Spectral Interpolation

The need for spectral interpolation comes up in many situations. For example, we always use the DFT in practice, while conceptually we often prefer the DTFT. For time-limited signals, that is, signals which are zero outside some finite range, the DTFT can be computed from the DFT via spectral interpolation.^3.5 Another application of DFT interpolation is spectral peak estimation; in this situation, we obtain a sampled spectral peak from a DFT, and interpolation is used to estimate the frequency of the peak more accurately than what is obtained by rounding to the nearest DFT bin frequency.

In this and the following section, we will discuss two types of spectral interpolation:

• Ideal spectral interpolation (zero-padding in the time domain)
• Parabolic interpolation (fitting a parabola at the sampled peak)

When these methods are used together, we have what we call the quadratically interpolated FFT (QIFFT) method [238,1]. The QIFFT method can be considered an approximate maximum likelihood method for spectral peak estimation, as we will see.