To represent practical FFT implementations, it is preferable to shift the frame back to the time origin:
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
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
, 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 ).