Stretch Theorem (Repeat Theorem)

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

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

Proof: Recall the stretch operator:

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

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

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

But

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

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