1030 24 Magnetic Resonance Spectroscopy
The experiment consists of many repetitions of the pulse sequence with different
values oft 1. Let us assume that 1000 repetitions are carried out with values oft 1 ranging
from 0 to a convenient maximum value that allows a number of precession cycles
(1 microsecond would allow 200 precession cycles in a 200 MHz instrument). The FID
signal is recorded numerically as a discrete set of values. We assume that 1000 (10^3 )
data points are taken for each value oft 1. This gives us a total of 10^6 data points, each
one corresponding to a specific value oft 1 and a specific value oft 2.
The Fourier transform for each FID signal is carried out (integrating overt 2 ), giving
one spectrum as a function of a frequencyν 2 for each value oft 1. For some values oft 1
the FID signal contains a strong oscillation only forνA, whereas for other values oft 1
the signal contains a strong oscillation only forνB. For some values oft 1 there will be
no strong oscillation. Each spectrum will contain no more than one strong peak. Since
the calculation is carried out numerically each transform consists of a set of numerical
values, one for each of a set of values ofν 2. Let us assume that each transform consists of
1000 points (this value doesn’t have to equal the number of values oft 2 and an actual
experiment might use more points than this). We now have a set of 10^6 numerical
values, each one for a specific value ofν 2 and a specific value oft 1.
For a given value ofν 2 , we have 1000 values of the FID signal intensity at different
values oft 1. This is equivalent to a FID signal as a function oft 1. We now carry
out another set of Fourier transforms (integrating overt 1 ) and obtain 1000 spectra in
terms of a frequencyν 1 , with one spectrum for each value ofν 2. We have carried out
106 Fourier transformations. Before the development of rapid computers and the fast
Fourier transform algorithm, this would have required a great deal of computer time,
but can now be done fairly quickly.
The results can be represented by a three-dimensional graph, withν 1 plotted on
one axis andν 2 plotted on another. The strength of the absorption is plotted in a
third dimension. Figure 24.12 depicts schematically a perspective view of a three-
dimensional figure depicting the absorptions of protons A and B. The diagonal from
lower left to upper right in theν 1 −ν 2 plane represents equal values ofν 1 andν 2. Thev 2v 0 vAvBvAv^1Figure 24.12 Schematic Perspective View of the Three-Dimensional Graph Repre-
senting the COSY Spectrum of Two Uncoupled Protons.