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.