The impulse is given as a sequence: $x: {10; -3; -1}$. To determine if it is minimum-phase I compute the $Z$-transform (the geophysical case, positive powers of $z$) of it: $X(z)=10-3z-z^2$. Then I factor $X(z)$ into dipoles $(a_0+a_1 z)$: $X(z)=10-3z-z^2=(5+z)(2-z)$. Then I observe that all the dipoles satisfy the $|a_0|>|a_1|$ thus being minimum phase, so the whole sequence $x$ is minimum phase.

Here I'm stuck about the zero-phase equivalent. I can evaluate the spectrum and it's magnitude by evaluating $X(z_k)$ at $z_k=e^{2 \pi i \frac{k}{N}}$, where $k=\{0,1,\dots,N-1\}$, and $N$ - length of the impulse.

From this point on how can I find the respective zero-phase impulse that has the same spectrum magnitude $|X(z_k)|$?


1 Answer 1


To find the zero-phase impulse response, simply take the Fourier transform of your signal, find magnitude and find the inverse Fourier transform of that. Mathematically speaking,


This essentially says that the entire Fourier transform of your new sequence is real-valued. All real-valued numbers have zero phase.

  • $\begingroup$ The inverse transform of the magnitude will yield infinite time domain signal. How could I determine the wavelet of the same size $N$ that is (a) zero-phase, (b) minimum phase? $\endgroup$
    – mbaitoff
    Commented Oct 29, 2013 at 9:23
  • $\begingroup$ @mbaitoff It will not be infinite, it's going to be the same length as your magnitude. I think you're mixing up continuous and discrete Fourier transforms. You need to do IDFT. The wavelet question is a separate question, and you should ask it separately. $\endgroup$
    – Phonon
    Commented Oct 29, 2013 at 18:13

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