Minimum Phase Filter

Question

Suppose we want to find the minimum phase filter of a causal system with system function $H(z)=z^{-1}−0.3$. The minimum phase filter is $H_1(z)=1−0.3z^{−1}$ (by taking the zero to its conjugate reciprocal location inside the unit circle). But could the minimum phase filter also be $H_2(z)=−H_1(z)=0.3z^{−1}−1$? Both $H_1$ and $H_2$ have the same phase response and thus the same group delay, so it seems like the minimum phase filter is not unique. Is this correct?

Answer 1

$H_1(z)$ and $H_2(z)$ do not have the same phase response. If $\phi_1(\omega)$ and $\phi_2(\omega)$ are the phase responses of $H_1$ and $H_2$, respectively, then $\phi_2(\omega)=\phi_1(\omega)+\pi$. The group delay functions are of course identical but here phase is important, not group delay. To answer your question, the minimum phase filter is unique, and it is $H_1(z)$.