Probability Theory As Logic Applied To Hypercomplex Two Dimensional Nuclear Magnetic Resonance Data

G. Larry Bretthorst
Washington University
Department of Chemistry
1 Brookings Drive
St. Louis, Missouri 63130


In two dimensional hypercomplex nuclear magnetic resonance data there are two time domains or precession periods in which the spins evolve. The spins are detected only in the second time domain. Their behavior in the first time domain is monitored by changes in the amplitude and phase of the resonances in the measured time domain. The measured free induction decay consists of measured pairs of numbers. The measured pairs are obtained by projecting the signal onto a cosine and sine respectively. Thus the measured pairs may be thought of as a pair of complex numbers. However, there is a second time domain and a second free induction decay may be obtained that is ninety degrees out of phase with the first free induction decay. Thus a single free induction decay may be thought of as a hypercomplex set of data, each element in the hypercomplex free induction decay consists of four numbers: the real-real, the real-imaginary, the imaginary-real, and the imaginary-imaginary. Thus at a fixed value of t_1 each hypercomplex free induction decay will consist of n_2 hypercomplex values. To monitor the spins in the second time domain t_1 is incremented and a new hypercomplex free induction decay is acquired. Thus there are a total of n_1 free induction decays. Each free induction decay having a total of n_2 hypercomplex data values. In this paper we report on the first tentative attempts to apply probability theory as logic to such data. The problem addressed is of estimating the value of an exponentially decaying hypercomplex two dimensional frequency. In one dimension, when this problem is addressed, there are two nuisance parameters: the amplitude and phase of the sinusoid. When these nuisance parameters are removed from the problem, probability theory leads one to a power spectrum as the sufficient statistic for estimating the frequency. However, in two dimensions the problem is more complex because there is an additional phase that must be dealt with as a nuisance parameter. Consequently, in two dimensional hypercomplex data the sufficient statistic is no longer just a power spectrum; rather there are two sufficient statistics --- a power spectrum, and a term that is best described as a cross correlation term. In addition to deriving the sufficient statistics probability theory as logic is used to derive an explicit estimate of the accuracy of the frequency estimates and these accuracy estimates explicitly show that both frequency estimates depend on the square root of the total number of data values.
MaxEnt 94 Abstracts /