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

,
![$ k\in[0,N-1]$](http://www.dsprelated.com/josimages_new/mdft/img1046.png)
. 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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