Tracking Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 285

by one data point. By shifting one data point, the peak correlations from the
early and late gates are slightly higher, but the tracking range decreases. Since
the weak signal is tracked under low dynamic environment, the tracking range
is not of primary concern. Higher peak correlation values can improve the S / N ;
therefore, the one data point shift is used to generate the early and late codes
for weak signal tracking. The early and late local codes can be obtained in two
steps: generation of the C/A code and the carrier frequency. The early C/A code
is obtained by shifting the last point of the prompt C/A code to the front. The
late C/A code is obtained by shifting the first point of the prompt local code to
the end. The same carrier frequency generated in Section 11.6 is used for the
early and late local codes.
After multiplying with 1 ms of input signal with both the early and late codes,
each will have 5000 outputs. The 5000 outputs are divided into two portions at
the initial C/A point and jointed with the outputs from two adjacent ms to form
the compressed early and late outputs. The approach of generating the early and
late compressed outputs is identical to the generation of the prompt point, as
discussed in Section 11.6 and Figure 11.8. The only difference is the input data
are multiplied with the local early and late C/A codes.
In this operation it is desirable to adjust the prompt correlation peak close to the
true peak. To achieve this goal, locally generated C/A must match closely to the
C/A code of the input signal to improve the S / N . When the prompt peak matches
the true peak, the early and late correlation peak values are about 813.7 relative to
the ideal peak value of 1023. These outputs are about 2 dB (20 log(813.7/1023))
below the prompt correlation peak. When the early and late codes are generated
by shifting two samples, the early and late correlation peaks are about 4.6 dB
below the prompt correlation peak. When weak signals are processed, higher
early and late correlation peaks can provide a more accurate result.
To find the fine time x , the time difference between the measured peak and
the true peak position, the same relation in Equation (8.45) can be used, which
can be rewritten as

y lsa
y esa =

1 − x − d

≡

r or

1 + x − d

( 11 . 8 )

( 1 − r)( 1 − d)

x =

1 + r

In this equation y esa and y lsa replace y e and y l in Equation (8.45). The values
of y esa and y lsa are the amplitude of the summed early and late peak correlation
outputs, respectively, and they will be further discussed in the next section. In
this equation, d is the time between the early and prompt or the late and the
prompt peaks. Both x and d are measured in unit of chip time of 0.9775 ms.
Its value is 0.2046 instead of 0.4092 used in Equation (8.45) because this choice
has slightly better noise performance but less tracking range.
In one second, 1000 compressed early outputs are generated. The 1000 com-
pressed early outputs are summed together in vector form to produce one point;