Ideal Spectral Interpolation

Using Fourier theorems, we will be able to show (§7.4.12) that zero padding in the time domain gives ideal bandlimited interpolation in the frequency domain.^7.4 In other words, for truly time-limited signals $x$, taking the DFT of the entire nonzero portion of $x$ extended by zeros yields exact interpolation of the complex spectrum--not an approximation (ignoring computational round-off error in the DFT itself). Because the fast Fourier transform (FFT) is so efficient, zero-padding followed by an FFT is a highly practical method for interpolating spectra of finite-duration signals, and is used extensively in practice.