MATLAB: Wavelet bandwith, centre frequency

continuous wavelet transformcwtwavelet

The Wavelet Toolbox provides the fucntion scal2frq which gives a so called pseudo-frequency to a corresponding scale.

As I understand it, each scale represents a daughter wavelet. In contrast to a periodic sine wave doesn´t a wavelet represent a frequency range rather than one frequency?

How can one calculate the bandwidth of a wavelet?

Is there a way of getting the actual function of the scaled wavelets used in cwt in order to perform an fft on them?

Best Answer

Hi Chris, there are a couple things you can do:

1.) If you want to see the support of wavelets at different scales, you can input a shifted Kronecker delta sequence

x = zeros(1000,1);
x(500) = 1;
cfs = cwt(x,1:2:64,'sym8');
% next part obviously not needed
for nn = 1:32
plot(cfs(nn,:)); pause(0.2);
end

2.) The other thing is that the Wavelet Toolbox now has a DFT-based continuous wavelet transform, implemented in cwtft.m and an inverse CWT in icwtft.m. The wavelets supported in cwtft have their Fourier transforms explicit. See the reference page for cwtft.m

Also see:

http://www.mathworks.com/help/toolbox/wavelet/ug/bs2gc05-1.html

for some examples.

Related Solutions

MATLAB: Inverse CWT using coif4

Hi Zozo, I'll give you a couple examples where you reconstruct an approximation to a time-localized 100 Hz component.

But first, I should say the following. Your time- and scale-localized reconstruction of a signal will be better if you use cwtft.m and icwtft.m with logarithmically-spaced scales. The reason for that is that you have to keep in mind the inverse CWT is a single integral approximation to a double integral problem. It turns out that the stability of the reconstruction is better when we use certain scales and for certain wavelets where the Fourier transform has a nice expression. For the coif4 wavelet, that is not the case.

   dt = 0.001;
   t = linspace(0,1,1000);
   x = cos(2*pi*100*t).*(t<0.5)+sin(2*pi*300*t).*(t>=0.75)+0.2*randn(size(t));
   scales = 1:30;
   cwtcoefs = cwt(x,scales,'coif4');
   cwtstruct = struct('wav','coif4','cfs',cwtcoefs,'scales',scales,'meanSIG',mean(x),'dt',dt);
   xrec = icwtlin(cwtstruct,'IdxSc',6:8);
   plot(t,x,'r-.'); 
   hold on;
   plot(t,xrec,'k'); 
   legend('Original Signal','Reconstruction with coif4 wavelet');
%Now using cwtft.m and icwtft.m
s0 = 2*dt;
ds = 0.4875;
NbSc = 20;
wname = 'morl';
sig = {x,dt};
sca = {s0,ds,NbSc};
wave = {wname,[]};
cwtsig = cwtft(sig,'scales',sca,'wavelet',wave);
newcfs = zeros(size(cwtsig.cfs));
indices = 4:6;
newcfs(indices,:) = cwtsig.cfs(indices,:);
cwtsig.cfs = newcfs;
xrec1 = icwtft(cwtsig);
figure;
plot(t,x,'r-.'); 
hold on;
plot(t,xrec1,'k');
legend('Original Signal','Reconstruction with Morlet Wavelet');

If you compare the two, you'll see that both techniques essentially isolate the 100-Hz component, but that the Morlet wavelet with logarithmically-spaced scales does a better job at the getting the amplitude right.

If you just want to find the localization of the signal feature at certain scales, then the coif4 wavelet works. If you are concerned about an accurate amplitude scaling, then cwtft and icwtft work better than cwt.m paired with icwtlin.m

Hope that helps, Wayne

MATLAB: Proper wavelet scaling / restricted frequency analysis

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');

Best Answer

Related Solutions

MATLAB: Inverse CWT using coif4

MATLAB: Proper wavelet scaling / restricted frequency analysis

Related Question