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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       Portnoff Windows

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Portnoff Windows

In 1976 [195], Portnoff observed that any window $ w$ of the form

$\displaystyle w(n) = w_0(n)$   sinc$\displaystyle (n/N), $

being $ \hbox{\sc Nyquist}(N)$ by construction, will obey the weak $ \hbox{\sc Cola}(2\pi/N)$ constraint, where $ N$ is the number of spectral samples. In this result, $ w_0(n)$ is any window function whatsoever, and the sinc function is defined as usual by

   sinc$\displaystyle (n) \mathrel{\stackrel{\Delta}{=}}\frac{\sin(\pi n)}{\pi n} $

(the unit-amplitude sinc function with zeros at all nonzero integers).

Portnoff suggested that, in practical usage, windowed data segments longer that the FFT size should be time-aliased about length $ N$ prior to taking the FFT. This result is readily derived from the definition of the time-normalized STFT introduced in Eq.$ \,$(8.3):

\begin{eqnarray*} {\tilde X}_m(\omega_k) &\isdef & \hbox{\sc Sample}_{\Omega_N,... ...T}_{N,k}\{\hbox{\sc Alias}_N[\hbox{\sc Shift}_{-m}(x)\cdot w]\}, \end{eqnarray*}

where $ \omega_k \isdef 2\pi k/N \isdef k\Omega_N$ as usual.

Choosing $ M\gg N$ 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 $ N$, the overall filter bank based on a Portnoff window remains oversampled in the time domain.

If we use a window which is $ \hbox{\sc Nyquist}(M)$ with $ M$ 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.

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Previous: The Nyquist Property on the Unit Circle
Next: Downsampled STFT Filter Banks
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.


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