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Periodic Interpolation (Spectral Zero Padding)

The dual of the zero-padding theorem states formally that zero padding in the frequency domain corresponds to periodic interpolation in the time domain:



Definition: For all $ x\in{\bf C}^N$ and any integer $ L\geq 1$ ,

$\displaystyle \zbox {\hbox{\sc PerInterp}_L(x) \isdef \hbox{\sc IDFT}(\hbox{\sc ZeroPad}_{LN}(X))} \protect$ (7.7)

where zero padding is defined in §7.2.7 and illustrated in Figure 7.7. In other words, zero-padding a DFT by the factor $ L$ in the frequency domain (by inserting $ N(L-1)$ zeros at bin number $ k=N/2$ corresponding to the folding frequency7.22) gives rise to ``periodic interpolation'' by the factor $ L$ in the time domain. It is straightforward to show that the interpolation kernel used in periodic interpolation is an aliased sinc function, that is, a sinc function $ \sin(\pi n/L)/(\pi n/L)$ that has been time-aliased on a block of length $ NL$ . Such an aliased sinc function is of course periodic with period $ NL$ samples. See Appendix D for a discussion of ideal bandlimited interpolation, in which the interpolating sinc function is not aliased.

Periodic interpolation is ideal for signals that are periodic in $ N$ samples, where $ N$ is the DFT length. For non-periodic signals, which is almost always the case in practice, bandlimited interpolation should be used instead (Appendix D).



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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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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