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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