Zero Padding in the Time Domain

Unlike time-domain interpolation [270], ideal spectral interpolation is very easy to implement in practice by means of zero padding in the time domain. That is,

$\parbox{0.8\textwidth}{\emph{Zero padding} in the time domain corresponds to \emph{ideal interpolation} in the frequency domain.}$

Since the frequency axis (the unit circle in the $z$ plane) is finite in length, ideal interpolation can be implemented exactly to within numerical round-off error. This is quite different from ideal (band-limited) time-domain interpolation, in which the interpolation kernel is sinc$(\pi f_st)$ ; the sinc function extends to plus and minus infinity in time, so it can never be implemented exactly in practice.^3.9