Digital Signal Processing Reference
In-Depth Information
i
the integration time;
r
0
any point on the surface where to integrate the
functions; the delay at which the correlation function is being evaluated; and
f
c
the central correlation frequency;
R.
a
;
b
/ the distance between points
a
and
b
;
.
r
0
/ is the delay of the ray-path from the transmitter to the surface point
r
0
and
from there to the receiver; and f
D
.
r
0
/ its Doppler frequency;
0
is the bi-static scattering coefficient, defined as the fraction of incident power
that can be scattered into certain direction and polarization state
pq
, normalized
by the incident power density and area.
Note that other sources of power attenuation or loss might also be intro-
duced, such as atmospheric attenuation; cabling/instrumental loss; quantification
(number of bit sampling) loss.... All these factors would simply multiply the
right-hand side of Eq.
8.49
The bi-static scattering coefficient for KGO is (e.g.
Ulaby et al. 1982
;
Zavorotny
and Voronovich 2000
;
Cardellach 2002
)
2
q
2
q
z
pq
D
k
2
j
R
pq
j
PDF.Z
x
;Z
y
/
(8.50)
where k is the electromagnetic wavenumber; R
pq
the scattering coefficients; and
PDF.Z
x
;Z
y
/ is the 2-D Probability Density Function of the surface slopes Z
(along the x-direction Z
x
, and y-direction Z
y
). In order to extend the bi-static radar
equation to other electromagnetic scattering models,
0
must be replaced by its
corresponding expression.
This bi-static radar equation can also be seen as the 2D-convolution between the
Woodward Ambiguity Function (WAF)
2
.; f
c
/ (impulse response from a single
delay-Doppler cell) and a function †.; f
c
/ to weight each delay-Doppler cell based
on the scattering coefficient, geometry, antenna patterns...(
Marchan-Hernandez
et al. 2008
)
2
>
D
2
.; f
c
/
†.; f
c
/
<
j
Y
pq
.; f
c
/
j
(8.51)
with
2
.; f
c
/
D
ƒ
2
. /
j
S.f
c
/
j
2
(8.52)
and
4
i
Z
G
T
.
r
0
/G
R
.
r
0
/
pq
.
r
0
/
R
2
.
T
;
r
0
/R
2
.
r
0
;
R
/
P
T
.4/
2
2
ı.
.
r
0
//ı.f
c
f
D
.
r
0
//d
2
r
0
(8.53)
†.; f
c
/
D
S
being ı.x
x
0
/ Dirac delta functions.
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