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Zero Padding Theorem (Spectral Interpolation)

A fundamental tool in practical spectrum analysis is zero padding. This theorem shows that zero padding in the time domain corresponds to ideal interpolation in the frequency domain (for time-limited signals):



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

$\displaystyle \zbox {\hbox{\sc ZeroPad}_{LN}(x) \;\longleftrightarrow\;\hbox{\sc Interp}_L(X)}
$

where $ \hbox{\sc ZeroPad}()$ was defined in Eq.$ \,$ (7.4), followed by the definition of $ \hbox{\sc Interp}()$ .



Proof: Let $ M=LN$ with $ L\geq 1$ . Then

\begin{eqnarray*}
\hbox{\sc DFT}_{M,k^\prime }(\hbox{\sc ZeroPad}_M(x))
&=& \sum_{m=0}^{M-1} x(m) e^{-j2\pi mk^\prime /M} \;\isdef \;\left<x,s_{\omega_{k^\prime }}\right>\\
&\isdef & X(\omega_{k^\prime }) = \hbox{\sc Interp}_{L,k^\prime }(X).
\end{eqnarray*}

Thus, this theorem follows directly from the definition of the ideal interpolation operator $ \hbox{\sc Interp}()$ . See §8.1.3 for an example of zero-padding in spectrum analysis.


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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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