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ô
i
defined by
(
a
i
,
φ
i
,
s
i
,
u
i
,
)
. The returned signal
a
t the
n
th
element due to fir-
ing from the
n
th
element and reflection from the
C
i
scatterer is
s
i
n
1
n
2
t
mn
1
(
,
;
)
=
a
i
G
2
(
sin
β
i
) ρ
k
t
()ρ
t
()
(
⁄
)
4
(12.91)
cos
[
2π
f
o
(
t
mn
1
τ
n
1
n
2
)
+
ζ
φ
i
]
where represents the two-way antenna gain, and the term
denotes the range attenuation at the
G
2
(
ρ
k
t
()ρ
t
()
⁄
)
4
k
th
range cell. The analysis in this paper
will assume hereon that
ζ
and
φ
i
are both equal to zeroes.
Suppose that there are
N
o
scatterers within the
k
th
range cell, with angular
locations given by
ô
i
{
(
a
i
φ
i
,
,
s
,
u
i
,
)
i
;
=
1 …
N
o
,
}
(12.92)
i
The composite r
e
turned signal at time
t
mn
1
within this range cell due to the
path
(
n
1
⇒⇒
all C
i
n
2
)
is
N
o
∑
sn
1
(
,
n
2
;
t
mn
1
)
=
s
i
(
n
1
,
n
2
;
t
mn
1
)
(12.93)
i
=
1
The platform motion synthesizes a rectangular array of size , where
only one column of elements exists at a time. However, if and the
real array is operated in the sequential mode, a square planar array of size
is synthesized. The element spacing along the flight path is
.
NM
×
N
M
=
2
N
2
N
×
2
N
d
y
=
vD
ob
⁄
M
Consider the range bin. The corresponding two-dimensional information
sequence consists of similar vectors.
The vector represents the returns due to the sequential firing of all ele-
ments during the subinterval. Each vector has rows, and it is
extended, by adding zeroes, to the next power of two. For example, consider
the
k
th
{
b
k
(
nm
,
)
;
(
nm
,
)
=
0 …2
N
,
2
}
2
N
m
th
N
m
th
(
2
N
1
)
m
th
subinterval, and let
M
==
2
N
4
.
Then, the elements of the extended
column
{
b
k
(
nm
,
)
}
are
{
b
k
(
0
m
,
)
b
k
,
(
1
m
,
)
b
k
,
(
2
m
,
)
b
k
,
(
3
m
,
)
b
k
,
(
4
m
,
)
b
k
,
(
5
m
,
)
,
(12.94)
b
k
(
6
m
,
)
b
k
,
(
7
m
,
)
}
=
{
s
00
t
mn
0
(
,
;
)
s
01
t
mn
0
,
(
,
;
)
+
s
10
t
mn
1
(
,
;
)
,
s
02
t
mn
0
(
,
;
)
+
s
11
t
mn
1
(
,
;
)
+
s
20
t
mn
2
(
,
;
)
s
03
t
mn
0
,
(
,
;
)
+
s
12
t
mn
1
(
,
;
)
+
s
21
t
mn
2
(
,
;
)
+
s
30
t
mn
3
(
,
;
)
s
13
t
mn
1
,
(
,
;
)
+
s
22
t
mn
2
(
,
;
)
+
s
31
t
mn
3
(
,
;
)
s
23
t
mn
2
,
(
,
;
)
+
s
32
t
mn
3
(
,
;
)
s
33
t
mn
3
,
(
,
;
) 0
,
}
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