Separating the Transfer Function Numerator and Denominator

From Eq.(6.5) we have that the transfer function of a recursive filter is a ratio of polynomials in $z$ :

$$H(z) = \frac{B(z)}{A(z)} \protect$$ (8.4)

where

$$\begin{eqnarray*} B(z) &=& b_0 + b_1 z^{-1}+ \cdots + b_M z^{-M}\\ A(z) &=& 1 + a_1 z^{-1}+ \cdots + a_N z^{-N}. \end{eqnarray*}$$

By elementary properties of complex numbers, we have

$$\begin{eqnarray*} G(\omega) &=& \frac{\left\vert B(e^{j\omega T})\right\vert}{\left\vert A(e^{j\omega T})\right\vert}\\ \Theta(\omega) &=& \angle B(e^{j\omega T}) - \angle A(e^{j\omega T}). \end{eqnarray*}$$

These relations can be used to simplify calculations by hand, allowing the numerator and denominator of the transfer function to be handled separately.