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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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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2016-07-18 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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