296
Biomedical Signal and Image Processing
transverse component of the magnetic field, T 2 * . In the x–y plane, the magnetization
is expressed in Equation 15.9:
M = M 0 e
−
*
t T
/ 2
xy
(15.9)
The declining axial magnetic moment is expressed similarly, except with the time constant of the relaxation of the longitudinal field, T 1 . The difference between the steadystate magnetic field and the changing axial component is provided in Equation 15.10:
M 0 − M z = M 0 e
−t T
/ 1
(15.10)
In general, the following holds true: T 1 > T *
2 . This means that the x–y projection of
the magnetization decays faster than the difference.
Based on this information, it is clear that the FID contains all the information of
the NMR signal.
The RF pulse is repeated many times, which provides a platform for averaging. The
averaging process will increase the signal-to-noise ratio and will give a representation
that is relatively true to the process of what the signal should look like, since physiological effects are canceled out this way as well. The frequency spectrum of the MRI
signal will have two characteristics of importance: line width and chemical shift.
One particularly interesting spectrum arises from water molecules in the neighborhood of proteins or other large molecules, commonly found in biological media.
The water with large molecules gives a very broad frequency spectrum. In addition,
the precession of the magnetization vector in the y-direction can be written as the
result of two influences acting on it:
M = M e
−
*
t T
/ 2
.
y
0
(15 11)
and
1
1 gΔH
=
+
0
(15.12)
T *
2
T 2
2
The term 1/T 2 represents the dephasing of spins as a result of the local magnetic
fields produced by neighboring spin systems, and the term γΔH 0 /2 represents the
dephasing due to the localized inhomogeneous magnetic field across the biological
sample.
An exponentially decaying magnetization vector in the time domain will display
a single broad frequency band in the frequency domain. A damped harmonic oscillation in the time domain will have a spectrum of two broad peaks in the frequency
domain. This can be seen from the complex FT:
f t
( ) =
∫
F(w)e
−i t
w dw
(15.13)
Biomedical Signal and Image Processing
transverse component of the magnetic field, T 2 * . In the x–y plane, the magnetization
is expressed in Equation 15.9:
M = M 0 e
−
*
t T
/ 2
xy
(15.9)
The declining axial magnetic moment is expressed similarly, except with the time constant of the relaxation of the longitudinal field, T 1 . The difference between the steadystate magnetic field and the changing axial component is provided in Equation 15.10:
M 0 − M z = M 0 e
−t T
/ 1
(15.10)
In general, the following holds true: T 1 > T *
2 . This means that the x–y projection of
the magnetization decays faster than the difference.
Based on this information, it is clear that the FID contains all the information of
the NMR signal.
The RF pulse is repeated many times, which provides a platform for averaging. The
averaging process will increase the signal-to-noise ratio and will give a representation
that is relatively true to the process of what the signal should look like, since physiological effects are canceled out this way as well. The frequency spectrum of the MRI
signal will have two characteristics of importance: line width and chemical shift.
One particularly interesting spectrum arises from water molecules in the neighborhood of proteins or other large molecules, commonly found in biological media.
The water with large molecules gives a very broad frequency spectrum. In addition,
the precession of the magnetization vector in the y-direction can be written as the
result of two influences acting on it:
M = M e
−
*
t T
/ 2
.
y
0
(15 11)
and
1
1 gΔH
=
+
0
(15.12)
T *
2
T 2
2
The term 1/T 2 represents the dephasing of spins as a result of the local magnetic
fields produced by neighboring spin systems, and the term γΔH 0 /2 represents the
dephasing due to the localized inhomogeneous magnetic field across the biological
sample.
An exponentially decaying magnetization vector in the time domain will display
a single broad frequency band in the frequency domain. A damped harmonic oscillation in the time domain will have a spectrum of two broad peaks in the frequency
domain. This can be seen from the complex FT:
f t
( ) =
∫
F(w)e
−i t
w dw
(15.13)