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
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 , sets the spectrum to zero everywhere outside a
zero-centered interval of samples. Note that
ideal lowpass filtering operation in the frequency domain.
The ``cut-off frequency'' is
For even , we allow to be ``passed'' by the window,
but in our usage (below), this sample should always be zero anyway.
With this notation defined we can efficiently restate
periodic interpolation in terms of the
consists of one or more periods from a periodic
In other words, ideal periodic interpolation of one period of by
the integer factor may be carried out by first stretching by
the factor (inserting zeros between adjacent samples of
), taking the DFT, applying the ideal lowpass filter as an
-point rectangular window in the frequency domain, and performing
the inverse DFT.
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
The last step follows by definition of ideal periodic interpolation
in Eq. (7.9).