Portnoff Windows

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

$$w(n) = w_0(n) (n/N),$$ (10.23)

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

$$(n) \mathrel{\stackrel{\Delta}{=}}\frac{\sin(\pi n)}{\pi n}$$ (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 $N$ prior to performing an FFT. This result is readily derived from the definition of the time-normalized STFT introduced in Eq.(8.21):

$$\begin{eqnarray*} {\tilde X}_m(\omega_k) &\isdef & \hbox{\sc Sample}_{\Omega_N,k}\left(\hbox{\sc DTFT}\left({\tilde x}_m\right)\right) \\ &\isdef & \hbox{\sc Sample}_{\Omega_N,k}\left(\hbox{\sc DTFT}\left(\hbox{\sc Shift}_{-m}(x)\cdot w\right)\right) \\ &=& \sum_{n=-\infty}^\infty x(n+m)w(n)e^{-j\omega_k n}\quad\hbox{(now let $n\isdef lN+i$)}\\ &=& \sum_{l=-\infty}^\infty \sum_{i=0}^{N-1}x(lN+i+m)w(lN+i) \underbrace{e^{-j\omega_k (lN+i)}}_{e^{-j\omega_k i}}\\ &=& \sum_{i=0}^{N-1}\left[\sum_{l=-\infty}^\infty x(lN+i+m)w(lN+i)\right] e^{-j\omega_k i}\\ &=& \sum_{i=0}^{N-1}\hbox{\sc Alias}_{N,i}[\hbox{\sc Shift}_{-m}(x)\cdot w] e^{-j\omega_k i}\\ &\isdef & \hbox{\sc DFT}_{N,k}\{\hbox{\sc Alias}_N[\hbox{\sc Shift}_{-m}(x)\cdot w]\}, \end{eqnarray*}$$

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

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