Maximum-Entropy Revisited

TL;DR

This paper revisits the maximum-entropy method, clarifies its theoretical basis, and demonstrates its effectiveness in spectral sharpening and noise filtering, especially for Lorentzian lineshapes, outperforming linear methods.

Contribution

It provides a theoretical correction to the maximum-entropy approach, showing it extends low-index Fourier coefficients correctly and offers superior noise filtering capabilities.

Findings

Maximum-entropy extension of low-index Fourier coefficients is theoretically justified.

Analytical solution for Lorentzian lineshape as AR(1) model.

Maximum-entropy noise filtering outperforms linear methods.

Abstract

For over five decades the procedure termed maximum-entropy (M-E) has been used to sharpen structure in spectra, optical and otherwise. However, this is a contradiction: by modifying data, this approach violates the fundamental M-E principle, which is to extend, in a model-independent way, trends established by low-index Fourier coefficients into the white-noise region. The Burg derivation, and indirectly the prediction-error equations on which sharpening is based, both lead to the correct solution, although this has been consistently overlooked. For a single Lorentzian line these equations can be solved analytically. The resultant lineshape is an exact autoregressive model-1 (AR(1)) replica of the original, demonstrating how the M-E reconstruction extends low-index Fourier coefficients to the digital limit and illustrating why this approach works so well for lineshapes resulting from first-order decay processes. By simultaneously retaining low-index coefficients exactly and eliminating Gibbs oscillations, M-E noise filtering is quantitatively superior to that achieved by any linear method, including the high-performance filters recently proposed. Examples are provided.