Zero Padding Theorem (Spectral Interpolation)

Chapter Contents:

Search Mathematics of the DFT

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

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

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

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

Proof: Let with $L\geq 1$ . Then

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

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.

Order a Hardcopy of Mathematics of the DFT

Previous: Illustration of the Downsampling/Aliasing Theorem in Matlab
Next: Periodic Interpolation (Spectral Zero Padding)

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Chapters

Mathematics of the DFT
    Fourier Theorems for the DFT
       Fourier Theorems
          Zero Padding Theorem (Spectral Interpolation)