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):

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