## Filter Order = Transfer Function Order

Recall that the
*order of a polynomial*
is defined as the highest
power of the polynomial variable. For example, the order of the
polynomial
is 2. From Eq.(8.1), we see that is
the order of the transfer-function numerator polynomial in .
Similarly, is the order of the denominator polynomial in .

A *rational function* is any ratio of polynomials. That is,
is a rational function if it can be written as

*order of a rational function*is defined as the maximum of its numerator and denominator polynomial orders. As a result, we have the following simple rule:

It turns out the transfer function can be viewed as a rational
function of either or without affecting order. Let
denote the order of a general LTI filter with transfer
function expressible as in Eq.(8.1). Then multiplying
by gives a rational function of (as opposed to )
that is also order when viewed as a ratio of polynomials in .
Another way to reach this conclusion is to consider that replacing
by is a *conformal map* [57] that inverts the
-plane with respect to the unit circle. Such a transformation
clearly preserves the number of poles and zeros, provided poles and
zeros at and are either both counted or both not
counted.

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Graphical Computation of Amplitude Response from Poles and Zeros

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Frequency Response Analysis Problems