The Transfer Function in the Z-domain
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$\newcommand{\op}[1]{\mathsf #1}$
$\newcommand{\ztarrow}{\stackrel{\op Z}{\longrightarrow}}$

A LTI system is completely characterized by its impulse response
$h[n]$ or equivalently the Z-transform of the impulse response $H(z)$ which
is called the *transfer function*. Remember:

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

In case the impulse response is given to define the LTI system we can
simply calculate the Z-transform to obtain $H(z).

In case the system is defined with a difference equation we could
first calculate the impulse response and then calculating the
Z-transform. But it is far easier to calculate the Z-transform of both
sides of the difference equation.

As an example consider the following difference equation:

.. math::
   y[n] = 1.5y [n - 1] - 0.5y [n - 2] + 0.5x[n].

Remember that $x[n-n_0]\ztarrow z^{-n_0}X(z)$ and knowing that the
Z-transform is a linear transform we can apply the Z-transform to both
sides of the above equation and obtain:

.. math::
   Y(z) = 1.5 z^{-1} Y(z) - 0.5 z^{-2} Y(z) + 0.5 X(z)

This can be rewritten as:

 .. math::
    H(z) = \frac{Y(z)}{X(z)} = \frac{0.5}{1-1.5z^{-1}+0.5z^{-2}} = \frac{z^2}{2z^2 - 3z + 1}