Digital Signal Processing Reference
In-Depth Information
Fig. 8.15
Stationary phase conditions (Eqs.
8.28
and
8.29
) apply at point A (
q
k
n
A
), but they do
not apply at point B (
q
6k
n
B
) (Figure modified from
Cardellach
(
2002
))
@Q
@x
0
@
z
0
@x
0
D
q
x
C
q
z
D
0
(8.26)
D
q
y
C
q
z
@
z
0
@y
0
@Q
@y
0
D
0
(8.27)
which leads to the relationships
@
z
0
@x
0
q
x
q
z
D
(8.28)
@
z
0
@y
0
q
y
q
z
D
(8.29)
The above conditions only hold when the local incident and the scattered angles are
the same:
q
k
n
, local specular reflection, as illustrated in Fig.
8.15
.
Under the stationary phase limit, it can be shown that (
Ulaby et al. 1982
, p. 930
and Appendix 12A) the received fields are
D
MŒR
k
.
h
s
k
i
/.
h
k
s
/
C
R
?
.
v
s
k
i
/.
v
k
s
/
E
hh
(8.30)
D
MŒR
k
.
v
s
k
i
/.
h
k
s
/
R
?
.
h
s
k
i
/.
v
k
s
/
E
v
h
(8.31)
D
MŒR
k
.
v
s
k
i
/.
v
k
s
/
C
R
?
.
h
s
k
i
/.
h
k
s
/
E
vv
(8.32)
D
MŒR
k
.
h
s
k
i
/.
v
k
s
/
R
?
.
v
s
k
i
/.
h
k
s
/
E
h
v
(8.33)
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