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



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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Frequency Response Analysis Problems