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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(\tilde{\omega}) \isdef \sum_{n=-\infty}^\infty x(n) e^{-j\tilde{\omega}n}

where $ \tilde{\omega}\isdef \omega T\in[-\pi,\pi)$ denotes the continuous normalized radian frequency variable,B.1 and $ x(n)$ is the signal amplitude at sample number $ n$ .

The inverse DTFT is

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

which can be derived in a manner analogous to the derivation of the inverse DFT (see Chapter 6).

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{\bf Z}$ . As a result, the DTFT frequencies form a continuum. That is, the DTFT is a function of continuous frequency $ \tilde{\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 (see Fig.6.1). Thus, as $ N\to\infty$ , a continuous frequency axis must result in the limit along the unit circle in the $ z$ plane. The axis is still finite in length, however, because the time domain remains sampled.

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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 © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University