Maximum-entropy revisited: Optimal filtering of spectra

Abstract

In spectroscopy, the objective is to obtain information by analyzing spectra that ideally are undistorted and noise-free. In standard Fourier-space filtering, this goal cannot be achieved because of apodization, which forces a trade-off among errors arising from distortion, noise leakage, and Gibbs oscillations. We show that low-order coefficients can be preserved and apodization, and its associated errors eliminated with the corrected maximum-entropy (M-E) filter obtained here. Although the Burg derivation begins as M-E, by making certain assumptions the Burg approach yields a procedure that deconvolves (sharpens) structure in spectra, thereby violating the basic M-E principle of leaving the low-order coefficients intact. The corrected solution preserves these data and projects the trends established by them into the white-noise region in a model-independent way, thereby eliminating apodization and its associated errors. For a single Lorentzian line, the corrected M-E approach has an exact analytic solution, which reveals not only how M-E performs its extension but also why it works particularly well for line shapes resulting from first-order decay processes. The corrected M-E filter is quantitatively superior to any previous filtering method, including recently proposed high-performance linear filters, yet requires only minimal computational effort. Examples, including multiple differentiation, are provided.