Wavelet question about the timing of coefficients compared to original signal

Hi, I am studying electrical power systems fault analysis using PSCAD. I export current and voltage fault signals to MATLAB and use wavelet analysis for my study. I have a basic question. For example I want to know the coefficients of d1 to d5 of an original signal in an specific time like 2.05 sec, but when i use wavelet analysis each of the coefficients and plot them, each of them has a different horizontal axe scale and I dont know how to find the value of coefficients in an specified time. For example in MATLAB examples about wavelets, which is on the link below https://www.mathworks.com/help/releases/R2017a/wavelet/ref/detcoef.html the original plot has a scale 1:4000 but the coefficients are until 500, 1000 and 2000 respectively. So how do I know the values of these coeficients in a specified time similar to original signal? Thanks

Answer 1

[Abhi Sundararaman]

Since the original plot has a scale of 1:4000, the first detail coefficient d1 must have a size of 1:2000. This is because calculating the detail coefficients involves removing half of the frequencies, half of the samples can be discarded thanks to Nyquist's rule. As a result, every subsequent transform (i.e getting d2 or d3) will have half the samples of the previous one. This is why the coefficient arrays have a lower number of samples. Every transform downsamples the signal by a factor of 2.

To find the value of each coefficient at each point in time, simply use the fact that each transform has half the number of samples as the previous transform. For example, d1 will have 2000 samples because we only keep the even indexed samples, so to find the value of d1 at sample 10 of the original signal, you would look to sample 20 of d1.

You can find more information on the algorithm used for detail coefficient analysis here: https://www.mathworks.com/help/releases/R2017a/wavelet/ref/wavedec.html#f11-37377

[Mani Ashouri]

so how the wavelet toolbox does the job? the 1-d wavelet transform from the toolbox, plots all coefficients,approximation and the priginal wave in one time Axis in a way that all have a value on each second or time step.