On the extreme accuracy of maximum entropy spectrum estimation from an error-free autocorrelation function

The Maximum Entropy Method (MEM) is compared to the periodogram method (DFT) for the estimation of line spectra given an error-free autocorrelation function (ACF). In one computer simulation run, a 250 lag ACF was generated as the sum of 63 cosinusoids with given amplitudes, Ai, and wave numbers, fi. The wave numbers cover a band from 0 to 89.239 cm^-1with a Nyquist of 90 cm^-1and are given to an accuracy of six significant figures. The amplitudes range over four orders of magnitude and are given to five significant figures. The resulting normalized ACF was truncated to five significant figures. The resulting MEM spectrum was remarkably accurate. The lowest 53 spectral peaks reproduced the input frequencies exactly (to six significant figures) while the largest error in the remaining 10 lines occurred at line 63: input = 89.2814; MEM = 89.2934. The relative amplitude of this tiny line is only 2×10^-4and yet its frequency was recovered to one part in 9000.