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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 is allowed to approach
infinity:

where

denotes the

*continuous*
normalized radian frequency variable,

^{B.1} and

is the

signal amplitude at sample
number

.

The inverse DTFT is

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 (like the DFT),
the DTFT operates on sampled signals defined over all integers
. As a result, the DTFT frequencies form a
*continuum*. That is, the DTFT is a function of
*continuous* frequency
, while the DFT is a
function of discrete frequency ,
. The DFT
frequencies
,
, are given by
the angles of points uniformly distributed along the unit circle
in the complex plane (see
Fig.6.1). Thus, as
, a continuous frequency axis
must result in the limit along the unit circle in the 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

**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 Associate Professor (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

http://ccrma.stanford.edu/~jos/ for details.