Compute the z-transform of each of the following:
a)$ \ x[n] = \delta[n] $
b)$ \ x[n] = \delta[n - 1] $
c)$ \ x[n] = \delta[n + 1] $
a)$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty \delta[n]z^{-n} = z^{0} = 1 $
The ROC is the entire finite Z-plane.
$ X(\frac{1}{z}) = 1 \rightarrow 1 $ as $ z \rightarrow 0 $ Therefore $ z = \infty $ is also in the ROC.
b)$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty \delta[n - 1]z^{-n} = z^{-1} = /frac{1}{z} $
The ROC is the entire finite Z-plane, except at $ z = 0 $.
$ X(\frac{1}{z}) = \frac{1}{\frac{1}{z}} = z \rightarrow 0 $ as $ z \rightarrow 0 $ Therefore $ z = \infty $ is also in the ROC.
c)$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty \delta[n + 1]z^{-n} = z^{-(-1)} = z $
The ROC is the entire finite Z-plane, except at $ z = \infty $.
$ X(\frac{1}{z}) = \frac{1}{z} = z \rightarrow \infty $ as $ z \rightarrow 0 $ Therefore $ z = \infty $ is NOT in the ROC.
