4.2.7. The Transfer Function in the Z-domain

\(\newcommand{\op}[1]{\mathsf #1}\) \(\newcommand{\ztarrow}{\stackrel{\op Z}{\longrightarrow}}\)

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:

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

In case the impulse response is given to define the LTI system we can simply calculate the Z-transform to obtain :math:`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:

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

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:

\[Y(z) = 1.5 z^{-1} Y(z) - 0.5 z^{-2} Y(z) + 0.5 X(z)\]

This can be rewritten as:

\[H(z) = \frac{Y(z)}{X(z)} = \frac{0.5}{1-1.5z^{-1}+0.5z^{-2}} = \frac{z^2}{2z^2 - 3z + 1}\]