Damping Ratio
The damping ratio controls how fast the filter reaches its settle point. The damping ratio also controls how much overshoot the filter can have. A smaller settling time results in a larger overshoot. This can be seen in Figure 7.4.
Noise Bandwidth
The second parameter in the PLL filter is the noise bandwidth BL. The noise bandwidth controls the amount of noise allowed in the filter. This parameter can also, as the damping ratio, control the settling time. As the tracking loop starts to track a signal the start frequency is the frequency found by the acquisition algorithm. (This phase is sometimes called the pull-in phase in the literature. It is in this phase the filter is trying to converge to the correct frequency and phase.) The start frequency from the acquisition algorithm can be off by some Hz. The tracking loop is then going to lock onto the correct frequency. To see the impact of various noise bandwidths, a real GPS signal is used where the acquisition algorithm found a frequency that is about 21 Hz off. Figure 7.5 shows the offset from the start frequency for three different noise bandwidths.
From the figure, it follows that if the noise bandwidth is 60Hz, the tracking loop immediately finds the correct frequency offset of about 21 Hz. It can also be seen that a lot of noise in the tracking frequency is allowed. In the second case, where the noise bandwidth is 30 Hz, the tracking loop also locks on the signal quite fast.
FIGURE 7.6. Basic GPS receiver tracking loop block diagram.
It can be seen that the noise on the tracking frequency is much smaller than with a noise bandwidth of 60 Hz. In the third case where the noise bandwidth is 10 Hz, the tracking loop is not fast enough to reach the real frequency before a phase shift occurs. Therefore, it is not likely to converge to the proper value. A large noise bandwidth implies that the tracking loop quickly locks to the real frequency but has a relatively large frequency noise in the locked state. A smaller noise bandwidth implies that it can take some time before the tracking loop can be locked to the frequency, but after the lock the frequency is stable. Some implementations split the PLL into two filters, often called pull-in and tracking filters.
For land applications, a typical value for noise bandwidth is about 20 Hz.
Carrier Tracking
To demodulate the navigation data successfully an exact carrier wave replica has to be generated. To track a carrier wave signal, phase lock loops (PLL) or frequency lock loops (FLL) are often used.
Figure 7.6 shows a basic block diagram for a phase lock loop. The two first multiplications wipe off the carrier and the PRN code of the input signal. To wipe off the PRN code, the
output from the early-late code tracking loop described above is used. The loop discriminator block is used to find the phase error on the local carrier wave replica. The output of the discriminator, which is the phase error (or a function of the phase error), is then filtered and used as a feedback to the numerically controlled oscillator (NCO), which adjusts the frequency of the local carrier wave. In this way the local carrier wave could be an almost precise replica of the input signal carrier wave.
The problem with using an ordinary PLL is that it is sensitive to 180° phase shifts. Due to navigation bit transitions, a PLL used in a GPS receiver has to be insensitive to 180° phase shifts.
Figure 7.7 shows a Costas loop. One property of this loop is that it is insensitive for 180° phase shifts and hereby a Costas loop is insensitive for phase transitions due to navigation bits. This is the reason for using this carrier tracking loop in GPS receivers. The Costas loop in Figure 7.7 contains two multiplications. The first multiplication is the product between the input signal and the local carrier wave and the second multiplication is between a 90° phase-shifted carrier wave and the input signal.
FIGURE 7.7. Costas loop used to track the carrier wave.
The goal of the Costas loop is to try to keep all energy in the I (in-phase) arm. To keep the energy in the I arm, some kind of feedback to the oscillator is needed. If it is assumed that the code replica in Figure 7.7 is perfectly aligned, the multiplication in the I arm yields the following sum:
where
is the phase difference between the phase of the input signal and the phase of the local replica of the carrier phase. The multiplication in the quadrature arm gives the following:
If the two signals are lowpass filtered after the multiplication, the two terms with the double intermediate frequency are eliminated and the following two signals remain:
To find a term to feed back to the carrier phase oscillator, it can be seen that the phase error of the local carrier phase replica can be found as
From Equation (7.24), it can be seen that the phase error is minimized when the correlation in the quadrature-phase arm is zero and the correlation value in the in-phase arm is maximum. The arctan discriminator in Equation (7.24) is the most precise of the Costas discriminators, but it is also the most time-consuming. Table 7.1 describes other possible Costas discriminators.
TABLE 7.1. Various types of Costas phase lock loop discriminators
|
Discriminator |
Description |
 |
The output of the discriminator is proportional to sin(^). |
 |
The discriminator output is proportional to sin(2^). |
 |
The discriminator output is the phase error. |
Figure 7.8 shows the responses corresponding to the different discriminators. The phase discriminator outputs in this figure are computed using expressions in Table 7.1 for all possible phase errors. In the same figure it can be seen that the discriminator outputs are zero when the real phase error is 0 and
This is why the Costas loop is insensitive to the 180° phase shifts in case of a navigation bit transition.
The behavior of the Costas loop when a 180° phase shift occurs is more clearly illustrated in Figure 7.9. In this figure the vector sum of
is shown as the vector in the coordinate system. If the local carrier wave were in phase with the input signal, the vector would be aligned to the I-axis, but in the present case a small phase error is illustrated. When the signal is tracked correctly the vector sum of
tends to remain aligned with the I-axis. This property ensures that if a navigation bit transition occurs, the vector on the phasor diagram will flip 180° (showed by the dashed vector in the figure). If a navigation bit transition occurs, the Costas loop will still track the signal and nothing will happen. This property does make Costas loop the commonly chosen phase lock loop in GPS receivers.
The output of the phase discriminator is filtered to predict and estimate any relative motion of the satellite and to estimate the Doppler frequency.
FIGURE 7.8. Comparison between the common Costas phase lock loop discriminator responses.
FIGURE 7.9. Phasor diagram showing the phase error between the phase of the input carrier wave and the phase of the local carrier wave replica.
