============================
 $\ZT$-Transform Properties
============================

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

   * - Time Domain

     - Z-domain

   * - Synthesis (Inverse Z=Transform)

       .. math::
          x[n] = \frac{1}{2\pi j}\oint_C X(z)z^{n-1}dz

     - Analysis (Z-Transform)

       .. math::
          X(z) = \sum_{n=-\infty}^{\infty} x[n] z^{-n}


   * - Complex Conjugate

       .. math::
          x[n]^\ast

     -
       .. math::
          X^\ast(z^\ast)


   * - Real Signal

       .. math::
          x[n]\in\setR

     -
       .. math::
          X(z^\star) = X^\star(z)

   * - Time Shift

       .. math::
          x[n-n_0]

     - 'Phase' factor

       .. math::
          z^{-n_0} X(z)

   * - Time difference

       .. math::
          x[n] - x[n-1]

     -
       .. math::
          (1-z^{-1}) X(z)

   * - Convolution

       .. math::
          x[n] \ast y[n]

     - Multiplication

       .. math::
          X(z) Y(z)

The properties in the above table are easy to prove. Some proofs are
given below.


.. rubric:: Complex Conjugate

Let calculate the Z-transform of $x[n]^\ast$:

.. math::
   \sum_{n=-\infty}^{\infty} x[n]^\ast z^{-n} &=
   \sum_{n=-\infty}^{\infty} \left(x[n] (z^{-n})^\ast \right)^\ast \\
   &= \left(\sum_{n=-\infty}^{\infty} x[n] (z^{-n})^\ast \right)^\ast \\
   &= \left(\sum_{n=-\infty}^{\infty} x[n] (z^\ast)^{-1} \right)^\ast \\
   &= \left( X(z^\ast) \right)^\ast = X^\ast(z^\ast)

.. rubric:: Time Shift

The Z-transform of $x[n-n_0]$ is given by

.. math::
   \sum_{m=-\infty}^{\infty} x[n-n_0] z^{-n} &= \sum_{k=-\infty}^{\infty} x[k] z^{-(k+n_0)}\\
   &= \sum_{k=-\infty}^{\infty} x[k] z^{-k} z^{-n_0}\\
   &= z^{-n_0} \sum_{k=-\infty}^{\infty} x[k] z^{-k}\\
   &= z^{-n_0} X(z)




.. rubric:: Convolution Property

The definition of the convolution is:

.. math::
   x[n] \ast y[n] = \sum_{m=-\infty}^{\infty} x[n-m] y[m]

Its Z-transform is:

.. math::
   \sum_{n=-\infty}^{\infty} \left(x[n]\ast y[n]\right)z^{-n} &=
   \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} x[n-m] y[m]\right) z^{-n}\\
   &= \sum_{m=-\infty}^{\infty} y[m] \sum_{n=-\infty}^{\infty} x[n-m] z^{-n}\\

In the last summation we recognize the Z-transform of the shifted signal $x$ so:

.. math::
    \sum_{n=-\infty}^{\infty} \left(x[n]\ast y[n]\right)z^{-n} &=
    \sum_{m=-\infty}^{\infty} y[m] z^{-m} X(z)\\
    &= X(z) Y(z)

Summarizing:

.. math::
   x[n]\ast y[n] \ZTright X(z) Y(z)