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

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.

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

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

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

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

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

• 1/ H(z) converges means it have some finite value?.2/ why 0 To +inf is the limit?For causal system? – Hasan Shuvo Oct 5 '18 at 16:36
• @HasanShuvo: Yes, if the sum converges, $H(z)$ has a finite value. If you look at Eq. (1) in my answer you see that a causal $h[n]$ vanishes for negative indices $n$, that's why there are no negative indices in the sum. – Matt L. Oct 5 '18 at 16:56