### 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*.^{7.9}In 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

*bandlimited interpolation*, as discussed further in Appendix D and in Book IV [70] of this series.

**Next Section:**

Interpolation Operator

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Zero Padding Applications