Frequency Response

GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Frequency Response

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Frequency Response

Beginning with Eq. (B.4.3), we have


where $X(z)$ is the $z$ transform of the filter input, $Y(z)$ is the $z$ transform of the output signal, and $H(z)$ is the filter transfer function.

Definition. The frequency response of a linear time-invariant digital filter is defined to be the transfer function, $H(z)$, evaluated on the unit circle, that is, $H(e^{j\.

The frequency response is a complex-valued function of a real variable. The response at frequency $f$ Hz, for example, is $H(e^{j2\, where $T$ is the sampling period in seconds.

Since every complex number can be represented as a magnitude and angle, the frequency response may be decomposed into two real-valued functions, the amplitude response and the phase response. Formally, we may define them as follows:

$\ $\ $\ (B.20)
$\ $\ $\ (B.21)

so that

Thus $G(\ is the magnitude (or complex modulus) of $H(e^{j\, and $\ is the phase (or complex angle) of $H(e^{j\.

Definition. The real valued function $G(\ in Eq. (B.21) is called the filter amplitude response and it specifies the amplitude gain that the filter provides at each frequency.

Definition. The function $G^2(\ is called the power response and it specifies the power gain at each frequency.

Definition. The real function $\ in Eq. (B.21) is thephase response and it gives the phase shift in radians that each input component sinusoid will undergo.

If the filter input and output signals are $x(n)$ and $y(n)$respectively, then

$\ $\ $\ (B.22)
$\ $\ $\ (B.23)

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