A fundamental tool in practical spectrum analysis is zero padding. This theorem shows that zero padding in the time domain corresponds to ideal interpolation in the frequency domain (for time-limited signals):
Theorem: For any
where was defined in Eq. (7.4), followed by the definition of .
Proof: Let
with
. Then
Thus, this theorem follows directly from the definition of the ideal interpolation operator . See §8.1.3 for an example of zero-padding in spectrum analysis.