Convolution in Z-domain
=======================
$\newcommand{\op}[1]{\mathsf #1}$
$\newcommand{\ztarrow}{\stackrel{\op Z}{\longrightarrow}}$

Consider a LTI system characterized with the impulse response function
$h[n]$. Given an input signal $x[n]$ the output of the system equals
$y[n]=x[n]\ast h[n]$ with:

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

The \Zt{} of the output is defined as:

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

Therefore:

.. math::
   x[n]\ast h[n] \ztarrow X(z) H(z)