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 truly time-limited signals):

Theorem: For any $x\in{\bf C}^N$

$$\zbox {\mbox{\sc ZeroPad}_{LN}(x) \leftrightarrow \mbox{\sc Interp}_L(X)}$$

where ZEROPAD$()$ was defined in Eq. (7.3), followed by the definition of INTERP$()$.

Proof: Let $M=LN$ with $L\geq 1$. Then

$$\begin{eqnarray*} \mbox{\sc DFT}_{M,k^\prime }(\mbox{\sc ZeroPad}_M(x)) &=& \su... ...ef & X(\omega_{k^\prime }) = \mbox{\sc Interp}_{L,k^\prime }(X). \end{eqnarray*}$$

Thus, this theorem follows directly from the definition of the ideal interpolation operator INTERP$()$. See §8.1.3 for an example of zero-padding in spectrum analysis.