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Introduction to Digital Filters
    Frequency Response Analysis
      
             Separating the Transfer Function Numerator and Denominator

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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}{\... ...a(\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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Previous: Polar Form of the Frequency Response
Next: Frequency Response as a Ratio of DTFTs
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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