Using Fourier theorems, we will be able to show (§7.4.12) that
zero padding in the time domain gives exact bandlimited interpolation in
the frequency domain.7.10In other words, for truly time-limited signals
,
taking the DFT of the entire nonzero portion of
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.
Before we can interpolate a spectrum, we must be clear on what a
``spectrum'' really is. As discussed in Chapter 6, the
spectrum of a signal
at frequency
is
defined as a complex number
computed using the inner
product
That is,