Digital Signal Processing Reference
In-Depth Information
−
t
4. The exponentially decreasing term
e
)
in (
2.64
) describes the micro-level
behaviour of the individual magnetic moments (spin-spin interactions) at (
x
T
2
(
x
,
y
y
).
Thus the equation can also bewrittenwithmicro-level behaviour of the individual
magnetic moments as follows.
,
e
j
(
−
γ(
B
0
)τ
p
+
φ
n
(
t
,
x
,
y
))
e
j
(
−
γ(
B
0
+
G
y
y
)τ
y
)
M
xy
(
x
,
y
,
t
,
n
)
=
2
M
0
sin
(α
τ
p
/
2
)
n
n
e
j
(
−
γ(
B
0
+
G
x
x
)τ
x
)
e
j
(
−
γ(
B
0
t
))
.
×
where,
M
xy
(
x
,
y
,
t
,
n
)
is the
n
th micro-level magnetic moment which is the func-
tion of
x
,
y
and
t
.
5. Now apply the 180
◦
RF pulse. This is not same as that of the RF pulse. This helps
in changing the phase component of the transverse component from arbitrary
ρ
. Note that the selection gradient(
G
z
) is applied while applying 180
◦
to
−
ρ
pulse.
6. After applying 180
◦
pulse,
the resultant
transverse magnetic moment
is
given as
e
j
(
+
γ(
B
0
)τ
p
−
φ
n
(
t
,
x
,
y
))
e
j
(
+
γ(
B
0
+
G
y
y
)τ
y
)
M
xy
(
x
,
y
,
t
,
n
)
=
2
M
0
sin
(α
τ
p
/
2
)
n
n
e
j
(
+
γ(
B
0
+
G
x
x
)τ
x
)
e
j
(
−
γ(
B
0
t
))
.
×
7. The magnitude of the signal
M
xy
(
at every pixel corresponding to the
transverse component in step 5 is decreasing gradully with time (due to dephas-
ing). Hence the magnitude of
M
xy
(
x
,
y
,
t
)
corresponding to the transverse com-
ponent in step 6 increases with time (refer Sect.
3.2.2
for illustration) and
reaches maximum after some time duration. This is known as spin-echo.
Spin-echo guarantees the existence of required amplitude of MRI signal for
sampling.
8. Read-out phase starts immediate after some time (required time for rephas-
ing) after applying 180
◦
pulse along with the positive
x
-gradient
G
x
for the
duration of
x
,
y
,
t
)
τ
x
. The resultant transverse component during read-out phase is
given as
e
j
(
+
γ(
B
0
)τ
p
−
φ
n
(
x
,
y
,
t
))
e
j
(
+
γ(
B
0
+
G
y
y
)τ
y
)
M
xy
(
x
,
y
,
t
,
n
)
=
2
M
0
sin
(α
τ
p
/
2
)
n
n
e
j
(
+
γ(
B
0
+
G
x
x
)τ
x
)
e
j
(
−
γ((
B
0
+
G
x
x
)
t
))
.
×
9. After time duration of
τ
x
, there is the cancellation of the phase introduced due
to
G
x
gradient (upto step 8). This is known as Gradient echo. This helps to
synchronize the hardware and sample the magnitude of
s
(
t
)
at the end of the
reading phase 2
τ
x
corresponding to the particular location in the K-space. This
step is same as that of the one used in proton-density imaging using phase
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