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If I have a causal system H(z) and I find the inverse of this system:

$$ G(z) = \frac{1}{H(z)} $$

Is G(z) also causal?

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It's not sufficient to only consider causality, you also need to check whether the inverse system is stable, otherwise it can't be implemented. If $G(z)$ has zeros on the unit circle, it cannot be inverted. If $G(z)$ has no zeros on the unit circle, but if there are zeros outside the unit circle, then there is no causal and stable inverse, because the zeros of $G(z)$ become the poles of the inverse system. There is, however, a stable inverse but it is not causal. If $G(z)$ has all its zeros (and poles) inside the unit circle, then there exists a causal and stable inverse (with all its poles and zeros inside the unit circle). Such a system is called strictly minimum-phase, and its inverse is also a strictly minimum-phase system.

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