COLA/Nyquist OLA/FBS Duality

Let $\hbox{\sc Cola}(N)$ denote constant overlap-add using hop size $N$ . Then we have (by the Poisson summation formula):

$$w \in \hbox{\sc Nyquist}(N) \;\Longleftrightarrow\; W \in \hbox{\sc Cola}(2\pi/N) \qquad \hbox{(FBS)}$$

$$w \in \hbox{\sc Cola}(R) \;\Longleftrightarrow\; W \in \hbox{\sc Nyquist}(2\pi/R) \qquad \hbox{(OLA)}$$

• For perfect-reconstruction (PR) STFT processing, we prefer both window conditions $w\in\hbox{\sc Cola}(R)$ and $w\in\hbox{\sc Nyquist}(N)$ , in which case the window transform satisfies $W\in\hbox{\sc Cola}(2\pi/N)$ and $W\in\hbox{\sc Nyquist}(2\pi/R)$

• Window lengths $M$ shorter than the FFT length $N$ trivially satisfy the $\hbox{\sc Nyquist}(N)$ property