MATLAB: Proper wavelet scaling / restricted frequency analysis

Question

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

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');