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 $x\in\mathbb{C}^N$

$$\zbox {\hbox{\sc ZeroPad}_{LN}(x) \;\longleftrightarrow\;\hbox{\sc Interp}_L(X)}$$

where $\hbox{\sc ZeroPad}()$ was defined in Eq.(7.4), followed by the definition of $\hbox{\sc Interp}()$ .

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

$$\begin{eqnarray*} \hbox{\sc DFT}_{M,k^\prime }(\hbox{\sc ZeroPad}_M(x)) &=& \sum_{m=0}^{M-1} x(m) e^{-j2\pi mk^\prime /M} \;\isdef \;\left<x,s_{\omega_{k^\prime }}\right>\\ &\isdef & X(\omega_{k^\prime }) = \hbox{\sc Interp}_{L,k^\prime }(X). \end{eqnarray*}$$

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