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

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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