Tracking GPS Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 169

navigation data and it can be considered as one type of a Costas loop. A Costas
loop is a phase-locked loop, which is insensitive to phase transition. The output
of the comparator is filtered again and generates a control signal. This control
signal is used to tune the oscillator to generate a carrier frequency to follow the
input cw signal. This carrier frequency is also used to strip the carrier from the
input signal.

8.7 USING THE PHASE-LOCKED LOOP TO TRACK GPS SIGNALS ( 6,7 )

In this section, the application of the equations derived in Sections 8.3 through 8.5
will be discussed. A tracking program using the phase-locked loop will be dis-
cussed. The input data to the tracking loop are collected from actual satellites. In
this discussion second-order phase-locked loops will be used. Several constants
must be determined such as the noise bandwidth, the gain factors of the phase
detector, and the VCO (or the digital frequency synthesizer). These constants are
determined through trial and error and are by no means optimized. This tracking
program is applied only on limited data length. Although it generates satisfactory
results, further study might be needed if it is used in a software GPS receiver
designed to track long records of data. The following steps can be applied to
both the code loop and the carrier loop:

1. Set the bandwidths and the gain of the code and carrier loops. The loop
gain includes the gains of the phase detector and the VCO. The bandwidth
of the code loop is narrower than the carrier loop because it tracks the
signal for a longer period of time. Choose the noise bandwidth of the code
loop to be 1 Hz and the carrier loop to be 20 Hz. This is one set of several
possible selections that the tracking program can operate or function.
2. Select the damping factor in Equation (8.25) to be ζ = . 707. This ζ value

is often considered close to optimum.
3. The natural frequency can be found from Equation (8.26).
4. Choose the code loop gain (k 0 k 1 ) to be 50 and the carrier loop gain to be
4 π 100. These values are also one set of several possible selections. The

constants C 1 and C 2 of the filter can be found from Equation (8.39).

×

These four steps provide the necessary information for the two loops. Once the
constants of the loops are known, the phase of the code loop and the phase of the
carrier frequency can be adjusted to follow the input signals. In this approach, the
loops usually update every millisecond, because the C/A code is one millisecond
long. At every millisecond the C/A code must be regenerated and the initial
phase of the C/A code must be continuous from the previous one. This initial
code phase can be related to fine time resolution. The phase of carrier frequency
is updated from the output of the arctangent phase comparator. The output is
obtained from the in-phase channel of the carrier loop as shown in Figure 8.3.
A typical output data set is shown in Figure 8.4. In this figure, the amplitude