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Practical Zero Padding

To interpolate a uniformly sampled spectrum $ X(\omega_k)$ , $ k=0,1,2,\ldots,N-1$ by the factor $ L$ , we may take the length $ N$ inverse DFT, append $ N\cdot(L-1)$ zeros to the time-domain data, and take a length $ NL$ DFT. If $ NL$ is a power of two, then so is $ N$ and we can use a Cooley-Tukey FFT for both steps (which is very fast):

$\displaystyle X(\omega_l) = \hbox{\sc FFT}_{NL,l}(\hbox{\sc ZeroPad}_{LN}(\hbox{\sc IFFT}_N(X))),\quad l=0,\ldots,LN-1$ (3.45)

This operation creates $ L-1$ new bins between each pair of original bins in $ X$ , thus increasing the number of spectral samples around the unit circle from $ N$ to $ LN$ . An example for $ L=2$ is shown in Fig.2.4 (compare to Fig.2.3).

In matlab, we can specify zero-padding by simply providing the optional FFT-size argument:

X = fft(x,N);  % FFT size N > length(x)

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2018-02-13 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University