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 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:
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.