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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.

Previous: Fourier Transforms for Continuous/Discrete Time/Frequency
Next: Fourier Transform (FT) and Inverse

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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