of Peter Nagy

This
animated figure demonstrates what kind of principles help us locate the
signal in the selected slice along the X and Y axes using a specific,
calculated example. The animation merely demonstrates how the combined
application of a phase- and a frequency-encoding gradient allows
determining the source of the signal, since the applied mathematical
approach is completely different from and much simpler than the one used
in clinical MRI devices.

There are four pixels in the selected slice, designated
by A-D. The length of the vectors is also displayed (A=1; B=0.9; C=0.8;
D=0.7). Similar to the previous figure, the vectors represent the
macroscopic magnetization rotated to the horizontal plane. Only the
precession of these vectors is simulated, and their relaxation is
neglected. During the animation the plot above “NMR signal” displays the **oscillating NMR signal**, i.e. **the projection of the sum of macroscopic magnetization vectors on one of the horizontal axes (X or Y)**.
Alternatively, the NMR signal can also be interpreted as the current
induced in a coil in the X-Y plane. The plots above this part display
the Fourier transforms of the NMR signal. Fourier transformation is a
mathematical procedure that **determines the contribution of different frequency components to an oscillating signal**.
The eight Fourier transforms calculated in the eight steps of the
simulation are displayed in fields FT1-FT8. It is worth noticing that if
the NMR signal is a single-component sine wave, then the Fourier
transform only contains a single peak. If, on the other hand, the NMR
signal is composed of two different frequency sine waves, the Fourier
transform also contains two peaks.

First, spins precess with identical frequency in a
homogeneous magnetic field, and consequently only a single peak appears
in the Fourier transform (1). Then, the phase-encoding gradient is
turned on, and **the spins in pixels on the left and right exhibit different precession frequencies**. Consequently, **there are two peaks in the Fourier transform** (2). After turning off the phase-encoding gradient, the frequencies are
again identical for every pixel, but a phase difference persists (3). **The phase difference generated between pixels on the left and right is determined and displayed (delta phi1)**.
The Fourier transform again contains only one peak in accordance with
identical precession frequencies for all pixels. Afterwards, the **frequency-encoding gradient is turned on along the vertical direction, and spins precess with two different frequencies,** and there are two peaks in the Fourier transform (4). **The two peaks in the Fourier transform are measured and displayed (P1, P2)**.
Peaks P1 and P2 are determined by variables A-B and C-D, respectively,
and by the phase difference according to the equations displayed in the
lower right corner. **The phase-encoding – frequency-encoding pair must be repeated once more** (5-8) so that four independent equations can be written for the four unknowns **(A-D). At the end of the animation the program calculates A-D by solving the equation set consisting of four equations.** The solutions for A-D are identical to the assumed values of the variables.