Relation to Stretch Theorem
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
![$\displaystyle \hbox{\sc Chop}_{M,k}(X) \isdef
\left\{\begin{array}{ll}
X(k), ...
...+1}{2} \leq \left\vert k\right\vert \leq \frac{N}{2}. \\
\end{array} \right.
$](http://www.dsprelated.com/josimages_new/mdft/img1461.png)
![$ X$](http://www.dsprelated.com/josimages_new/mdft/img55.png)
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
![$ \hbox{\sc Chop}_M(X)$](http://www.dsprelated.com/josimages_new/mdft/img1462.png)
![$ \omega_c = 2\pi[(M-1)/2]/N$](http://www.dsprelated.com/josimages_new/mdft/img1463.png)
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
![$ X(-M/2)$](http://www.dsprelated.com/josimages_new/mdft/img1464.png)
![$ \hbox{\sc Stretch}()$](http://www.dsprelated.com/josimages_new/mdft/img1465.png)
Theorem: When
consists of one or more periods from a periodic
signal
,
![$\displaystyle \zbox {\hbox{\sc PerInterp}_L(x) = \hbox{\sc IDFT}(\hbox{\sc Chop}_N(\hbox{\sc DFT}(\hbox{\sc Stretch}_L(x)))).}
$](http://www.dsprelated.com/josimages_new/mdft/img1467.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![$ L-1$](http://www.dsprelated.com/josimages_new/mdft/img1218.png)
![$ x$](http://www.dsprelated.com/josimages_new/mdft/img25.png)
![$ N$](http://www.dsprelated.com/josimages_new/mdft/img35.png)
Proof: First, recall that
. That is,
stretching a signal by the factor
gives a new signal
which has a spectrum
consisting of
copies of
repeated around the unit circle. The ``baseband copy'' of
in
can be defined as the
-sample sequence centered about frequency
zero. Therefore, we can use an ``ideal filter'' to ``pass'' the
baseband spectral copy and zero out all others, thereby converting
to
. I.e.,
![$\displaystyle \hbox{\sc Chop}_N(\hbox{\sc Repeat}_L(X)) = \hbox{\sc ZeroPad}_{LN}(X)
\;\longleftrightarrow\;\hbox{\sc Interp}_L(x).
$](http://www.dsprelated.com/josimages_new/mdft/img1472.png)
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Bandlimited Interpolation of Time-Limited Signals
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Illustration of the Downsampling/Aliasing Theorem in Matlab