## 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

*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.

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Frequency Response Analysis Problems