Using FFT to find CWT

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Ameen Bin Obadi 2020-5-26

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https://ww2.mathworks.cn/matlabcentral/answers/533918-using-fft-to-find-cwt

编辑： Ameen Bin Obadi 2020-5-26

I am trying to calculate CWT not using the function "cwt" in matlab , but from the definition of CWT itself. I am doing this for the purpose of implemening the same mathematical steps on FPGA board.

So, I have a signal "I" of length (39062) and I have the morlet wavelet of length (39062). I find FFT for "I" , then I find FFT for morlet wavelet at each of the 124 scales , so I get amatrix of 124 by 39062 each row containing the FFT of each scale let us call it (psi1_f). Then I multipled the FFT of (I) with first row of psi1_f element by element and take IFFT, then multiply by second row and so on until I get a new matrix which should contain the CWT coeffiecints. My code is as follows:

fs=325.520833333333; % sampling frequncy
[cw1,freq1,~,fb1,scalingcfs]=cwt(I,'amor',fs); % this step is just to extract the fb and then extract the wavelets, I is my signal
[psi1,t1] = wavelets(fb1);  % to extract psi1
%%%%%%%%%%
I_f=fft(I);
psi1_f=fft(psi1,39062,2);  % finding FFT of each row (FFT of wavelet at each scale)
for i=1:124
    temp=i*psi1_f(i,:);  % this is just to increase magnitude of each wavelet (not sure if this is necessary)
    cw11(i,:)=(I_f'.*temp); 
end
cw11=ifft(cw11,39062,2); % now this cw11 should be the cwt coeffiecients

According to my known signal (I), I should get some very high frequency components at time 8 seconds and 104 seconds (total signal time is 120 seconds), howevr, I got these components delayed. They happen to be at 24 seconds and 112 seconds instead.

Any explanation and solution to this issue?

By the way: When I implement the same method but in time domain (definition of cwt is convolution in time doamin rather than multiplication) I got accurate results.