The procedure for an interpolation can be summarized as follows:

- 1.
- I pad zeros in the time domain with the same number of original time samples, and Fourier transform over time to .
- 2.
- For each frequency from zero to half Nyquist, I find
a minimum phase wavelet whose inverse spectrum is the spectrum of
the signal along the
*x*axis. Here, I used Kolmogoroff spectral factorization for finding minimum phase filters. - 3.
- For each frequency, I find missing data by minimizing the filtered output in the least-squares sense using a conjugate-gradient algorithm. The filter applied is the wavelet which comes from the half frequency.

Another approach for finding filters is to obtain a prediction-error filter for each frequency, because the prediction-error filter has a spectrum that is inverse to the input (Claerbout, 1991). If you want to limit the length of filter, the prediction-error filter is more attractive than spectral factorization.

12/18/1997