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 $x$ .