Maximum-likelihood estimation for nuclear magnetic resonance spectroscopy

Abstract

A new method for spectroscopy which relies on the application of maximum likelihood for the estimation of the parameters of the free-induction decay signal in noise is proposed, where the FID is composed of multiple sinusoids of unknown amplitudes, frequencies, phases, and exponential decays. An iterative expectation-maximization algorithm is introduced for maximizing the nonlinear likelihood function, which has the properties that (i) the iteration monotonically improves the parameter values and (ii) at each iteration the coupled maximization across the sinusoids is reduced to separate simple parallel maximizations making it ideal for parallel processor implementation. The major contribution of the new method is that at signal / noise ratios where the noise peaks at a particular frequency in the spectrum may be larger than the corresponding Larmor frequency components, the maximum-likelihood method provides low-variance, high-resolution estimates of the signal amplitudes, decays, and frequency parameters. The fundamental information measure generated by the new maximum-likelihood method is the application of an energy-transform criterion for the determination of frequency components.