Amplitude Response

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Amplitude Response

We can isolate the filter amplitude response $G(\omega)$ by squaring and adding the above two equations:

$\begin{eqnarray*} a^2(\omega) + b^2(\omega) &=& G^2(\omega)\cos^2[\Theta(\omega)... ...ta(\omega)] + \sin^2[\Theta(\omega)]\right\}\\ &=& G^2(\omega). \end{eqnarray*}$

This can then be simplified as follows:

$\begin{eqnarray*} G^2(\omega) &=& a^2(\omega) + b^2(\omega)\\ &=& [1 + \cos(\... ... \cos(\omega T) \\ &=&4 \cos^2\left(\frac{\omega T}{2}\right). \end{eqnarray*}$

So we have made it to the amplitude response of the simple lowpass filter :

$\displaystyle G(\omega) = 2 \left\vert\cos\left(\frac{\omega T}{2}\right)\right\vert$

Since $\cos(\pi fT)$ is nonnegative for $-f_s/2 \leq f \leq f_s/2$ , it is unnecessary to take the absolute value as long as

is understood to lie in this range:

$\displaystyle \zbox {G(\omega) = 2 \cos(\pi f T)} \qquad \left\vert f\right\vert \leq \frac{f_s}{2} \protect$

(2.6)

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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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Chapters

Introduction to Digital Filters
    The Simplest Lowpass Filter
       Finding the Frequency Response
          Amplitude Response