Global Positioning System Reference
In-Depth Information
As in the two-dimensional case, the goal is to find the projection
P
s
y
of
y
on
S
. This is obtained by the following matrix operation:
S(S
T
S)
−
1
S
T
y
=
P
s
y
(
12
.
13
)
Since the GPS satellite codes are nearly orthogonal, and also because of
a
w
a
s
,
a
w
S
T
W
a
s
S
T
·
S
(
12
.
14
)
Therefore,
S(S
T
S)
−
1
S
T
y
S(S
T
S)
−
1
S
T
(a
w
W
P
s
y
≈
=
+
a
s
S
+
n)
≈
a
s
S
+
P
s
n
12
.
15
)
where
P
s
n
represents the noise projection on the strong signal
S
. This means
that the result from Equation (12.13) is the sum of the nominal signals and the
projected noise. The weak signals can be obtained as
a
w
W
=
y
−
P
s
y
(
12
.
16
)
In this operation the nominal signals are obtained from the input
y
through con-
ventional acquisition, and they form the
S
matrix. Equations (12.13) and (12.16)
are used to find the weak signals. Simulated data are used to compare the results
from direct subtraction and the projection methods. The input signal contains
three strong signals (
C/N
0
=
44
∼
51 dB,) one weak signal (
C/N
0
=
36 dB),
and the proper amount of noise. Random numbers within realistic ranges are used
for each satellite C/A code initial phase offset, carrier frequency, and phase. The
three strong signals are acquired first and written in the time domain according
to Equation (12.3). The residues are found through both the direct and subspace
projection methods. At each input power level from
C/N
0
=
44
∼
51 dB, 30
runs with different noise are performed. The purpose is to find the residue after
the signals are subtracted. The results are shown in Figure 12.19. The dotted line
represents the direct subtraction method, and the solid line represents the sub-
space projection method. The vertical bars show the standard deviations of the
30 runs. It appears that the subspace projection method produces small residue
values, and the values do not increase with increasing of strong signal power.
Another set of simulated data are used to acquire the weak signal. The fol-
lowing results are obtained:
1. In the absence of any strong signal, the lowest input
S
/
N
that can be
acquired is
−
40 dB
(C/N
0
=
23
)
, which is one dB below the minimum
required
S
/
N
of
−
39 dB.
2. In the presence of strong signals, the weak signal that can be acquired
depends on the strong signal level. Table 12.4 shows three sets of test
results. The first column is the strong signal
S
/
N
level, and the second
column is the weakest signal can be acquired. In this table the
S
/
N
is
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