Graphics Programs Reference
In-Depth Information
β
∗
β
∗
Oe
n
=
D
n
cos
a
x
++
vta
y
(
D
n
sin
+
h
)
a
z
(12.77)
1
2
N
d
;
D
n
=
-------------
+
n
=
0 …
N
,
1
(12.78)
The range between a scatterer
C
within the
k
th
range cell and the
n
th
element
of the real array is
r
2
t
εµ
ô
h
ô
h
ô
)
2
β
∗
(
,,,
;
)
=
D
2
++
v
2
t
2
(
+
2
D
n
sin
(
)
+
(12.79)
n
β
∗
h
tan
(
β
k
+
ε
)
h
[
tan
(
β
k
+
ε
)
2
D
n
cos
cos
µ
2
vt
sin
µ
]
It is more practical to use the scatterer's elevation and azimuth direction-
sines rather than the corresponding increments. Therefore, define the scat-
terer's azimuth and elevation direction-sines as
s
=
sin
µ
(12.80)
u
=
sin
ε
(12.81)
Then, one can rewrite Eq. (12.79) as
r
2
tsu
ô
h
ô
)
2
(
,,,
;
D
n
)
=
D
2
++
v
2
t
2
(
+
h
2
f
2
()
+
(12.82)
h
ô
β
∗
β
∗
2
D
n
sin
(
)
(
2
D
n
h
cos
f
()1
s
2
2
vhtf
()
s
)
f
()
=
tan
(
β
k
+
asin
u
)
(12.83)
Expanding as a third order Taylor series expansion about incremental
yields
r
n
tsu
ô
(
,,,
)
ô
2
2
s
2
----
rtsu
ô
ô
ô
r
ô
u
ô
u
-----
(
,,,
;
D
n
)
=
r
+++
r
u
u
+
+ + +
r
st
st
(12.84)
ô
ô
ss
r
ô
ô
ô
ô
3
ô
2
u
2
ô
u
2
2
t
2
----
u
2
2
r
ô
st
ô
st
-----
-----
--------
--------
r
tt
+
r
uu
+
+
r
ô
ô
u
+
+
r
ô
uu
+
6
ô
s
2
2
r
t
ô ô
t
ô
2
r
uss
us
2
r
suu
su
2
r
utt
ut
2
r
uuu
u
3
r
ô
ss
--------
+
--------
+
r
stu
stu
+
--------
+
-------
+
-------
+
-----
2
2
2
2
6
where subscripts denote partial derivations, and the over-bar indicates evalua-
tion at the state
tsu
ô
(
,,,
)
=
(
0000
,,,
)
. Note that
{
r
s
====== = = = =
=====
r
t
r
ô
s
r
ô
t
r
su
r
tu
r
ô ô
s
r
ô ô
t
r
ô
su
r
ô
tu
(12.85)
r
sss
r
sst
r
stt
r
ttt
r
tsu
0
}
Section 12.9.8 has detailed expressions of all non-zero Taylor series coeffi-
cients for the
k
th
range cell.
ô
mx
Even at the maximum increments
t
mx
,
s
mx
,
u
mx
,
, the terms:
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