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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\mathbb{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).

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\mathbb{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), & -\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\mathbb{C}^N$ consists of one or more periods from a periodic signal $ x^\prime\in \mathbb{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 theorem (§7.4.12).

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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 © 2016-05-31 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University