Since *z* transforming the convolution representation for digital filters was
so fruitful, let's apply it now to the general difference equation,
Eq.
(5.1). To do this requires two properties of the *z* transform,
*linearity* (easy to show) and the *shift theorem*
(derived in §6.3 above). Using these two properties, we
can write down the *z* transform of any difference equation by inspection, as
we now show. In
§6.8.2, we'll show how to *invert* by inspection as well.

Repeating the general difference equation for LTI filters, we have (from Eq. (5.1))

Let's take the *z* transform of both sides, denoting the transform by
. Because
is a linear operator,it may be distributed through the terms on the right-hand side as
follows:^{7.3}
where we used the superposition and scaling properties of linearity
given on page , followed by use of the shift
theorem, in that order. The terms in
may be grouped together
on the left-hand side to get

Factoring out the common terms and gives

Defining the polynomials

the *z* transform of the difference equation yields

Finally, solving for , which is by definition the transfer function , gives

Thus, taking the

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