## STFT Summary and Conclusions

The Short-Time Fourier Transform (STFT) may be viewed either as an OverLap-Add (OLA) processor, or as a Filter-Bank Sum (FBS). We derived two conditions for perfect reconstruction which are Fourier duals of each other:

- For OLA, the window must overlap-add to a constant in the time
domain. By the Poisson summation formula, this is equivalent
to having window transform nulls at all nonzero multiples of
the frame rate
.
- For FBS, the window
*transform*must overlap-add to a constant in the frequency domain, and this is equivalent to having window nulls in the time domain at all nonzero multiples of the transform size .

In general, STFT filter banks are *oversampled* except when using
the rectangular window of length
and a hop size
. Critical
sampling is desirable for compression systems, but this can be
problematic when spectral modifications are contemplated
(adjacent-channel aliasing no longer canceled).

STFT filter banks are *uniform* filter banks, as opposed
``constant Q''. In some audio applications, it is preferable to use
non-uniform filter banks which approximate the *auditory filter
bank*. Approximate constant-Q filter banks are easily synthesized
from STFT filter banks by *summing* adjacent frequency channels,
as detailed in §10.7 below. Additional pointers can be found in
Appendix E. We will look at a particular octave filter bank when we
talk about wavelet filter banks (§11.9).

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STFT with Modifications