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Thread Subject:
Wavelet: Is this still a cone of influence?

Subject: Wavelet: Is this still a cone of influence?

From: Sebastian Gatzka

Date: 11 Mar, 2011 13:45:07

Message: 1 of 6

Hello World.

As I'm digging more into Wavelets I get more excited.
And it raises more questions.

Here you find a 3D plot of a wavelet transformation:
http://therealsega.th.funpic.de/glr/wavelet-comparison.png

In the small scales (the plot at the bottom) I can locate the scale corresponding to the characteristic frequency of the signal.

I'm investigating a transient physical process in which this characteristic frequency changes at one time. You can see these changes in the small scales.

So now there are three questions:

1) Are these coefficient maxima traveling from the origins of the frequency change the so called "cones of influence"?

2) In much larger scales i can still see many ripples traveling from smaller to larger scales. Are these still cones of influence?

3) As a matter of fact the change of characteristic frequency is accompanied by another physical process, which will introduce a second dominant frequency into the system. It should be located at the scales around 2080.
The plot for the bigger scales shows the relevant scale range. Maybe there is a rise in local coefficient maxima correspond to the new frequency at scales around 2080, but I can't see them because of these (suspected) cones of influences in the scale range?

Hope to hear your thoughts about this!
Thanks, Sebastian

Subject: Wavelet: Is this still a cone of influence?

From: Wayne King

Date: 11 Mar, 2011 15:18:05

Message: 2 of 6

"Sebastian Gatzka" <sebastian.gatzka.NOSPAM@stud.tu-darmstadt.de> wrote in message <ild913$sd8$1@fred.mathworks.com>...
> Hello World.
>
> As I'm digging more into Wavelets I get more excited.
> And it raises more questions.
>
> Here you find a 3D plot of a wavelet transformation:
> http://therealsega.th.funpic.de/glr/wavelet-comparison.png
>
> In the small scales (the plot at the bottom) I can locate the scale corresponding to the characteristic frequency of the signal.
>
> I'm investigating a transient physical process in which this characteristic frequency changes at one time. You can see these changes in the small scales.
>
> So now there are three questions:
>
> 1) Are these coefficient maxima traveling from the origins of the frequency change the so called "cones of influence"?
>
> 2) In much larger scales i can still see many ripples traveling from smaller to larger scales. Are these still cones of influence?
>
> 3) As a matter of fact the change of characteristic frequency is accompanied by another physical process, which will introduce a second dominant frequency into the system. It should be located at the scales around 2080.
> The plot for the bigger scales shows the relevant scale range. Maybe there is a rise in local coefficient maxima correspond to the new frequency at scales around 2080, but I can't see them because of these (suspected) cones of influences in the scale range?
>
> Hope to hear your thoughts about this!
> Thanks, Sebastian

Hi Sebastian, I haven't had a chance to look at your image, but you may be observing cone of influence effects, you can use conofinf to see what the cone of influence is for a particular wavelet at a particular set of points. For example:

Fs = 1e3;
t = linspace(0,1,1e3);
x = cos(2*pi*200*t).*(t>0.25 & t<0.5);
wav = 'sym4';
scales = 1:.25:10;
coeffs = cwt(x,scales,'sym4','plot');
axis xy;
hold on;
cone = conofinf('sym4',scales,1e3,375,'plot');

What you have to keep in mind (and this is illustrated in the MATLAB documentation), is that the CWT coefficients are the result of integrating the product of the signal with the dilated and translated version of the wavelet. Since a sine wave integrates to zero over any period, as the wavelet becomes much bigger than the period of the sine wave, you get wavelet coefficients of zero, except beginning and end of the oscillation.

This is illustrated here where I pick the biggest scale where the length of the wavelet is much bigger than the period of the oscillation. I scale the cone of influence down for visualization purposes only and I illustrate the cone of influence at the beginning and end points of the oscillation.

figure;
cone = conofinf('sym4',scales,1e3,[250 500]);
plot(cone{1}(end,:).*max(coeffs(end,:)));
hold on;
plot(cone{2}(end,:).*max(coeffs(end,:)));
plot(coeffs(end,:));

You seem to be learning a lot. Keep it up, MATLAB has some great wavelet tools.

Wayne

Subject: Wavelet: Is this still a cone of influence?

From: Sebastian Gatzka

Date: 20 Mar, 2011 10:13:04

Message: 3 of 6

"Wayne King" <wmkingty@gmail.com> wrote in message <ildefd$s21$1@fred.mathworks.com>
>
> Hi Sebastian, I haven't had a chance to look at your image, but you may be observing cone of influence effects, you can use conofinf to see what the cone of influence is for a particular wavelet at a particular set of points. For example:
>
> Fs = 1e3;
> t = linspace(0,1,1e3);
> x = cos(2*pi*200*t).*(t>0.25 & t<0.5);
> wav = 'sym4';
> scales = 1:.25:10;
> coeffs = cwt(x,scales,'sym4','plot');
> axis xy;
> hold on;
> cone = conofinf('sym4',scales,1e3,375,'plot');
>
> What you have to keep in mind (and this is illustrated in the MATLAB documentation), is that the CWT coefficients are the result of integrating the product of the signal with the dilated and translated version of the wavelet. Since a sine wave integrates to zero over any period, as the wavelet becomes much bigger than the period of the sine wave, you get wavelet coefficients of zero, except beginning and end of the oscillation.
>
> This is illustrated here where I pick the biggest scale where the length of the wavelet is much bigger than the period of the oscillation. I scale the cone of influence down for visualization purposes only and I illustrate the cone of influence at the beginning and end points of the oscillation.
>
> figure;
> cone = conofinf('sym4',scales,1e3,[250 500]);
> plot(cone{1}(end,:).*max(coeffs(end,:)));
> hold on;
> plot(cone{2}(end,:).*max(coeffs(end,:)));
> plot(coeffs(end,:));
>
> You seem to be learning a lot. Keep it up, MATLAB has some great wavelet tools.
>
> Wayne

Dear Wayne.

I have been thinking a lot about your reply.

Just recently I got access to the new release of MATLAB which was able to compute the conofinf command and so it took me a while to see the results of your command suggestions.

From the investigation of the conofinf output it may be possible that I see cone of influences at these high scales in my transformations.

So what I am currently worried about is this:
What is the use of transformation which is able to detect peak-like signal changes with high frequencies (= high scales) if the cones of influence from signal changes at low frequencies (= small scales) mess up the transformation on the high scales?!

Sebastian

Subject: Wavelet: Is this still a cone of influence?

From: Wayne King

Date: 20 Mar, 2011 11:11:04

Message: 4 of 6

"Sebastian Gatzka" <sebastian.gatzka.NOSPAM@stud.tu-darmstadt.de> wrote in message <im4jvg$olf$1@fred.mathworks.com>...
> "Wayne King" <wmkingty@gmail.com> wrote in message <ildefd$s21$1@fred.mathworks.com>
> >
> > Hi Sebastian, I haven't had a chance to look at your image, but you may be observing cone of influence effects, you can use conofinf to see what the cone of influence is for a particular wavelet at a particular set of points. For example:
> >
> > Fs = 1e3;
> > t = linspace(0,1,1e3);
> > x = cos(2*pi*200*t).*(t>0.25 & t<0.5);
> > wav = 'sym4';
> > scales = 1:.25:10;
> > coeffs = cwt(x,scales,'sym4','plot');
> > axis xy;
> > hold on;
> > cone = conofinf('sym4',scales,1e3,375,'plot');
> >
> > What you have to keep in mind (and this is illustrated in the MATLAB documentation), is that the CWT coefficients are the result of integrating the product of the signal with the dilated and translated version of the wavelet. Since a sine wave integrates to zero over any period, as the wavelet becomes much bigger than the period of the sine wave, you get wavelet coefficients of zero, except beginning and end of the oscillation.
> >
> > This is illustrated here where I pick the biggest scale where the length of the wavelet is much bigger than the period of the oscillation. I scale the cone of influence down for visualization purposes only and I illustrate the cone of influence at the beginning and end points of the oscillation.
> >
> > figure;
> > cone = conofinf('sym4',scales,1e3,[250 500]);
> > plot(cone{1}(end,:).*max(coeffs(end,:)));
> > hold on;
> > plot(cone{2}(end,:).*max(coeffs(end,:)));
> > plot(coeffs(end,:));
> >
> > You seem to be learning a lot. Keep it up, MATLAB has some great wavelet tools.
> >
> > Wayne
>
> Dear Wayne.
>
> I have been thinking a lot about your reply.
>
> Just recently I got access to the new release of MATLAB which was able to compute the conofinf command and so it took me a while to see the results of your command suggestions.
>
> From the investigation of the conofinf output it may be possible that I see cone of influences at these high scales in my transformations.
>
> So what I am currently worried about is this:
> What is the use of transformation which is able to detect peak-like signal changes with high frequencies (= high scales) if the cones of influence from signal changes at low frequencies (= small scales) mess up the transformation on the high scales?!
>
> Sebastian

Hi Sebastian, Just to be clear about the use of terminology here, high frequencies correspond to small scales and low frequencies correspond to long (high) scales.

If you want to localize a discontinuity in your signal or a peak-like change, then you use the small scale wavelet coefficients because those will localize it most exactly. You would not localize that type of signal phenomena using long scale coefficients.

On the other hand if you want to characterize long-scale phenomena, use the longer-scale coefficients.

I don't think "mess up" is the way to look at it. You have to realize that as the wavelet stretches and then slides along the position axis, that has to affect the number of wavelet coefficients at a particular scale affected by a particular signal feature.

Wayne

Subject: Wavelet: Is this still a cone of influence?

From: Sebastian Gatzka

Date: 30 Mar, 2011 09:45:04

Message: 5 of 6

"Wayne King" <wmkingty@gmail.com> wrote in message <im4nc8$fjg$1@fred.mathworks.com>...
> Hi Sebastian, Just to be clear about the use of terminology here, high frequencies correspond to small scales and low frequencies correspond to long (high) scales.
>
> If you want to localize a discontinuity in your signal or a peak-like change, then you use the small scale wavelet coefficients because those will localize it most exactly. You would not localize that type of signal phenomena using long scale coefficients.
>
> On the other hand if you want to characterize long-scale phenomena, use the longer-scale coefficients.
>
> I don't think "mess up" is the way to look at it. You have to realize that as the wavelet stretches and then slides along the position axis, that has to affect the number of wavelet coefficients at a particular scale affected by a particular signal feature.
>
> Wayne

Thank you Wayene.

Just to make sure:
With my signal recorded at 500kHz I have a sampling period of 1/500e3 seconds.
With this information I used the scal2frq function for the appropriate relation between the wavelet type's scale and pseudo frequency:

fs = 500e3; % Sample Frequenz
sp = 1/fs; % Sample Periode
wavelet = 'gaus1';
scales = 1:2100;
pFrequencies = scal2frq(scales,wavelet,sp);

Plotting the pseudo frequencies over scales results in this plot:
http://therealsega.th.funpic.de/glr/scal2frq.png

Am I right so far?!

Sebastian

Subject: Wavelet: Is this still a cone of influence?

From: Wayne King

Date: 31 Mar, 2011 15:30:21

Message: 6 of 6

"Sebastian Gatzka" <sebastian.gatzka.NOSPAM@stud.tu-darmstadt.de> wrote in message <imuu30$gd5$1@fred.mathworks.com>...
> "Wayne King" <wmkingty@gmail.com> wrote in message <im4nc8$fjg$1@fred.mathworks.com>...
> > Hi Sebastian, Just to be clear about the use of terminology here, high frequencies correspond to small scales and low frequencies correspond to long (high) scales.
> >
> > If you want to localize a discontinuity in your signal or a peak-like change, then you use the small scale wavelet coefficients because those will localize it most exactly. You would not localize that type of signal phenomena using long scale coefficients.
> >
> > On the other hand if you want to characterize long-scale phenomena, use the longer-scale coefficients.
> >
> > I don't think "mess up" is the way to look at it. You have to realize that as the wavelet stretches and then slides along the position axis, that has to affect the number of wavelet coefficients at a particular scale affected by a particular signal feature.
> >
> > Wayne
>
> Thank you Wayene.
>
> Just to make sure:
> With my signal recorded at 500kHz I have a sampling period of 1/500e3 seconds.
> With this information I used the scal2frq function for the appropriate relation between the wavelet type's scale and pseudo frequency:
>
> fs = 500e3; % Sample Frequenz
> sp = 1/fs; % Sample Periode
> wavelet = 'gaus1';
> scales = 1:2100;
> pFrequencies = scal2frq(scales,wavelet,sp);
>
> Plotting the pseudo frequencies over scales results in this plot:
> http://therealsega.th.funpic.de/glr/scal2frq.png
>
> Am I right so far?!
>
> Sebastian

Hi Sebastian, yes, you are right.

Wayne

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