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:

The inverse DTFT is

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.

[How to cite this work] [Order a printed hardcopy]

Copyright ©

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

[Automatic-links disclaimer]