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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 $ p(x)=1+2x+3x^2$ is 2. From Eq.(8.1), we see that $ M$ is the order of the transfer-function numerator polynomial in $ z^{-1}$ . Similarly, $ N$ is the order of the denominator polynomial in $ z^{-1}$ .

A rational function is any ratio of polynomials. That is, $ R(z)$ is a rational function if it can be written as

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

for finite-order polynomials $ P(z)$ and $ Q(z)$ . 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

It turns out the transfer function can be viewed as a rational function of either $ z^{-1}$ or $ z$ without affecting order. Let $ K=\max\{M,N\}$ denote the order of a general LTI filter with transfer function $ H(z)$ expressible as in Eq.(8.1). Then multiplying $ H(z)$ by $ z^K/z^K$ gives a rational function of $ z$ (as opposed to $ z^{-1}$ ) that is also order $ K$ when viewed as a ratio of polynomials in $ z$ . Another way to reach this conclusion is to consider that replacing $ z$ by $ z^{-1}$ is a conformal map [57] that inverts the $ z$ -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 $ z=0$ are either both counted or both not counted.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University