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

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.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University