Overcoming the Multiplex Disadvantage by Using Maximum-Likelihood Inversion

A maximum-likelihood estimator, derived under quantum-noise-limited measurement conditions, is used to obtain wavenumber-or-dered spectra produced by a model Michelson interferometer. The estimator is tested on a number of synthetic interferograms, and results are compared to similar spectra obtained by using the Fourier (cosine) transform. It is found that the maximum-likelihood inversion method does not result in white noise in the spectrum estimate when the spectrum is sparse. It thus may be used to circumvent the main disadvantage in multiplexed spectrometer measurements using quantum-noise-limited detectors for emission-based measurements. It is also found that maximum-likelihood inversion methods can be used to obtain spectrum estimates with greater peak-width resolution than those found by using Fourier transforms. The methods produce optical spectrum estimates that are relatively free of artifacts associated with the Fourier transform. The method is extended to include measurements with both quantum noise and additive white noise. Results obtained by using the quantum-noise-limited and normal noise maximum-likelihood spectrum estimation methods suggest that both the multiplex-disadvantage and the Gibbs phenomenon effects may be reduced by limiting the parameter space. The main problem with the maximum-likelihood method is the relatively long times required to obtain spectrum estimates.