Amplitude Response

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Amplitude Response

Since the frequency response is a complex-valued function, it has a magnitude and phase angle for each frequency. The magnitude of the frequency response is called the amplitude response (or magnitude frequency response), and it gives the filter gain at each frequency $\omega$ .

In this example, the amplitude response is

$\displaystyle \left\vert H(e^{j\omega T})\right\vert = \left\vert\frac{1 + g_1 e^{-jM_1\omega T}}{1 + g_2 e^{-jM_2\omega T}}\right\vert \protect$

(4.5)

which, for

, reduces to

$\displaystyle \left\vert H(e^{j\omega T})\right\vert = \frac{\left\vert\cos\lef... ...a T/2\right)\right\vert}{\left\vert\cos\left(M_2\omega T/2\right)\right\vert}.$

Figure 3.10a shows a graph of the amplitude response of one case of this filter, obtained by plotting Eq. $\,$ (3.5) for $\omega T \in[-\pi,\pi]$ , and using the example settings , , , and .

**Figure 3.10:** Frequency response of the example filter . (a) Amplitude response. (b) Phase response.
$\includegraphics[width=\textwidth]{eps/efr}$

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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
Analysis of a Digital Comb Filter
Amplitude Response