Ideal Spectral Interpolation
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.
7.9In 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,

is the unnormalized
coefficient of projection of

onto the
sinusoid 
at frequency

. When

, for

, we obtain the
special set of spectral samples known as the DFT. For other values of

, we obtain spectral points in between the DFT samples.
Interpolating DFT samples should give the same result. It is
straightforward to show that this ideal form of interpolation is what
we call
bandlimited interpolation, as discussed further in
Appendix
D and in Book IV [
70] of this series.
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