# How to obtain time-averaged wavelet spectral density by the new version of CWT?

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Shen Shen on 28 Jan 2024
Commented: Shen Shen on 9 May 2024
I know how to obtain time-averaged wavelet spectral density by the old version of CWT, and the result is similar to the power spectral density (PSD) by function PWELCH. But how to obtain wavelet spectral density by the new version of CWT? Given that the old version is no longger recommended. Thank you. Here is the code and signal I used.
Fs=800; % sampling frequency of signal 'u', in Hz;
% PSD by pwelch
SegOvlp=75;
windowL=1024;
window=hamming(windowL);
noverlap=fix(SegOvlp/100*windowL);
nfft=windowL;
[pu_F,f]=pwelch(u,window,noverlap,nfft,Fs);
Pu_F=pu_F*var(u)/trapz(f,pu_F); %normalization
% WSD by old version of CWT, which gives similar results to pwelch
wavename='gaus3';
wcf=centfrq(wavename);
fre=10.^(linspace(-0.11,2.6,100));
scal=wcf*Fs./fre;
coefs=cwt(u,scal,wavename);
pu_w=sum(abs(coefs).^2,2)/Fs;
Pu_w=pu_w*var(u)/abs(trapz(fre,pu_w)); %normalization
% figure
loglog(f,Pu_F,'k');
hold on
loglog(fre,Pu_w,'r');
xlabel('\itf/Hz');ylabel('\itPSD');
legend('PSD','WSD by old CWT')
set(gcf,'Units','centimeters','Position',[10 5 15 8]);
set(gcf,'Color','w');
set(gcf, 'PaperPositionMode', 'auto');

Wayne King on 6 May 2024
Hi Shen, you can use cwtfilterbank and then use the timeSpectrum method (function). If you want scale-averaged power, then scaleSpectrum is also available.
Note both methods accept either the raw time series data, or the CWT coefficient matrix.
For example:
% Construct filterbank object. Only required input is length of data
% but see the reference page for all the properties you can specify
fb = cwtfilterbank(SignalLength=length(kobe));
[tavgp,f] = fb.timeSpectrum(kobe);
semilogx(f,tavgp)
grid on
xlabel('Frequency (Hz)')
ylabel('Power')
% There is also a convenience plot syntax which may provide a convenient
% visualization for you. Call the methods with no output arguments
figure
fb.timeSpectrum(kobe);
% As I said, if you want you, you can compute the CWT coefficients separately and
% input those
cfs = fb.wt(kobe);
figure
fb.timeSpectrum(cfs);
I would recommend the following help pages:
Note there are normalization options under timeSpectrum. Hope that helps,
Wayne
Shen Shen on 9 May 2024
Wayne, thanks so much for your prompt reply. It does help. However, a new problem appears. And I will have to bother you.
I have tried 'timeSpectrum' with both my data and the 'kobe' data. I found that the spectral results obtained by 'timeSpectrum' and 'pwelch' are so different, whatever normalization options are used under timeSpectrum. The 'spectrum' by 'timeSpectrum' seems different from the power spectral density? I'm confused. Is it possible that you may give a clue on this?
figure(1)
fb = cwtfilterbank(SignalLength=length(kobe));
fb.timeSpectrum(kobe);
figure(2)
pwelch(kobe);

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