It is instructive to interpret the periodic interpolation theorem in
terms of the stretch theorem,
.
To do this, it is convenient to define a ``zero-centered rectangular
window'' operator:
Definition: For any
and any odd integer
we define the
length
even rectangular windowing operation by
Thus, this ``zero-phase rectangular window,'' when applied to a spectrum
Theorem: When
consists of one or more periods from a periodic
signal
,
In other words, ideal periodic interpolation of one period of
Proof: First, recall that
. That is,
stretching a signal by the factor
gives a new signal
which has a spectrum
consisting of
copies of
repeated around the unit circle. The ``baseband copy'' of
in
can be defined as the
-sample sequence centered about frequency
zero. Therefore, we can use an ``ideal filter'' to ``pass'' the
baseband spectral copy and zero out all others, thereby converting
to
. I.e.,
The last step is provided by the zero-padding theorem (§7.4.12).