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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
\left\{\begin{array}{ll}
X(k), & -\frac{M-1}{2}\leq k \leq
\frac{M-1}{2} \\ [5pt]
0, & \frac{M+1}{2} \leq \left\vert k\right\vert \leq \frac{N}{2}. \\
\end{array} \right.
$

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)\leftrightarrow \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 is provided by the zero-padding theorem7.4.12).


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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