Why must the Region Of Convergence (ROC) contain infinity and the system function be a right-sided sequence for it to be causal?

Question

I need the simple logic on the condition a system becomes causal. We know that causal contains only past values. I can't relate this with the Region Of Convergence (ROC) concept.

Answer 1

A causal impulse response is zero for negative argument:

$$h[n]=0,\qquad n<0\tag{1}$$

Hence its $\mathcal{Z}$-transform is given by

$$H(z)=\sum_{n=-\infty}^{\infty}h[n]z^{-n}=\sum_{n=0}^{\infty}h[n]z^{-n}=h[0]+h[1]z^{-1}+h[2]z^{-2}+\ldots\tag{2}$$

Note that there are no positive powers of $z$ in Eq. $(2)$. Consequently, $H(z)$ converges for $|z|\to\infty$, which means that infinity is inside the region of convergence.

Specifically you have

$$\lim_{z\to\infty}H(z)=h[0]\tag{3}$$

Eq. $(3)$ is called the initial value theorem of the $\mathcal{Z}$-transform.