Global Positioning System Reference
In-Depth Information
where
C
s
(
t
) represents the C/A code of satellite
s
. The delayed version of this
signal can be written as
τ)e
j
2
πf (t
−
τ)
s(t
−
τ)
=
C
s
(t
−
(
7
.
14
)
where
τ
is the delay time. The product of
s
(
t
) and the complex conjugate of the
delayed version
s(t
−
τ)
is
τ)
∗
=
τ)
∗
e
j
2
πf t
e
−
j
2
πf (t
−
τ)
C
n
(t)e
j
2
πf τ
s(t)s(t
−
C
s
(t)C
s
(t
−
≡
(
7
.
15
)
where
C
n
(t)
≡
C
s
(t)C
s
(t
−
τ)
(
7
.
16
)
can be considered as a “new code,” which is the product of a Gold code and its
delayed version. This new “new code” belongs to the same family as the Gold
code.
(
5
)
Simulated results show that its autocorrelation and the cross correlation
can be used to find its beginning point of the “new code.” The beginning point
of the “new code” is the same as the beginning point of the C/A code. The
interesting thing about Equation (7.15) is that it is frequency independent. The
term
e
j
2
πf τ
is a constant, because
f
and
τ
are both constant. The amplitude
of
e
j
2
πf τ
is unity. Thus, one only needs to search for the initial point of the
“new code.” Although this approach looks very attractive, the input signal must
be complex. Since the input data collected are real, they must be converted to
complex. This operation can be achieved through the Hilbert transform discussed
in Section 6.13 or down converted into a complex signal; however, additional
calculations are required.
A slight modification of the above method can be used for a real signal.
(
4
)
The approach is as follows. The input signal is
s(t)
=
C
s
(t)
sin
(
2
πf t)
(
7
.
17
)
where
C
s
(
t
) represents the C/A code of satellite
s
. The delayed version of the
signal can be written as
s(t
−
τ)
=
C
s
(t
−
τ)
sin[2
πf (t
−
τ)
]
(
7
.
18
)
The product of
s
(
t
) and the delayed signal
s
(
t
−
τ
)is
s(t)s(t
−
τ)
=
C
s
(t)C
s
(t
−
τ)
sin
(
2
πf t)
sin[2
πf (t
−
τ)
]
C
n
(t)
2
≡
{
cos
(
2
πf τ )
−
cos[2
πf (
2
t
−
τ)
]
}
(7.19)
where
C
n
(
t
) is defined in Equation (7.16). In the above equation there are two
terms: a dc term and a high-frequency term. Usually the high frequency can
be filtered out. In order to make this equation usable, the
must be
close to unity. Theoretically, this is difficult to achieve, because the frequency
f
|
cos
(
2
πf τ )
|
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