Frequency Response

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Frequency Response

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>

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)

<< Previous page  TOC  INDEX  Next page >>


© 1998-2017 – Nicola Asuni - - All rights reserved.
about - disclaimer - privacy