Digital Signal Processing Reference
In-Depth Information
If the scattered fields were reflected off the Earth surface in a purely specular
way, the scattered signal s would come from a small area around the specular point
(Fresnel zone), where the Doppler frequency factor could be assumed constant.
Therefore, s could reduce to s
D
E
pq
.
r
R
I
r
spec
/e
i2f
D
.
r
spec
/t
, being f
D
.
r
spec
/ the
Doppler frequency of the ray-path reflected at the specular point, and E
pq
.
r
R
I
r
spec
/
the field from the ray-path reflected at the specular point, reaching the receiver
coordinates, as modelled in the previous sections around the specular point, but
with the phase-shifts introduced by the GNSS modulation (we use E for fields with
phase-shift modulations, and E for fields as modelled in the previous sections).
Then, the result of the cross-correlation
Z
i
Y.;f
c
/
D
s.t/r.t
C
; f
c
/dt
(8.45)
0
could be simplified
Z
i
E
pq
.
r
R
;t
I
r
spec
/e
i2f
D
.
r
spec
/t
c.t
C
/e
i2f
c
t
dt
Y.;f
c
/
D
0
i
E
pq
.
r
R
I
r
spec
/ƒ. /S.f
c
f
D
/
(8.46)
note that the field E has become E, and S is the
sinc-exponential
function
introduced in Eq.
8.7
.
Z
i
1
i
sin.ıf
i
/
ıf
i
e
i2ıf t
dt
D
e
iıf
i
S.ıf /
D
(8.47)
0
If the correlation modelled frequency f
c
were close enough to the real Doppler
frequency of the signal, f
D
, then
j
S
j!
1. If no instrumental noise is considered,
then the waveform reduces to the triangle function centered at the signal receiving
time, with maximum amplitude given by the scattered field
j
E
pq
.
r
R
I
r
spec
/
j
and an
overall phase driven by propagation, scattering phase shifts, and correlation process.
It is important to highlight that Eq.
8.46
assumes purely specular reflections only.
For diffuse scattering, E
pq
.
r
R
I
r
spec
/ must be replaced by any of its integral forms. If
we re-shape the integral forms given in the previous sections and add the frequency
factor:
E
pq
.
r
R
/
R
S
E
pq
.
r
R
I
r
0
/e
i2f
D
.
r
0
/t
d
2
r
0
, then:
Z
i
dt
Z
E
pq
.
r
R
I
r
0
/e
i2f
D
.
r
0
/t
d
2
r
0
c.t
C
/e
i2f
c
t
Y.;f
c
/
D
0
S
i
Z
E
pq
.
r
R
I
r
0
/ƒ. .
r
0
//S.ıf .
r
0
//d
2
r
0
(8.48)
S
Note that both local delay .
r
0
/ and local frequency shift ıf
D
f
c
f
D
.
r
0
/ are
functions of the surface point under integration
r
0
, as explained in Sect.
8.4
. Y.;f
c
/
corresponds to the complex waveform as coherently correlated during
i
, and central
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