Global Positioning System Reference
In-Depth Information
where
τ
(
t
)
is the time-variant propagation delay.
With a first-order expansion of the
τ
(
)=
τ
0
+
·
+
time-variant delay, i.e.
t
a
t
..., the carrier phase becomse:
[
− τ
(
)] =
(
− τ
0
−
·
)=
2
π
f
RF
t
t
2
π
f
RF
t
a
t
=
2
π
f
RF
t
−
2
π
f
RF
τ
0
−
2
π
f
RF
a
·
t
=
(23)
=[
2
π
(
f
RF
+
f
D
)
t
+
ϕ
0
]=
Let us denote
ϕ
0
=
−
2
π
f
RF
τ
0
and
f
D
=
−
2
π
f
RF
a
then
2
f
RF
1
t
+
ϕ
0
f
D
f
RF
2
π
f
RF
[
t
−
τ
(
t
)] =
π
+
(24)
)
dt
is the usual Doppler frequency shift. Due to the Doppler
effect, the observed carrier frequency is different from the nominal RF carrier frequency. With
a second-order expansion for
τ
(
f
RF
d
t
where
f
D
=
−
af
RF
=
−
τ
(
)
, we could see that also
f
D
changes in time and we could take
into account a Doppler-rate term
r
D
. For a ground GPS receiver in low-dynamics conditions,
the typical intervals are
f
D
t
=
−
5 kHz
÷
5 kHz and
r
D
=
−
1 Hz/s
÷
0 Hz/s.
The IF down-conversion leaves unmodified the Doppler frequency, as the IF carrier results:
{
[
π
(
+
)
+
φ
0
]
·
[
π
(
−
)
]
}
=
[
π
(
+
)
+
φ
0
+
φ
RX
]
BPF
cos
2
f
RF
f
D
t
2 cos
2
f
RF
f
IF
t
cos
2
f
IF
f
D
t
(25)
{}
where BPF
refers to the front-end filtering operation performed by the down-conversion
stage and
ϕ
RX
is the related additional phase contribution.
The code component is theoretically periodic with fundamental frequency equal to the inverse
of the code period. When propagating from the satellite to the receiver, the same time-variant
delay impacts on all the harmonic components:
∞
h
=
0
μ
h
e
j
2
h
π
f
0
[
t
−τ
(
t
)]
c
[
t
−
τ
(
t
)] =
h
=
0
μ
h
e
j
2
h
π
f
0
1
+
t
f
D
f
RF
∞
+
ϑ
0
=
(26)
Due to the Doppler effect, each harmonic is shifted of the same relative frequency offset
1
. Thus the fundamental frequency of the delayed code is now
f
0
1
and
f
D
f
RF
f
D
f
RF
+
+
its period duration is:
T
code
T
code
=
1
(27)
f
D
f
RF
+
where
T
code
is the nominal one. Consequently, the true chip rate is
R
c
1
f
D
f
RF
R
c
=
+
(28)
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