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Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Stretch Theorem (Repeat Theorem)

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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 \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

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

But

$\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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Next: Downsampling Theorem (Aliasing Theorem)
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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