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

for finite-order polynomials and . The

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University