Filter Order = Transfer Function Order

Free Books Introduction to Digital Filters

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 $z^{-1}$ . Similarly, is the order of the denominator polynomial in $z^{-1}$ .

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

$\displaystyle R(z)\eqsp \frac{P(z)}{Q(z)}$

for finite-order polynomials

and

. The 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:

$\textstyle \parbox{0.9\textwidth}{\emph{The order of an LTI filter is the order of its transfer function.}}$

It turns out the transfer function can be viewed as a rational function of either $z^{-1}$ or without affecting order. Let $K=\max\{M,N\}$ 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 $z^{-1}$ ) that is also order when viewed as a ratio of polynomials in . Another way to reach this conclusion is to consider that replacing by $z^{-1}$ 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 $z=\infty$ and are either both counted or both not counted.