Maximum entropy for spectral analysis revisited

For many years the mathematical procedure termed maximum-entropy (M-E) has been used to nominally enhance resolution by sharpening (“whitening”) features in spectra, optical and otherwise. However, details of how it works, and why it fails its primary goal, which is to extrapolate trends in low-order Fourier coefficients into the white-noise region, remain unknown. To solve these problems we re-examine M-E derivations, obtaining a first-order analytic solution that exhibits both properties depending on assumptions made: a correct solution that indeed extrapolates trends, and an incorrect solution that follows from a particular assumption. In the latter case, the first-order solution allows the amount of sharpening to be calculated quantitatively. The results are significant because the correct solution can eliminate Gibbs oscillations that result when high-performance linear filtering is used to reduce noise, thereby allowing spectra to be reconstructed essentially without noise or distortion. Taken together with recently developed methods of eliminating endpoint-discontinuity artifacts, it is now possible to access underlying information essentially unimpeded, thereby changing the way spectra are processed. Examples are given.