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 .

• STFT filter banks are oversampled except when using the rectangular window of length and a hop size .

• 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 E.
  • We will study a particular case (an octave filter bank) when we talk about wavelet filter banks in §12.2.

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

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

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Subsections

• Overlap-Add
• Filter Bank Summation

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Order a Hardcopy of Spectral Audio Signal Processing

Previous: Time Varying Modifications in FBS
Next: Overlap-Add

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.