# 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

### Abstract

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 / mas@mrao.cam.ac.uk