Duality of COLA and Nyquist Conditions

Let denote constant overlap-add using hop size . Then we have (by the Poisson summation formula Eq. (8.30))

Specific Windows

• Recall that the rectangular window transform is , implying the rectangular window itself is , which is obvious.

• The window transform for the Hamming family is , implying that Hamming windows are , which we also knew.

• The rectangular window transform is also for any integer , implying that all hop sizes given by for are COLA.

• Because its side lobes are the same width as the sinc side lobes, the Hamming window transform is also ,for any integer , implying hop sizes are good, for . Thus, the available hop sizes for the Hamming window family include all of those for the rectangular window except one ( ).

The Nyquist Property on the Unit Circle

As a degenerate case, note that is COLA for any window, while no window transform is 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 ?

Intuitively, the condition on the window transform ensures that all nonzero multiples of the time-domain-frame-rate will be zeroed out over the interval along the frequency axis. When the frame-rate equals the sampling rate ( ), there are no frame-rate multiples in the range . (The range gives the same result.) When , there is exactly one frame-rate multiple at . When , there are two at . When , they are at and , and so on.

We can cleanly handle the special case of by defining all functions over the unit circle as being when there are no frame-rate multiples in the range . Thus, a discrete-time spectrum is said to be if , for all , where (the floor function'') denotes the greatest integer less than or equal to .

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