To represent practical
FFT implementations, it is preferable
to shift the

frame back to the time origin:

 |
(9.20) |
This is summarized in Fig.
8.11. Zero-based frames are needed
because the leftmost input sample is assigned to time zero by FFT algorithms.
In other words, a hopping FFT effectively redefines
time zero on each hop. Thus, a practical
STFT is a sequence of FFTs
of the zero-based frames

. On the other hand,
papers in the literature (such as [
7,
9]) work with the fixed time-origin case (

). Since they differ only by a time shift, it is not hard to
translate back and forth.
Note that we may
sample the
DTFT of both

and

,
because both are
time-limited to

nonzero samples. The
minimum information-preserving
sampling interval along the unit circle
in both cases is

. In practice, we often
oversample to some extent, using

with

instead. For

, we get
where

. For

we have
Since

, their transforms are related by the
shift theorem:
where

denotes modulo

indexing (appropriate since the
DTFTs have been sampled at intervals of

).
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