12. Frequency Response

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Frequency Response

NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

<< Previous page  TOC  INDEX  Next page >>

Frequency Response

Beginning with Eq. (2.2.1), 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:

$\$\$\(48)
$\$\$\(49)

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. (49) 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. (49) 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

$\$\$\(50)
$\$\$\(51)

<< Previous page  TOC  INDEX  Next page >>