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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
      
             The Nyquist Property on the Unit Circle

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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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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