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I am a beginner in signal processing. I am learning about the Fourier transform. I was working on the zero-phase Ricker wavelet. As I understand If I extract the phase spectrum I should get something similar to that image. This is the reference

However, my result is totally different. This is what I get

Does anyone know the reason? Thank you in advance,

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  • $\begingroup$ It looks like you are seeing the effect of a very large delay (each sample delay will linearly go from 0 to -2 pi as you sweep from DC to the sampling rate) $\endgroup$ Commented Apr 22, 2022 at 19:37
  • $\begingroup$ Thank you very much for your comment. How can I solve that problem? $\endgroup$ Commented Apr 22, 2022 at 22:59
  • $\begingroup$ If you want to make it zero phase, multiply the result by the opposite phase versus frequency (using e^{j omega T}) where omega is your frequency axis and adjust T to match the opposite of your phase slope. $\endgroup$ Commented Apr 22, 2022 at 23:47

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I was working on the zero-phase Ricker wavelet.

Chances you were working on the "linear-phase" wavelet. Assuming a real input, zero-phase requires time symmetry, i.e. $x[-n] = x[n]$. If you just take the wavelet (which is centered around $N/2$) and apply an FFT you get a linear phase behavior.

How can I solve that problem? Two suggestions:

  1. Circularly shift the time domain wavelet before the DFT. Recall that the DFT is periodic, i.e. $x[-n] = x[n]$ is equivalent to $x[N-n] = x[n]$
  2. Apply the the circular shift in the frequency domain after the DFT by multiplying with $e^{j2\pi \frac{N}{2} \frac{k}{N_{FFT}}}$

Exact details depend a bit on whether the length of the wavelet is odd or even.

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