalization. "b" of course translates (moves back and forth) the gaussian. If your gaussian is in the time domain, then "b" does not affect the frequency content other than its phase info. If your gaussian is in the freq domain, then "b" clearly translates in the frequency domain and thus sets the center frequency. Back in the time domain, this results in a modulated gaussian. Look up Heaviside's shifting theorem for details. Your "c" does not affect your center frequency - it sets the variance of the gaussian and the normalization. Try to find Ingrid Daubechies' book "Ten Lectures on Wavelets," She really does cover the similarities and differences between the Mexican Hat Function and using a Windowed Fourier transform. You question strikes me as asking something akin to "What is the frequency of a parabola?" There is not a good answer because the question is ill posed.