### Interpolating a DFT

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.

**Next Section:**

Zero Padding in the Time Domain

**Previous Section:**

Ideal Spectral Interpolation