### 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

**Previous Section:**

Zero Padding Applications