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Stretch Theorem (Repeat Theorem)

Theorem: For all $ x\in{\bf C}^N$,

$\displaystyle \zbox {\hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X).}

Proof: Recall the stretch operator:

$\displaystyle \hbox{\sc Stretch}_{L,m}(x) \isdef
...=\mbox{integer} \\ [5pt]
0, & m/L\neq \mbox{integer} \\
\end{array} \right.

Let $ y\isdeftext \hbox{\sc Stretch}_L(x)$, where $ y\in{\bf C}^M$, $ M=LN$. Also define the new denser frequency grid associated with length $ M$ by $ \omega^\prime_k \isdeftext 2\pi k/M$, and define $ \omega_k=2\pi k/N$ as usual. Then

$\displaystyle Y(k) \isdef \sum_{m=0}^{M-1} y(m) e^{-j\omega^\prime_k m}
= \sum_{n=0}^{N-1}x(n) e^{-j\omega^\prime_k nL}$   $\displaystyle \mbox{($n\isdef m/L$).}$


$\displaystyle \omega^\prime_k L \isdef \frac{2\pi k}{M} L = \frac{2\pi k}{N} = \omega_k .

Thus, $ Y(k)=X(k)$, and by the modulo indexing of $ X$, $ L$ copies of $ X$ are generated as $ k$ goes from 0 to $ M-1 = LN-1$.

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Downsampling Theorem (Aliasing Theorem)
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Rayleigh Energy Theorem (Parseval's Theorem)