Biomedical Engineering Reference
In-Depth Information
z
j
=
π
y
x
t >T
E
/2
(a)
t
(b)
T
E
f
L
f
(c)
FIGURE 16.40
pulse rotation causes a reversal in the rotation direction of the magnetization vectors so
they move toward realignment: (b) spin echo waveform versus time, and (c) spin echo spectral magnitude versus
frequency.
(a) A
p
To find the spectrum of the spin echo, we use the following transform pair
2
`
½
exp
ðj
t
jÞ ¼
ð
16
:
71a
Þ
2
1
þð
2p
f
Þ
along with the transform pair for the exponential (note in Eq. (16.70),
o ¼ o
L
¼
2
p
f
L
,a
constant, the electrical frequency corresponding to the physical Larmor frequency),
`
ð
e
jw
L
t
Þ¼dð
f
f
L
Þ
ð
16
:
71b
Þ
and the scaling theorem, Eq. (16.4a), to obtain the echo spectrum of Eq. (16.70)
!
(
)
0
xy
o
L
V
oxel
e
j
t
=
T
2
j
e
jw
L
t
2
0
jB
1
M
jB
1
M
xy
o
L
V
oxel
T
2
2
1
V
ð
f
Þ¼
`
¼
ð
16
:
71c
Þ
2
I
I
1
þ½
2
p
T
2
ð
f
f
L
Þ
and the magnitude of which is shown in Figure 16.39c. If a function
(
) is defined as
S
f
(
)
0
S
ð
f
Þ¼
jB
M
xy
o
L
V
T
1
1
oxel
2
ð
16
:
71d
Þ
2
I
2
1
þ½
2
p
T
f
2
then the right-hand side of Eq. (16.71c) becomes
V
ð
f
Þ¼
S
ð
f
f
n
Þ
ð
16
:
71e
Þ
