=======================
 Pairs of Z-Transforms
=======================

.. list-table::
   :widths: 10 10
   :header-rows: 1
   :class: tablefullwidth

   * - Time Domain

     - Z-domain$\hspace{4em}$(ROC)

   * - Pulse

       .. math::
          \delta[n]

     - Constant

       .. math::
          1\qquad \left(z\in\setC\right)

   * - Step

       .. math::
          u[n] = \begin{cases}1 &: n\geq 0\\ 0&: \text{elsewhere}\end{cases}

     -
       .. math::
          \frac{z}{z-1} \qquad\left( |z|>1 \right)


   * - Exponential

       .. math::
          a^n u[n]

     -
       .. math::
          \frac{1}{1-a z^{-1}} \qquad \left( |z|>|a| \right)


   * - Complex Exponential

       .. math::
          e^{j\W n} u[n]

     -
       .. math::
          \frac{z}{z-e^{-j\W}} \qquad\left( |z|>1 \right)


.. rubric:: Exponential

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

.. math::
   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}}

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


.. rubric:: 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.