Biomedical Engineering Reference
In-Depth Information
When the inverse Fourier transform of the impulse function is taken, the result is an
exponential
1
Þ
e
i
2p
ft
df
¼
e
i
2p
f
0
t
g
ð
t
Þ¼
dð
f
f
ð
16
:
77b
Þ
0
1
which has the forward Fourier transform
1
1
e
i
2p
tf
dt
¼
e
i
2p
ft
0
e
i
2p
t
ð
f
f
0
Þ
dt
¼ dð
f
f
0
Þ
G
ð
f
Þ¼
ð
16
:
77c
Þ
1
1
If
w
(
t
) has the Fourier transform W(
f
), then
1
1
e
i
2p
ft
dt
¼
w
ð
t
Þ
e
i
2p
ft
0
w
ð
t
Þ
e
i
2p
t
ð
f
f
0
Þ
dt
¼
W
ð
f
f
0
Þ
ð
16
:
77d
Þ
1
1
dimension, phase encoding according
to Eq. (16.76) is applied and expressed with the constant phase term suppressed after
mixing,
The simplest place to start is the
y
-axis. For the
y
jð
y
Þ¼g
G
y
t
p
y
ð
16
:
78a
Þ
The phase associated with a particular location is
j
m
ð
y
m
Þ¼ðg
G
y
,
m
t
p
y
m
Þ
ð
16
:
78b
Þ
where the phase is indexed to a position
y
m
as well as a specific gradient slope coefficient,
Gy
,
. The encoded time signal can be described by
m
G
y
Þ¼
e
i
g
t
p
y
m
G
y
s
m
ð
0,
ð
16
:
78c
Þ
The Fourier transform pair of variables for the
y
dimension will be the varying amplitude
slope,
G
y
, as indicated in Figure 16.45 and a new variable called
u.
Analogous to the paired
variables
t
and
f
, which have reciprocal units,
u
will have the upside-down or inverse units
of
m
/
T
. These variables,
G
y
and
u
, can be used to construct a specific phase term that later
will be associated with
y
m
in a Fourier transform format:
1
1
e
i
2p
uG
y
dG
y
¼
e
i
2p
u
m
G
y
e
i
2pð
u
u
m
Þ
G
y
dG
y
¼ dð
u
u
m
Þ
S
m
ð
0,
u
Þ¼
ð
16
:
79
Þ
1
1
Here,
appears in the argument of the second exponent in the integrand. The overall argu-
ment is recognized as a phase according to Eqs. (16.77c) and (16.78a) so an explicit expres-
sion for
u
u
can be obtained by equating phases:
2
p
uG
y
¼g
t
p
yG
y
ð
16
:
80a
Þ
u
¼g
t
p
y
=ð
2
pÞ
ð
16
:
80b
Þ
