Interpolating a DFT

Starting with a sampled spectrum $X(\omega_k)$, $k=0,1,\ldots,N-1$, typically obtained from a DFT, we can interpolate by taking the DTFT of the IDFT which is not periodically extended, but instead zero-padded:

$$\begin{eqnarray*} X(\omega) &=& \hbox{\sc DTFT}(\hbox{\sc ZeroPad}_{\infty}(\hbo... ...sc Sample}_N\{\hbox{\sc Shift}_{\omega}(\hbox{asinc}_N)\}\right> \end{eqnarray*}$$

(The aliased sinc function, $\hbox{asinc}_N(\omega)$, was defined in Eq.$\,$(1.4) and repeated in Eq.$\,$(1.7).) Thus, zero-padding in the time domain interpolates a spectrum consisting of $N$ samples around the unit circle by means of `` $\hbox{asinc}_N$ interpolation.'' This is ideal, time-limited interpolation in the frequency domain using the aliased sinc function as an interpolation kernel. We can almost rewrite the last line above as $X(\omega)=(X\ast \hbox{asinc}_N)_\omega$, but such an expression would normally be defined only for $\omega=\omega_l=2\pi l/N$, where $l$ is some integer.

Figure F.1 lists a matlab function for performing ideal spectral interpolation directly in the frequency domain. Such an approach is normally only used when non-uniform sampling of the frequency axis is needed.