How to properly multiply the Wavelet transform of a signal by a function of frequency and then reconstruct the signal?

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Larissa Perez 2019-3-19

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https://ww2.mathworks.cn/matlabcentral/answers/450865-how-to-properly-multiply-the-wavelet-transform-of-a-signal-by-a-function-of-frequency-and-then-recon

Hi everyone! So, I have been struggling with this for a few days.. I am not an expert on signal processing, but I am using it as a tool for an analysis, so I'm a bit unsure.

I have a signal x(t) and a function t(f), where f are the frequencies present in my spectrum. I need to multiply y(f) for the amplitudes of x(t) corresponding to f for each time stamp I have. So I have tried a few things:

% a. Obtain CWT of signal;
% fs = sampling frequency
[wt, f] = cwt(x,fs,'amor');
% b. Obtain the value of t(f) for each f from my spectrogram
    for i = 1:length(f);
        
    y(i) = Y*f;  
    end
    
% c. Multiply t(f) for wt for each time stamp
    y = y';
    wt_2 = [];
    for i = 1:length(wt)
        wt_2(:,i) = y.*wt(:,i);
    end
    
% d. Reconstruct the multiplied transform wt_2
x_2 = icwt(wt_2,'amor','SignalMean',nanmean(x));

However, this ends up giving me an underestimated result. Should I make any changes to t(f) when performing step c? I have tried

abs(t(f))^2

but that underestimates my results even more as t(f) < 1.

2.

% a. Obtain CWT of signal;
% fs = sampling frequency
[wt, f] = cwt(x,fs,'amor');
% b. Obtain the value of t(f) for each f from my spectrogram
    for i = 1:length(f);
        
    y(i) = Y*f;  
    end
    
% c. Multiply t(f) for wt for each time stamp
    y = y';
    wt_2 = [];
    for i = 1:length(wt)
        wt_2(:,i) = y.*wt(:,i);
    end
    
% d. Sum the real part of the transform for all the frequency bands in each time stamp. 
  x_2 = nansum(real(wt_2));
  

This second method gives me more reasonable results. However, I am not sure this is mathematically correct. I feel like this ignores scaling, but I am not sure how to overcome this.

When I plot cwt(x,fs,'amor');, the magnitude that shows up is not the amplitude of my signal at that given frequency, right? It is the transform coefficients?

Many thanks! :)

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How to properly multiply the Wavelet transform of a signal by a function of frequency and then reconstruct the signal?

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How to properly multiply the Wavelet transform of a signal by a function of frequency and then reconstruct the signal?

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