Graphics Programs Reference
In-Depth Information
where
k
=
2πλ
⁄
is the wave-number, and the operator
(
•
)
indicates dot
product. Therefore, the two-way geometric phase between the
(
n
x
1
,
n
y
1
)
and
(
n
x
2
,
n
y
2
)
elements is
ϕ
n
x
1
(
,
n
y
1
,
n
x
2
,
n
y
2
)
=
ku
[
•
{
en
x
1
(
,
n
y
1
)
+
en
x
2
(
,
n
y
2
)
}
]
(12.108)
The cumulative two-way normalized electric field due to all transmissions is
E
()
E
t
=
()
E
r
()
(12.109)
where the subscripts and , respectively, refer to the transmitted and
received electric fields. More precisely,
t
r
N
∑
1
N
∑
1
E
t
()
=
wn
xt
(
,
n
yt
)
exp jk u
[
{
•
e n
xt
(
,
n
yt
)
}
]
(12.110)
n
xt
=
0
n
yt
=
0
N
∑
1
N
∑
1
E
r
()
=
wn
xr
(
,
n
yr
)
exp jk u
[
{
•
e n
xr
(
,
n
yr
)
}
]
(12.111)
n
xr
=
0
n
yr
=
0
In this case, denotes the tapering sequence. Substituting Eqs.
(12.108), (12.110), and (12.111) into Eq. (12.109) and grouping all fields with
the same two-way geometric phase yields
wn
x
(
,
n
y
)
N
a
∑
1
N
a
∑
1
E
()
e
j
δ
=
w
'
mn
(
,
)
exp jkd
[
sin
β
(
m
cos
α
+
n
sin
α
)
]
(12.112)
m
=
0
n
=
0
N
a
=
2
N
1
(12.113)
m
=
+
n
xr
;
=
0 …2
N
,
2
(12.114)
xt
n
=
+
n
yr
;
=
0 …2
N
,
2
(12.115)
yt
d
sin
β
δ
=
-----------------
(
N
1
) α
(
cos
+
sin
α
)
(12.116)
2
The two-way array pattern is then computed as
N
a
∑
1
N
a
∑
1
E
()
=
w
'
mn
(
,
)
exp jkd
[
sin
β
(
m
cos
α
+
n
sin
α
)
]
(12.117)
m
=
0
n
=
0
Consider the two-dimensional DFT transform,
W
'
pq
(
,
)
, of the array
w
'
n
x
(
,
n
y
)
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