Comparison of Fourier and Bayesian analysis of nmr signals. I. Well-separated resonances (the single-frequency case)

Analysis of nuclear magnetic resonance data requires an estimation of the parameters (e.g., frequencies and amplitudes) that characterize the NMR free-induction-decay data. Bayesian probability theory provides a rigorous formalism for optimal parameter estimation and use of prior information. Although specific algorithms for time-efficient implementation of Bayesian methods have been presented (G. L. Bretthorst, J. Magn. Reson. 88, 533, 552, 571, 1990; G. L. Bretthorst, J. Magn. Reson., in press), acceptance of such methods requires demonstration of substantial improvements in the accuracy of parameter estimates. Toward this end, we report a comparison between the discrete Fourier transform (DFT) method and Bayesian analysis for estimating the signal frequency and amplitude of a single-frequency NMR resonance as a function of the signal-to-noise ( S N) ratio of the FID data. The results and methods are also applicable to data composed of multiple, well-separated frequency components. Parameter estimates are made both with and without prior knowledge of the decay rate and/or phase of the NMR signal. Under conditions where prior information about the signal phase and decay rate constant is not available, Bayesian analysis provides more precise estimates of the signal frequency and continues to do so considerably after the DFT method fails due to poor S N levels. In accordance with theory, the Bayesian and DFT methods yield identical frequency estimates when the DFT estimates are obtained from a zero-padded absorption spectrum when prior information about both the decay rate constant (i.e., matched exponential filter) and the signal phase is available. In all cases, Bayesian analysis is substantially more precise than the DFT method for estimating signal amplitudes. Reasons for the differences observed between the two analysis techniques are discussed in detail. At this level of validation, Bayesian analysis offers distinct advantages over DFT procedures for NMR parameter estimation.