Allpass Filters - Causal and Stable

Question

So I have been learning about how to test systems for causality and stability but I am confused about the implications on their unit circle representation.

Would it be safe to say that a causal and stable allpass filter does not have a zero inside the unit circle?

Answer 1

You're right, a causal and stable discrete-time system must have all its poles inside the unit circle. Since an allpass system with a zero $z_0$ inside the unit circle ($|z_0|<1$) must have a pole at $1/z_0^*$, i.e., outside the unit circle, it cannot be causal and stable.