Biomedical Engineering Reference
In-Depth Information
This evaluation of the Fourier transform of phase encoding follows directly from the dis-
cussion of the impulse function. As a consequence of this change in variables, the spectrum
S
m
in Eq. (16.79) can be restated in terms of the scaled variable from Eq. 16.80b,
S
m
ð
0,
u
Þ¼dð
u
u
m
Þ¼d½ðg
t
p
=ð
2
pÞÞð
y
y
m
Þ
ð
16
:
81a
Þ
S
m
ð
0,
y
Þ¼½
2
p=g
t
p
Þdð
y
y
m
Þ
ð
16
:
81b
Þ
where use has been made of the property
from Section 16.2.
The derivation of this key equation reveals an important principle in magnetic resonance
imaging. First, a local value of a gradient slope,
d
(
au
)
¼ d
(
u
)/
j
a
j
y
m
, into a phase
according to Eq. (16.78b). Second, this phase, in terms of Fourier transforms, ensures that
the signal function has a spectrum centered on the value of
G
y
,
m
, encodes a position,
associated with the location
of the corresponding net magnetization. Through simple scaling, the scaled spectra are
translated into spatial locations along the
u
-axis (Figure 16.48a). This process maps a scaled
spectral magnitude into its locations along the
y
-axis.
To formulate the Fourier transform relation for the
y
x
-axis, a single spin echo waveform is
generated, for example, at the position
y
¼
0inFigure 16.40b, so that
G
y
¼
0. From
Eq. (16.70), the signal waveform, delayed by time
T
E
and decaying with a unique time con-
stant
T
2
n
associated with location
x
n
can be expressed as
e
jð
t
T
E
Þ=
T
2
n
j
e
j
o
n
ð
t
T
E
Þ
s
n
ð
t
,0
Þ¼
2
S
ð
16
:
82
Þ
0
|S
m
|
u
m
u
y
y
m
(a)
|S
n
|
f
n
f
x
n
x
(b)
f
n
f
u
y
u
m
y
m
x
n
x
(c)
FIGURE 16.48
(a) Spectrally phase-encoded magnitudes plotted versus multiple scaled
y
- and
u
-axes. (b) Spec-
tral frequency-encoded magnitudes plotted versus multiple scaled
x
- and
f
-axes. (c) Resulting pattern in
xy
or
uf
plane. Scale exaggerated for clarity.
