# Closed form solution for the minimum phase of a continuous magnitude response

Let's say I have a continuous real function $F(\omega)$ defined in the region $\omega = [-\pi, \pi]$. Let's also say that I have a minimum phase $z$-domain transfer function $H(z)$ defined as:

$$\left| H(z) \right| = F(\omega)\quad\text{where}\quad z=e^{-i\omega}$$

i.e. $F(\omega)$ defines the magnitude of $H(z)$.

Is there a closed form expression in terms of $F(\omega)$ for the minimum phase component of $H(z)$? Even if the equation is not solvable in the majority of cases!

I know we can use the real cepstrum to reconstruct the minimum phase of a system from the log magnitude spectrum, however this is limited to equally spaced points on the unit circle.

• Does this answer your question? – Matt L. May 31 '17 at 9:03
• @Matt L. Yes I think it does :-) – keith May 31 '17 at 9:13
• @Matt L. Does it different for z-domain, as that looks like it's in the s-domain? – keith May 31 '17 at 9:17