4.2.3. Pairs of Z-Transforms

Time Domain	Z-domain$\hspace{4em}$(ROC)
Pulse $$\delta[n]$$	Constant $$1\qquad \left(z\in\setC\right)$$
Step $$\begin{split}u[n] = \begin{cases}1 &: n\geq 0\\ 0&: \text{elsewhere}\end{cases}\end{split}$$	$$\frac{z}{z-1} \qquad\left( |z|>1 \right)$$
Exponential $$a^n u[n]$$	$$\frac{1}{1-a z^{-1}} \qquad \left( |z|>|a| \right)$$
Complex Exponential $$e^{j\W n} u[n]$$	$$\frac{z}{z-e^{-j\W}} \qquad\left( |z|>1 \right)$$

Exponential

Let $x[n] = a^n u[n]$ then we can calculate its Z-transform as:

$$\begin{split}X(z) &= \sum_{n=-\infty}^{\infty} x[n] z^{-n}\\ &= \sum_{n=0}^{\infty} a^n z^{-n}\\ &= \sum_{n=0}^{\infty} (a z^{-1})^n\\ &= \frac{1}{1-az^{-1}}\end{split}$$

Note that this geometric series only converges for $|az^{-1}| < 1$ which can be reshufled and leads to $| z | > | a |$ for the ROC.

Complex Exponential

This result in the table follows directly from the result above.

Take a look at Wikipedia for many more examples of Z-transform pairs. Some of them you should be able to prove yourself.