Stretch Theorem (Repeat Theorem)

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Stretch Theorem (Repeat Theorem)

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Stretch Theorem (Repeat Theorem)



Theorem: For all $x\,

\

Proof: Recall that

\

Let $y\, where $y\, $M=LN$. Also define the new denser frequency grid associated with length $M$ by $\, with $\ as usual. Then

\

But

\

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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