The dual of the zero-padding theorem states formally that
zero padding in the frequency domain corresponds to periodic
interpolation in the time domain:

Definition: For all
and any integer ,

(7.9)

where zero padding is defined in §7.2.7 and illustrated in
Figure 7.7. In other words, zero-padding a DFT by the factor in
the frequency domain
(by inserting zeros at bin number corresponding to
the folding frequency^{7.14})
gives rise to ``periodic interpolation'' by the factor in the time
domain. It is straightforward to show that the interpolation kernel
used in periodic interpolation is a sinc function proportional to
which has been time-aliased on a block of
length . Such an ``aliased sinc function'' is of course periodic
with period samples. See Appendix D for a discussion of
idealbandlimited interpolation, in which the interpolating sinc
function is not aliased.

Periodic interpolation is ideal, however, for signals which are
periodic in samples, where is the DFT length.