Biomedical Engineering Reference
In-Depth Information
Δ
f
F. T.
t
f
f
1
sinc(
ft)
Δ
1/(
Δ
f)
[(f-f
1
)/
Δ
f]
FIGURE 16.42
Sinc modulated excitation pulse with a selected Larmor center frequency and its Fourier trans-
form representing its spectrum or the shape of the applied field.
domain rf pulse excitation is shaped to obtain a flatter field. A sinc-shaped pulse would
have a flat rectangular spectral shape, a consequence of its Fourier transform relation,
but, ideally, it would have to be infinitely long. This pulse and the corresponding field-
spectral shape are shown in Figure 16.42. A sinc-time pulse has a Fourier transform,
, that
`
can be expressed as a rect function of width
D
f
:
`
½
sin
c
ðD
ft
Þ
exp
ð
j
2
p
f
t
Þ ¼ ð
1
=D
f
ÞP½ð
f
f
Þ=D
f
ð
16
:
73
Þ
1
1
As a compromise, a truncated or tapered or modified sinc pulse of finite length is applied
instead, such as the tapered shape shown in Figures 16.40b and 16.40c.
EXAMPLE PROBLEM 16.15
Find the bandwidth needed to achieve a
z
-slice thickness of 10 mm. Assume
G
z
¼
0.352
T
/
m
.
For
B
0
¼
1T, determine the frequencies needed to move 10 cm and 20 cm from the isocenter.
Solution
From a version of Eq.
g
0
,
(16.72b) with
the difference in frequency is proportional
to
thickness, or
D
f
¼
42
:
58
ð
MHz
=
T
Þ
0
:
352
ð
T
=
m
Þ:
010
m
¼
0
:
15 MHz
The same equation can be used with a value of the offset from the isocenter,
f
1
¼
42.58
(MHz) [1
T
þ
0.352 (
T
/
m
)
0.1
m
]
¼
44.08 MHz. Similarly,
f
2
¼
45.58 MHz corresponds
to the frequency needed to move 20 cm from the isocenter.
In order to scan the
-axis gradient control, as seen in
Figure 16.43, the second step is to locate data points along the axis. A gradient is also
applied along the
x
-
y
slice plane located by the
z
x
-axis for spatial localization,
o ¼ g
B
x
g
B
0
¼ g
G
x
x
ð
16
:
74a
Þ
Again, with mixing,
o
mix
¼ o
0
for
B
¼
B
0
, the difference frequency is proportional to the
position
x
,
o
D
¼ o o
¼ g
G
x
x
ð
16
:
74b
Þ
mix
slice selection, a specific frequency is applied that corresponds to a particular
desired location, according to Eq. (16.72b). Although a similar principle is used for the loca-
tions along
For
z
x
,
the implementation is somewhat different. Here,
the detected signal
