Comparison of Fourier and Bayesian Analysis of NMR Signals. II. Examination of Truncated Free Induction Decay NMR Data

Abstract

In a recent report [J, J. Kotyk et al., J. Magn. Reson, 98, 483 (1992)], estimates of NMR frequency and amplitude parameters obtained from Bayesian probability theory were shown to be more precise and more accurate than those obtained from the discrete Fourier transform (DFT). This previous study examined the effects of varying signal-to-noise ratio and did not address the performance of either probability theory or the DFT as a function of acquisition time, i.e., truncation. Herein, a quantitative comparison between probability theory and the DFT is presented and discussed in terms of the accuracy and precision they provide in estimating the frequency and amplitude from truncated free induction decay data. For simplicity, data containing only a single frequency are examined. For frequency estimation, Bayesian probability theory gives either more precise and accurate estimates or exactly the same estimates as the DFT. This latter result only occurs when the theory indicates that the Bayesian procedure is functionally identical to the DFT. For amplitude estimation, the results presented herein are consistent with those cited in previous work. Even for the best DFT procedure, probability theory outperforms the DFT by a factor of two or more depending on the signal-to-noise ratio and the prior information supplied in the analysis. The amplitude estimates from probability theory are more precise and/or more accurate than the DFT results for all levels of truncation. Additionally, the only time the DFT amplitude estimates are more precise (less uncertain) are when they are inaccurate (give an incorrect answer). Thus, the use of probability theory offers significant improvement over the use of DFT in highly truncated NMR free induction decay data.