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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$ :

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


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2015-04-22 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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