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See Also

Embedded SystemsFPGAElectronics

Introduction to Digital Filters
    Frequency Response Analysis
       Frequency Response as a Ratio of DTFTs

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Frequency Response as a Ratio of DTFTs

From Eq.$ \,$(6.5), we have $ H(z)=B(z)/A(z)$, so that the frequency response is

$\displaystyle H(e^{j\omega T})=\frac{B(e^{j\omega T})}{A(e^{j\omega T})}, $

and

\begin{eqnarray*} B(e^{j\omega T}) &=& \mbox{{\sc DTFT}}_{\omega T}({\underline{... ...^{j\omega T}) &=& \mbox{{\sc DTFT}}_{\omega T}({\underline{a}}), \end{eqnarray*}

where

\begin{eqnarray*} {\underline{b}}&\isdef & [b_0,b_1,\ldots,b_M,0,\ldots]\\ {\underline{a}}&\isdef & [1,a_1,\ldots,a_N,0,\ldots], \end{eqnarray*}

and the DTFT is as defined in Eq.$ \,$(7.1).

From the above relations, we may express the frequency response of any IIR filter as a ratio of two finite DTFTs:

$\displaystyle H(e^{j\omega T}) \eqsp \frac{\mbox{{\sc DTFT}}_{\omega T}({\under... ...^M b_m e^{-j\omega mT}}{\displaystyle\sum_{n=0}^N a_n e^{-j\omega nT}} \protect$ (8.5)

This expression provides a convenient basis for the computation of an IIR frequency response in software, as we pursue further in the next section.


Subsections
Previous: Separating the Transfer Function Numerator and Denominator
Next: Frequency Response in Matlab

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About the Author: 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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