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:

(3.2) |

where denotes the

The inverse DTFT is

(3.3) |

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

Instead of operating on sampled signals of length (like the DFT), the DTFT operates on sampled signals defined over all integers .

Unlike the DFT, 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. Thus, as
, 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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University