Acquisition of Weak Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 267

transition exists. There is probability, however, of false detection and missing
detection. It appears that at least 20 positive identifications of data transitions
are needed to find the exact location of the data transition. In actual collected
data sometimes there is no data transition in 400 ms. From this observation it is
difficult to use a predetermined amount of data to find the data transition location.
The following steps show the detailed procedure to find the location of the data
transition. In some of the steps the approaches are obtained through experience.

Step 1 . The first step is to perform correlations with the input signals. For
each transition test two references are used: one with a transition at the mid-
dle and the other one without transition. Figure 10.22 shows two input signals,
each containing 80 points of data. One input has a phase transition in the mid-
dle, and the other one does not, as shown in Figures 10.22a and 10.22b. The
reference contains 40 points with a phase transition in the middle, as shown
in Figure 10.22c. Figure 10.22d is obtained by linearly correlating the input in
Figure 10.22a against the reference in Figure 10.22c. The correlation output has
41 points, and each point is obtained by 40 input data points. The correlation
output is obtained by shifting the 0 point of the reference against the input data
from 0 to 40 points. Figure 10.22e is obtained in a similar way by correlating the
input in Figure 10.22b with the reference. When the input has a phase transition,
the output has a peak with amplitude 40; otherwise, the output is zero, as shown
in Figures 10.22d and 10.22e.

1 1

Inputs

0 0

− 1 − 1

0 0 0 0 0 0 20 40 60 80

(a) Input with transition. (b) Input without transition.

1

Reference

0

(c) Reference with transition

− 1

0 10 20 30 40

40
20

40
20

0 0

Outputs

− 20 − 20

− 40 − 40

0 10 20 30 40 0 10 20 30 40

(d) Output with transition. (e) Output without transition.

FIGURE 10.22 Correlation results with transition in reference.