Determine if an impulse is minimum phase and compute a zero-phase equivalent 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 $|a_0|&gt;|a_1|$ 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 $|X(z_k)|$?
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,

$$\mathscr{F}^{-1}\{\text{abs}\left(\mathscr{F}\{x[n]\}\right)\}$$

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