Digital Signal Processing Reference
In-Depth Information
The signal contribution by reflections off areas away from the specular also have
different Doppler frequency than the specular itself.
1
.
k
i
.x; y/
V
R
k
s
.x; y//
V
T
f
D
.x; y/
D
(8.6)
being the GNSS carrier wavelength; V
T
and V
R
the transmitter and receiver
velocities; and k
i
and k
s
the incident and scattering direction unitary vectors. If
R and T are the receiver and transmitter positions, and P is the position of a point
on the reflecting surface, then: k
i
D
. P
T/=
j
P
T
j
; k
s
D
. R
P/=
j
R
P
j
.The
iso-frequency lines on the surface, under the simplified assumption of flat surface
(
z
D
0); receiver along the YZ plane; and only tangential velocity
of the
receiver
considered (V
R
z
p
B
2
k
i
V
T
B˙
4AC
D
0;
Const:), reduces to x
D
; where
2A
V
R
x
f
D
2
; B
V
R
y
.R
y
y/
2
A
D
D
2V
R
x
V
R
y
.R
y
y/; and C
D
f
D
2
.R
y
y/
2
f
D
2
H
2
.
Figure
8.10
shows the distribution of delay and Doppler parameters across the
reflecting surface, considering also the effects of the curvature of the Earth.
The coherent integration process, characterized by the coherent integration time,
i
, acts as a band-pass frequency filter given by the sinc
exponential function,
sin. ıf
i
/
ıf
i
e
iıf
i
(8.7)
where ıf
D
f
f
central
is the difference between the frequency of the signal-
component and the frequency of the replica used to correlate against the signal
(
correlation central frequency
, f
central
). This effectively blocks out frequencies
beyond
1
i
f
central
˙
(8.8)
Typically, the central frequency is chosen as the best estimate of the frequency at
the specular point. In that case ıf
D
f
f
spec
, thus the integration process filters
the contributions away from the central iso-Doppler line in Fig.
8.10
. An example is
given in the left panel of Fig.
8.11
. Nevertheless, the correlation against the signal
model can be done at other frequencies. Then, the correlation process might filter
out the signal around the specular and
select
instead contributions from a given
Doppler belt (see examples in Fig.
8.10
and right panel in Fig.
8.11
).
This section has shown that, under diffuse scattering regime, the waveform maps
the power reflected from areas delayed with respect to the specular point. These
areas correspond to elliptical rings. Moreover, tuning the frequency of the replica
in the code-correlation process, it is possible to select a given Doppler belt on the
surface: the scattering contributions from the selected Doppler-belt region will be
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