### 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

 (9.21)

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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