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 $2\pi/R$ .

  • 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 $N$ .

• STFT filter banks are oversampled except when using the rectangular window of length $M=N$ and a hop size $R=N$ .

• Critical sampling is desired for compression systems, but it is problematic in conjunction with spectral modifications. (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.
  • Some pointers can be found in Appendix F.
  • We will study a particular case (an octave filter bank) when we talk about wavelet filter banks in §10.9.2.

• Approximate constant-Q filter banks are easily synthesized from STFT filter banks by summing adjacent frequency channels. However,
  • when $K$ adjacent FFT bins are summed, the hop size in the time domain should be reduced by at least $K$ .

  • A refinement of bin-summing is to multiply the $K$ FFT bins by a $K\times K$ matrix which produces output samples from a $K$ -bin-wide filter band over $K$ successive time steps. (This is perhaps a research suggestion since we are not aware of any published papers describing this technique.)