# Is the inverse of a causal system also causal?

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?

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.