### STFT of COLA Decomposition

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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