Digital Signal Processing Reference
In-Depth Information
of the 3-D coordinate
r
that can be manipulated spatially by external
radio-frequency excitations and magnetic fields.
At a given voxel, the value of an MR image is characterized by two
important factors: the tissue properties and the scanner imaging proto-
col. The most relevant tissue properties are the relaxation parameters
T
1
and
T
2
and the proton density. The proton density is defined as the
number of targeted nuclei per unit volume. The scanner software and
hardware manipulate the magnetization vector
M
over time and space
based on the so-called pulse sequence.
In the following text, we will focus on a particular voxel and give the
equations of motion for
M
(
t
) as a function of time
t
. These equations
are based on the Bloch equations and describe a precession of the
magnetization vector around the external applied magnetic field with
a frequency
ω
0
, which is known as the resonance or Larmor frequency.
The magnetization vector
M
(
t
) has two components:
1. The longitudinal magnetization given by
M
z
(
t
), the
z
-component of
M
(
t
)
2. The transverse magnetization vector
M
xy
(
t
), a complex quantity, which
combines two orthogonal components:
M
xy
(
t
)=
M
x
(
t
)+
jM
y
(
t
)
(1.10)
where
ϕ
is the angle of the complex number
M
xy
, known as the phase
angle, given as
ϕ
=tan
−1
M
x
M
y
(1.11)
Since
M
(
t
) is a magnetic moment, it will have a torque if an external
time-varying magnetic field
B
(
t
) is applied. If this field is static and
oriented parallel to the
z
-direction, then
B
(
t
)=
B
0
.
The magnetization vector
M
precesses if it is initially oriented away
from the
B
0
. The spin system can also be excited by using RF signals,
such that RF signals are produced as output by the stimulated system.
This RF excitation is achieved by applying
B
1
at the Larmor frequency
rather than keeping it constant, and allows tracking the position of
M
(
t
).
However, the precession is not perpetual, and we will show that there
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