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Discrete Time Fourier Transform (DTFT)

The Discrete Time Fourier Transform (DTFT) can be viewed as the limiting form of the DFT when its length $ N$ is allowed to approach infinity:

$\displaystyle X(\omega) \isdefs \sum_{n=-\infty}^\infty x(n) e^{-j\omega n},$ (3.2)

where $ \omega\in[-\pi,\pi)$ denotes the continuous radian frequency variable,3.3 and $ x(n)$ is the signal amplitude at sample number $ n$ .

The inverse DTFT is

$\displaystyle x(n) \eqsp \frac{1}{2\pi}\int_{-\pi}^\pi X(\omega) e^{j\omega n} d\omega,$ (3.3)

which can be derived in a manner analogous to the derivation of the inverse DFT [264].

Instead of operating on sampled signals of length $ N$ (like the DFT), the DTFT operates on sampled signals $ x(n)$ defined over all integers $ n\in\mathbb{Z}$ .

Unlike the DFT, the DTFT frequencies form a continuum. That is, the DTFT is a function of continuous frequency $ \omega\in[-\pi,\pi)$ , while the DFT is a function of discrete frequency $ \omega_k$ , $ k\in[0,N-1]$ . The DFT frequencies $ \omega_k = 2\pi k/N$ , $ k=0,1,2,\ldots,N-1$ , are given by the angles of $ N$ points uniformly distributed along the unit circle in the complex plane. Thus, as $ N\to\infty$ , a continuous frequency axis must result in the limit along the unit circle. The axis is still finite in length, however, because the time domain remains sampled.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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