Proper wavelet scaling / restricted frequency analysis

Dear all, I sample a signal with 20000 Hz and I'm using e.g. the morlet wavelet to analyze the frequency content of short time periods (e.g. 500 ms). Two questions: The scaling os correlated to the pseudo-freuency range of the wavelet transfrom. How can I determine the optimal scaling factor? I'm just interested in the frequency range of 100-300 Hz. Is there a possibility - e.g. through the factors - to focus the wavelet analyses on this frequency range? Looking forward to your advice. Best, Peter

Answer 1

[Wayne King]

Yes, it is possible, but you have to remember that the CWT is not as localized in frequency as the Fourier basis, so you have to give yourself some wiggle room. A sine wave will not be localized at one scale in the CWT. The way to do it best depends on whether you are using cwt.m or cwtft.m.

I'll create a signal consisting of 100-Hz and 300-Hz sine waves in noise. The 100-Hz signal will occur over the first 500 msec, the 300-Hz sine wave over the interval from 500 msec to 1 second.

Here is cwt.m:

 fs = 2e4;
 t = 0:1/fs:2-1/fs;
 x = ...
 2*cos(2*pi*100*t).*(t<0.500)+3*cos(2*pi*300*t).*(t>0.500 & t <1)+randn(size(t));

 dt = 1/fs;
 minscale = centfrq('morl')/(300*dt);
 maxscale = centfrq('morl')/(100*dt);

 scales = minscale-10:maxscale+10;
 cfs = cwt(x,scales,'morl','plot');

Now for cwtft (using an analytic Morlet wavelet)

MorletFourierFactor = 4*pi/(6+sqrt(2+6^2));

s0 = (0.002*MorletFourierFactor);
smax = (1/100*MorletFourierFactor);
ds =0.2;
nb = ceil(log2(smax/s0)/ds+1);

scales = struct('s0',s0,'ds',0.5,'nb',nb,'type','pow','pow',2);

cwtS1 = cwtft({x,dt},'scales',scales,'wavelet','morl');
sc = cwtS1.scales;
freq = 1./(sc*MorletFourierFactor);
contour(t,freq,abs(cwtS1.cfs));
xlabel('Time'); ylabel('Pseudo-Frequency');

[Peter]

Thanks a lot. I will stick to the cwt.m version with the morlet transform. Is the morlet transformation of the cwtft somehow superior? I also added the pseudo-frequency axis for the cwt.m. Is this correct? Can I use other wavelets like the dB2, 3 or 4 in the same manner? Last problem: the y-axis is from 350 to 100 - how can I reverse the plot?

Kind regards, Peter

fs = 2e4;
t = 0:1/fs:2-1/fs;
x = ...
2*cos(2*pi*100*t).*(t<0.500)+3*cos(2*pi*300*t).*(t>0.500 ...
  & t <1)+randn(size(t));
dt = 1/fs;
minscale = centfrq('morl')/(300*dt);
maxscale = centfrq('morl')/(100*dt); 
scales = minscale-10:maxscale+10;
cfs = cwt(x,scales,'morl'); 
S = (abs(cfs) .^ 2);
F = scal2frq(scales,'morl', 1 / fs);
figure
imagesc(t, F, S)

[Wayne King]

yes, you can use a similar approach with other wavelets. The approach based on the Fourier transform is not substantively different, there may be some computational advantages with large datasets.

use

    axis xy

to flip the axis.

[Peter]

Thanks

[vinod naidu]

I wanted to do in the same way for my recorded audio signal I have a [x, fs]=audioread('rec2.wav'); which loads x [81410x1] double and fs=44100

I want to scale the signal 1.65 to 2 & with the decrement of 0.005 for choosing high frequency contents ,and i want to know which one represents the a,b coefficents ? in

[Akshay Ravindran]

Hi ,

Can i use the scale generated after performing cwtft as input to cwt function ( older version of cwt) .

I am a bit confused as you seem to assign linear scaling based on frequency of interest as input. How exactly does the scale factor in the older version of cwt work?