Global Positioning System Reference
In-Depth Information
S
2
y
(a) Strong signal measured correctly.
S
2
y
S
2
′
S
1p
S
1
(b) Amplitude of strong signal measured wrongly.
FIGURE 12.18
Strong and weak signals.
will be
S
2
, which is different from the correct signal
S
2
. The projection
S
1
p
, how-
ever, still equals to the true signal
S
1
, even if the amplitude is off. Thus the weak
signal obtained through the following relation can provide the correct answer:
S
2
=
y
−
S
1
p
(
12
.
11
)
This simple illustration demonstrates that the projection method is more robust
than the direct subtraction.
In practice, calculating the projection involves more than two signals. To
simplify, one divides the signals arbitrarily into two groups: the strong (
S
)andthe
weak (
W
) signals. First, the input signal is down-converted to baseband. Based
on the vector space notion, the down-converted input signal can be written as
(
5
)
y
=
a
w
W
+
a
s
S
+
n
(
12
.
12
)
where
1
)
]
T
, with
N
being the number of samples
in the down-converted input data and []
T
the transpose of a matrix.
W:
a matrix whose columns are unit vectors containing samples of the down-
converted weak signals.
a
w
:
amplitude vector for the weak satellite signals.
S:
a matrix whose columns are unit vectors containing samples of the down-
converted satellite signals with nominal power.
a
s
:
amplitude vector for the nominal satellite signals.
n:
white Gaussian noise vector with variance of
σ
n
.
y:
[
y
(0),
y
(1),
y
(2),
...
,
y(N
−
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