Starting with a sampled spectrum
,
,
typically obtained from a DFT, we can interpolate by taking the DTFT
of the IDFT which is not periodically extended, but instead
zero-padded [264]:3.8
(The aliased sinc function,
, is derived in
§3.1.)
Thus, zero-padding in the time domain interpolates a spectrum
consisting of
samples around the unit circle by means of ``
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
,
but such an expression would normally be defined only for
, where
is some integer, since
is
discrete while
is continuous.
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. For uniform spectral upsampling, it is more typical to take an inverse FFT, zero pad, then a longer FFT, as discussed further in the next section.