Global Positioning System Reference
In-Depth Information
cos
−
1
(
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
is found when
∂
ψ
/∂
θ =
0. This occurs at
θ
(
ψ
max
)
=±
−α
)
, the maximum
sin
−
1
value being
. The maximum multipath carrier phase error is only
a function of the amplitude attenuation
ψ
max
=±
α
α
in this particular case. The largest value is
±
90° and occurs for
α =
1. This maximum corresponds to
λ
/
4. If
α
1 then
ψ
can
be approximated by
.
The multipath effect on pseudoranges depends among other things on the chipping
rate
T
of the codes and the receiver's internal sampling interval
S
. A necessary step
for each receiver is to correlate the received signal with an internally generated code
replica. The offset in time that maximizes the correlation is a measure of the pseudo-
range. Avoiding the technical details, suffice it to say that time-shifting the internal
code replica and determining the correlation for early, prompt, and late delays even-
tually determines the offset. The early and late delays differ from the prompt delay
by
α
sin
θ
S
and
S
, respectively. When the early minus late correlation are zero, i.e., they
have the same amplitude, the prompt delay is used as a measure of the pseudorange.
Consult Kaplan (1996, p. 148) for additional details on the topic of code tracking
loops and correlation. For a single multipath signal, the correlation function consists
of the sum of two triangles, one for the direct signal and one for the multipath signal.
This is conceptually demonstrated in Figure 7.6. The solid thin line and the dashed
line represent the correlation functions of the direct and multipath signals, respec-
tively. The thick solid line indicates the combined correlation function, i.e., the sum
of the thin line and dashed line. The left figure refers to destructive reflection when the
reflected signal arrives out of phase with respect to the direct signal. The right figure
refers to constructive reflection when the reflected and direct signals are in phase. Let
the combined signal be sampled at the early and late delays. The figure shows that the
prompt delay would coincide with the maximum correlation for the direct signal and
indicate the correct pseudorange but will be in error by the multipath-induced range
error
q
for the combined signal. The resulting pseudorange measurement errors are
negative for destructive reflection and positive for constructive reflection even though
the reflected signal always arrives later than the direct one.
The pseudorange multipath error further depends on whether the sampling interval
is greater or smaller than half the chipping period. Byun et al. (2002) provide the
following expressions. If
S>T/
2 (wide sampling) then
−
[23
Lin
—
0.9
——
No
*PgE
[23
destructive
constructive
-T
T
0
-T
0
T
q p
q
p
Figure 7.6 Correlation function in the presence of multipath.
p
denotes the time delay
of the multipath signal and
q
is the multipath induced pseudorange error.
