Finite duration vs infinite duration sequences?

Question

What are difference between finite duration and infinite duration sequences? As far as i am able to google and know only one difference that is finite duration sequence ROC exist on xy plane while infinite duration sequence ROC exist on unit circle plane

I have also attached two snapshots in this regard

Answer 1

There is a slight misunderstanding here. There is no such plane called unit circle plane. Unit circle is in the xy plane (or we call it z plane). The difference is, rather, in the shape of the ROC. For infinite duration sequences, the ROC is in the form of a circular strip in the xy plane (as shown in the second figure).

In the case of finite duration sequences, ROC is the entire xy plane except at z=0 or at z = $\infty $ or at z=0 and $\infty$.

e.g. $x(n) = \{1,\underset{\uparrow}{2},3\} $

$X(z) = \sum \limits_{-\infty}^{+\infty} x(n)z^{-n}$

$ = x(-1)z^1+x(0)z^0+x(1)z^{-1} = z+2+\frac{3}{z}$

This Z transform will not exist only if $z=\infty$ (for the first term) and $z=0$ (for the last term). This can be generalised for all finite duration signals that their ROC consists of entire z plane (xy plane) except at z=0 or z=$\infty$ or at both.