Stretch Theorem (Repeat Theorem)
Theorem: For all
,
![$\displaystyle \zbox {\hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X).}
$](http://www.dsprelated.com/josimages_new/mdft/img1429.png)
Proof:
Recall the stretch operator:
![$\displaystyle \hbox{\sc Stretch}_{L,m}(x) \isdef
\left\{\begin{array}{ll}
x(...
...=\mbox{integer} \\ [5pt]
0, & m/L\neq \mbox{integer} \\
\end{array} \right.
$](http://www.dsprelated.com/josimages_new/mdft/img1430.png)
![$ y\isdeftext \hbox{\sc Stretch}_L(x)$](http://www.dsprelated.com/josimages_new/mdft/img1431.png)
![$ y\in{\bf C}^M$](http://www.dsprelated.com/josimages_new/mdft/img1432.png)
![$ M=LN$](http://www.dsprelated.com/josimages_new/mdft/img1250.png)
![$ M$](http://www.dsprelated.com/josimages_new/mdft/img244.png)
![$ \omega^\prime_k \isdeftext 2\pi k/M$](http://www.dsprelated.com/josimages_new/mdft/img1433.png)
![$ \omega_k=2\pi k/N$](http://www.dsprelated.com/josimages_new/mdft/img1310.png)
![$\displaystyle Y(k) \isdef \sum_{m=0}^{M-1} y(m) e^{-j\omega^\prime_k m}
= \sum_{n=0}^{N-1}x(n) e^{-j\omega^\prime_k nL}$](http://www.dsprelated.com/josimages_new/mdft/img1434.png)
![$\displaystyle \mbox{($n\isdef m/L$).}$](http://www.dsprelated.com/josimages_new/mdft/img1435.png)
![$\displaystyle \omega^\prime_k L \isdef \frac{2\pi k}{M} L = \frac{2\pi k}{N} = \omega_k .
$](http://www.dsprelated.com/josimages_new/mdft/img1436.png)
![$ Y(k)=X(k)$](http://www.dsprelated.com/josimages_new/mdft/img1437.png)
![$ X$](http://www.dsprelated.com/josimages_new/mdft/img55.png)
![$ L$](http://www.dsprelated.com/josimages_new/mdft/img1214.png)
![$ X$](http://www.dsprelated.com/josimages_new/mdft/img55.png)
![$ k$](http://www.dsprelated.com/josimages_new/mdft/img20.png)
![$ M-1 = LN-1$](http://www.dsprelated.com/josimages_new/mdft/img1438.png)
Next Section:
Downsampling Theorem (Aliasing Theorem)
Previous Section:
Rayleigh Energy Theorem (Parseval's Theorem)
Theorem: For all
,
Proof:
Recall the stretch operator: