Periodic Interpolation (Spectral Zero Padding)

The dual of the zero-padding theorem states formally that zero padding in the frequency domain corresponds to periodic interpolation in the time domain:

Definition: For all $x\in{\bf C}^N$ and any integer $L\geq 1$,

$$\zbox {\hbox{\sc PerInterp}_L(x) \isdef \hbox{\sc IDFT}(\hbox{\sc ZeroPad}_{LN}(X))} \protect$$ (7.6)

where zero padding is defined in §7.2.6 and illustrated in Figure 7.5. In other words, zero-padding a DFT by the factor $L$ in the frequency domain (by inserting $N(L-1)$ zeros at bin number $k=N/2$ corresponding to the folding frequency^7.12) gives rise to ``periodic interpolation'' by the factor $L$ in the time domain. It is straightforward to show that the interpolation kernel used in periodic interpolation is a sinc function proportional to $\sin(\pi n/L)/(\pi n/L)$ which has been time-aliased on a block of length $NL$. Such an ``aliased sinc function'' is of course periodic with period $NL$ samples. See Appendix G for a discussion of ideal bandlimited interpolation, in which the interpolating sinc function is not aliased.

Periodic interpolation is ideal, however, for signals which are periodic in $N$ samples, where $N$ is the DFT length.