##

Transfer Function

As we cover in Chapter 6, the transfer function of a digital filter
is defined as
where is the *z* transform of the input
signal , and is the *z* transform of the output signal . We
may find from Eq.(3.1) by taking the *z* transform of both sides
and solving for :

Some principles of this analysis are as follows:

- The
*z*transform is a*linear operator*which means, by definition, -
. That is, the
*z*transform of a signal*delayed*by samples, , is . This is the*shift theorem*for*z*transforms, which can be immediately derived from the definition of the*z*transform, as shown in §6.3.

*z*transform are all we really need to find the transfer function of any linear, time-invariant digital filter from its difference equation (its implementation formula in the time domain).

In matlab, difference-equation coefficients are specified as
transfer-function coefficients (vectors `B` and `A` in Fig.3.9).
This is why a minus sign is needed in Eq.(3.3).

Ok, so finding the transfer function is not too much work. Now, what
can we do with it? There are two main avenues of analysis from here:
(1) finding the *frequency response* by setting
, and
(2) *factoring* the transfer function to find the *poles and
zeros* of the filter. One also uses the transfer function to generate
different implementation forms such as cascade or parallel
combinations of smaller filters to achieve the same overall filter.
The following sections will illustrate these uses of the transfer
function on the example filter of this chapter.

**Next Section:**

Frequency Response

**Previous Section:**

Impulse Response