Duality of COLA and Nyquist Conditions

Let $ \hbox{\sc Cola}(N)$ denote constant overlap-add using hop size $ N$ . Then we have (by the Poisson summation formula Eq.$ \,$ (8.30))

\begin{eqnarray*}
w &\in& \hbox{\sc Nyquist}(N) \Leftrightarrow W \in \hbox{\sc Cola}(2\pi/N) \qquad \hbox{(FBS)} \\ [10pt]
w &\in& \hbox{\sc Cola}(R) \Leftrightarrow W \in \hbox{\sc Nyquist}(2\pi/R) \qquad \hbox{(OLA)}
\end{eqnarray*}

Specific Windows

  • Recall that the rectangular window transform is $ \hbox{\sc Nyquist}(2\pi/M)$ , implying the rectangular window itself is $ \hbox{\sc Cola}(M)$ , which is obvious.

  • The window transform for the Hamming family is $ \hbox{\sc Nyquist}(4\pi/M)$ , implying that Hamming windows are $ \hbox{\sc Cola}(M/2)$ , which we also knew.

  • The rectangular window transform is also $ \hbox{\sc Nyquist}(K2\pi/M)$ for any integer $ 1\leq K\leq M/2$ , implying that all hop sizes given by $ R=M/K$ for $ K=1,2,\ldots,M/2$ are COLA.

  • Because its side lobes are the same width as the sinc side lobes, the Hamming window transform is also $ \hbox{\sc Nyquist}(K2\pi/M)$ ,for any integer $ 2\leq K\leq M/2$ , implying hop sizes $ R=M/K$ are good, for $ K=2,\ldots,M/2$ . Thus, the available hop sizes for the Hamming window family include all of those for the rectangular window except one ($ R=M$ ).


The Nyquist Property on the Unit Circle

As a degenerate case, note that $ R=1$ is COLA for any window, while no window transform is $ \hbox{\sc Nyquist}(2\pi)$ except the zero window. (since it would have to be zero at dc, and we do not consider such windows). Did the theory break down for $ R=1$ ?

Intuitively, the $ \hbox{\sc Nyquist}(2\pi/R)$ condition on the window transform $ W(\omega)$ ensures that all nonzero multiples of the time-domain-frame-rate $ 2\pi/R$ will be zeroed out over the interval $ [-\pi,\pi)$ along the frequency axis. When the frame-rate equals the sampling rate ($ R=1$ ), there are no frame-rate multiples in the range $ [-\pi,\pi)$ . (The range $ [0,2\pi)$ gives the same result.) When $ R=2$ , there is exactly one frame-rate multiple at $ -\pi$ . When $ R=3$ , there are two at $ \pm 2\pi/3$ . When $ R=4$ , they are at $ -\pi$ and $ \pm\pi/2$ , and so on.

We can cleanly handle the special case of $ R=1$ by defining all functions over the unit circle as being $ \hbox{\sc Nyquist}(2\pi)$ when there are no frame-rate multiples in the range $ [-\pi,\pi)$ . Thus, a discrete-time spectrum $ W(\omega), \omega\in[-\pi,\pi)$ is said to be $ \hbox{\sc Nyquist}(2\pi/K)$ if $ W(r 2\pi/K)=0$ , for all $ \vert r\vert=1,2,\ldots,\left\lfloor K/2\right\rfloor $ , where $ \left\lfloor x\right\rfloor $ (the ``floor function'') denotes the greatest integer less than or equal to $ x$ .


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