Causality, Region of Convergence and which side is the system on

Question

I want to try to summarize some relationships between z transform, ROC and causality. But I am getting confused between left/right sided sequences, the ROC and causal.

1. If you need stability then the ROC must contain the unit circle. I can understand that.

2. If you need a causal system then the ROC must contain infinity and the system function will be a right-sided sequence. I am fine with that too.

3. If you need an anticausal system then the ROC must contain the origin and the system function will be a left-sided sequence. Yes that's fine.

4. If you need both, stability and causality, all the poles of the system function must be inside the unit circle.

But #4, doesn't it violate number 2 because we don't include infinity?

Got it from here: http://en.wikipedia.org/wiki/Z-transform#Example_2_.28causal_ROC.29

Thanks

Answer 1

If you combine the two requirements (stability => the unit circle must be inside the ROC), and causality (the ROC is given by $|z|>r$ with some real-valued radius $r$), then it follows that $r<1$ must be satisfied. This means that the ROC is outside a circle with radius $r<1$, and, consequently, all poles must be inside this circle with radius $r<1$. So all poles are inside the unit circle (because, by definition, there are no poles in the ROC).