## Portnoff Windows

In 1976 [212], Portnoff observed that any window of the form

sinc | (10.23) |

being by construction, will obey the weak constraint, where is the number of spectral samples. In this result, is any window function whatsoever, and the sinc function is defined as usual by

sinc | (10.24) |

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

Portnoff suggested that, in practical usage, windowed data segments
longer than the FFT size should be *time-aliased* about length
prior to performing an FFT. This result is readily derived from the
definition of the time-normalized STFT introduced in Eq.
(8.21):

where as usual.

Choosing
allows multiple side lobes of the sinc function to
alias in on the main lobe. This gives channel filters in the
frequency domain which are sharper bandpass filters while remaining COLA.
*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.

**Next Section:**

Downsampled STFT Filter Banks

**Previous Section:**

Duality of COLA and Nyquist Conditions