## 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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