Let's say that I have a system $H(z)$. What is causal inverse and how do I compute the causal inverse of $H(z)$?
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The inverse $G(z)$ of a system $H(z)$ is simply
$$G(z) = \frac{1}{H(z)}$$
The poles of $G(z)$ are the zeros of $H(z)$ and the zeros of $G(z)$ are the poles of $H(z)$.
$G(z)$ is only stable if the poles are inside the unit circle, which implies that for $G(z)$ to be stable, $H(z)$ needs have all it's zero inside the unit circle, i.e. it's minimum phase.
So if $H(z)$ is causal, stable and minimum phase, it's inverse is also causal stable and minimum phase.
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1$\begingroup$ $G(z)$ is not causal if $H(z)$ has more poles than zeros. Namely, $G(z)$ would then have more zeros than poles. One could compensate for this using a delay (poles at zero). $\endgroup$ Aug 27, 2022 at 17:07