MATLAB: Determining periods using Continuous Wavelet Transform

signal processingWavelet Toolbox

Hi, I have a signal which contains some quasi-periodic patterns which I would like to determine.

As its spectral content changes with time, I think that Wavelet analysis is the method which best fits to my purpose. So that, I was wondering if there exists a canonical way to detect reasonable periods in this signal by looking to CWT coefficients.

Best Answer

Hi Richard, you can use the approximate relationship between scale and frequency to do this.

Create a signal to illustrate this:

Fs = 1000;
t = 0:1/Fs:1-1/Fs;
x = zeros(size(t));
x([625,750]) = 2.5;
x = x+ cos(2*pi*100*t).*(t<0.25)+cos(2*pi*50*t).*(t>=0.5)+0.15*randn(size(t));
plot(t,x);

Set up the scale vector and spacing:

ds = 0.15;
J = fix((1/ds)*log2(length(x)/8));
dt = 1/Fs;
scales = 2*dt*2.^((0:J).*ds);

Obtain the CWT and plot the response:

    cwtstruct = cwtft({x,0.001},'Scales',scales,'Wavelet','morl');
    periods = cwtstruct.scales.*(4*pi)/(6+sqrt(38));
    freq  = 1./periods;
    cfs = cwtstruct.cfs;
    contour(t,freq,abs(cfs));
    set(gca,'xtick',[0 0.25 0.4 0.5 0.6 0.75 1]); grid on;
    xlabel('Time (seconds)'); ylabel('Hz');

Related Solutions

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');

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

Best Answer

Related Solutions

MATLAB: Proper wavelet scaling / restricted frequency analysis

MATLAB: Inverse CWT using coif4

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