### Zero Padding Theorem (Spectral Interpolation)

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

*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.

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Periodic Interpolation (Spectral Zero Padding)

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Downsampling Theorem (Aliasing Theorem)