## 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 .

*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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