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Relation to Stretch Theorem

It is instructive to interpret the periodic interpolation theorem in terms of the stretch theorem, $ \hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X)$. To do this, it is convenient to define a ``zero-centered rectangular window'' operator:

Definition: For any $ X\in{\bf C}^N$ and any odd integer $ M<N$ we define the length $ M$ even rectangular windowing operation by

$\displaystyle \hbox{\sc Chop}_{M,k}(X) \isdef
X(k), &...
...{M+1}{2} \leq \left\vert k\right\vert \leq \frac{N}{2}. \\

Thus, this ``zero-phase rectangular window,'' when applied to a spectrum $ X$, sets the spectrum to zero everywhere outside a zero-centered interval of $ M$ samples. Note that $ \hbox{\sc Chop}_M(X)$ is the ideal lowpass filtering operation in the frequency domain. The ``cut-off frequency'' is $ \omega_c = 2\pi[(M-1)/2]/N$ radians per sample. For even $ M$, we allow $ X(-M/2)$ to be ``passed'' by the window, but in our usage (below), this sample should always be zero anyway. With this notation defined we can efficiently restate periodic interpolation in terms of the $ \hbox{\sc Stretch}()$ operator:

Theorem: When $ x\in{\bf C}^N$ consists of one or more periods from a periodic signal $ x^\prime\in{\bf C}^\infty$,

$\displaystyle \zbox {\hbox{\sc PerInterp}_L(x) = \hbox{\sc IDFT}(\hbox{\sc Chop}_N(\hbox{\sc DFT}(\hbox{\sc Stretch}_L(x)))).}

In other words, ideal periodic interpolation of one period of $ x$ by the integer factor $ L$ may be carried out by first stretching $ x$ by the factor $ L$ (inserting $ L-1$ zeros between adjacent samples of $ x$), taking the DFT, applying the ideal lowpass filter as an $ N$-point rectangular window in the frequency domain, and performing the inverse DFT.

Proof: First, recall that $ \hbox{\sc Stretch}_L(x) \;\longleftrightarrow\;\hbox{\sc Repeat}_L(X)$. That is, stretching a signal by the factor $ L$ gives a new signal $ y=\hbox{\sc Stretch}_L(x)$ which has a spectrum $ Y$ consisting of $ L$ copies of $ X$ repeated around the unit circle. The ``baseband copy'' of $ X$ in $ Y$ can be defined as the $ N$-sample sequence centered about frequency zero. Therefore, we can use an ``ideal filter'' to ``pass'' the baseband spectral copy and zero out all others, thereby converting $ \hbox{\sc Repeat}_L(X)$ to $ \hbox{\sc ZeroPad}_{LN}(X)$. I.e.,

$\displaystyle \hbox{\sc Chop}_N(\hbox{\sc Repeat}_L(X)) = \hbox{\sc ZeroPad}_{LN}(X)
\;\longleftrightarrow\;\hbox{\sc Interp}_L(x).

The last step follows by definition of ideal periodic interpolation in Eq. (7.9).

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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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