Global Positioning System Reference
In-Depth Information
rapidly changes the amplitude of the sum signal, we say that the received signal
is
fading
. A signal component reaching the receiver via a different path than the
direct path is called a
multipath component
.
In general, the received signal
x
(
t
)
is composed of the direct signal and
M
−
1
multipath components. Let
A
i
(
)
denote the amplitude of the
i
th multipath com-
ponent, let
D
denote the navigation message, let
C
denote the code, let
t
τ
denote
the multipath error, let the frequency change be
v
i
, the phase offset be
ϕ
i
, and,
finally, add a noise term
n
(
t
)
; then the signal can be described as
M
D
t
)
C
t
)
cos
2
)
+
x
(
t
)
=
A
i
(
t
)
−
τ
i
(
t
−
τ
i
(
t
π(
f
0
+
v
i
(
t
))
t
+
ϕ
i
(
t
n
(
t
).
i
=
1
To simplify the discussion we consider a two-path scenario (
M
=
2) and make
the following assumptions:
A
1
(
)
=
A
1
;
A
2
(
)
=
A
2
;
t
t
τ
1
(
)
=
τ
1
;
τ
2
(
)
=
τ
2
;
t
t
v
1
(
t
)
=
v
1
;
v
2
(
t
)
=
v
2
;
n
(
t
)
=
0
.
In other words we have assumed that the parameters (amplitudes, delays, and
Doppler shifts) are constant over the time period we consider. In this case,
x
(
t
)
can be reduced to
cos
2
+
ϕ
1
x
(
t
)
=
A
1
D
(
t
−
τ
1
)
C
(
t
−
τ
1
)
π(
f
0
+
v
1
)
t
cos
2
+
ϕ
2
.
+
A
2
D
(
t
−
τ
2
)
C
(
t
−
τ
2
)
π(
f
0
+
v
2
)
t
(7.26)
(ϕ)
=
exp
ϕ)
to rewrite (7.26) as
We now exploit cos
(
j
A
1
D
exp
j
2
+
ϕ
1
x
(
t
)
=
(
t
−
τ
1
)
C
(
t
−
τ
1
)
π(
f
0
+
v
1
)
t
+
ϕ
2
exp
j
2
+
(
−
τ
2
)
(
−
τ
2
)
π(
f
0
+
v
2
)
A
2
D
t
C
t
t
A
1
D
exp
j
2
+
ϕ
1
=
(
t
−
τ
1
)
C
(
t
−
τ
1
)
π(
f
0
+
v
1
)
t
exp
j
2
+
A
2
D
(
t
−
τ
2
)
C
(
t
−
τ
2
)
π(v
2
−
v
1
)
t
+
(ϕ
2
−
ϕ
1
)
+
ϕ
1
exp
j
2
×
π(
f
0
+
v
1
)
t
A
1
D
=
(
t
−
τ
1
)
C
(
t
−
τ
1
)
+
(ϕ
2
−
ϕ
1
)
exp
j
2
+
A
2
D
(
t
−
τ
2
)
C
(
t
−
τ
2
)
π(v
2
−
v
1
)
t
+
ϕ
1
exp
j
2
×
π(
f
0
+
v
1
)
t
.
Defining the instantaneous phase difference between the two signal components
ψ(
t
)
=
2
π(v
2
−
v
1
)
t
+
(ϕ
2
−
ϕ
1
)
, the output of the integrators in the DLL can be
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