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Frequency Response

Definition: The frequency response of an LTI filter may be defined as the Fourier transform of its impulse response. In particular, for finite, discrete-time signals $ h\in\mathbb{C}^N$ , the sampled frequency response may be defined as

$\displaystyle H(\omega_k) \isdef \hbox{\sc DFT}_k(h).

The complete (continuous) frequency response is defined using the DTFT (see §B.1), i.e.,

$\displaystyle H(\omega) \isdefs \hbox{\sc DTFT}_\omega(\hbox{\sc ZeroPad}_\infty(h)) \isdefs \sum_{n=0}^{N-1}h(n) e^{-j\omega n}

where the summation limits are truncated to $ [0,N-1]$ because $ h(n)$ is zero for $ n<0$ and $ n>N-1$ . Thus, the DTFT can be obtained from the DFT by simply replacing $ \omega_k$ by $ \omega$ , which corresponds to infinite zero-padding in the time domain. Recall from §7.2.10 that zero-padding in the time domain gives ideal interpolation of the frequency-domain samples $ H(\omega_k)$ (assuming the original DFT included all nonzero samples of $ h$ ).

Definition: The amplitude response of a filter is defined as the magnitude of the frequency response

$\displaystyle G(k) \isdef \left\vert H(\omega_k)\right\vert.

From the convolution theorem, we can see that the amplitude response $ G(k)$ is the gain of the filter at frequency $ \omega_k$ , since

$\displaystyle \left\vert Y(\omega_k)\right\vert = \left\vert H(\omega_k)X(\omega_k)\right\vert
= G(k)\left\vert X(\omega_k)\right\vert,

where $ X(\omega_k)$ is the $ k$ th sample of the DFT of the input signal $ x(n)$ , and $ Y$ is the DFT of the output signal $ y$ .

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``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 © 2019-01-09 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University