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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.

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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.

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  • $\begingroup$ 1/ H(z) converges means it have some finite value?.2/ why 0 To +inf is the limit?For causal system? $\endgroup$ – Hasan Shuvo Oct 5 '18 at 16:36
  • $\begingroup$ @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. $\endgroup$ – Matt L. Oct 5 '18 at 16:56

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