The Filter Bank Summation (FBS) Interpretation of the Short Time
Fourier Transform (STFT)
Portnoff WindowsSearch Spectral Audio Signal Processing
Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?
In 1976 [195], Portnoff observed that any window
of the form
Portnoff suggested that, in practical usage, windowed data segments
longer that the FFT size should be time-aliased about length
prior to taking the FFT. This result is readily derived from the
definition of the time-normalized STFT introduced in Eq.
(8.3):
where
as usual.
Choosing
allows multiple sidelobes of the sinc function to
alias in on the main lobe. This gives channel filters in the
frequency domain which are sharper bandpass filters. I.e., there is
less channel
cross-talk in the frequency domain. However, the time-aliasing
corresponds to undersampling in the frequency domain, implying less
robustness to spectral modifications, since such modifications can
disturb the time-domain aliasing cancellation. Since the hop size
needs to be less than
, the overall filter bank based on a Portnoff
window remains oversampled in the time domain.
If we use a window which is
with
larger than the DFT
size, then the sinc main-lobe is wider than the DFT length in the time
domain, and the corresponding filter bands have gaps between them in
the frequency domain. This can be used in classical vocoder
applications which require the signal to be periodic: The channel
filters are tuned to the harmonics of the input signal, and each
filter only has to isolate the harmonic coming in at its center
frequency. It can also be pushed to the point that the vocoded signal
has a ``ringing'' quality--an effect often heard in commercial
vocoders.
