In this section, we evaluated the performance of the proposed algorithm by computer simulations. Simulations are divided into two parts. At the first part, signal overlapping algorithm is verified using Spirent simulator output, and at the second part, the tracking performance of the proposed RLL algorithm is verified under the several values of signal strength using the software-based signal generator output.
3.1. Verifying the Signal Overlapping Algorithm
In order to verify the improvement of SNR by the overlapping algorithm, a Spirent GSS7700 GPS simulator (Spirent, Crawley, West Sussex, UK) was used to collect 10 GPS satellites signals for a vehicle rotating at 5 Hz with an antenna which has a beam width of 180 degree as shown in
Figure 9. The strength of the received GPS satellites signals was 41.7–50.6 dB–Hz (mean: 46.5 dB–Hz, standard deviation: 3.2 dB–Hz).
The collected signals were stored as a computer file. The navigation data were decoded by processing the stored file with the software-based GPS receiver developed by Chungnam National University, and the satellites’ and vehicle’s positions were obtained from the receiver. Next, the roll angles of the satellites were calculated and the time delay for the rotation frequency of 5 Hz were obtained as the pseudocode of
Table 1. Finally, synchronization was performed by delaying and overlapping the correlator outputs of each satellite signals.
Correlator outputs before and after time synchronization are shown in
Figure 10a,b, in which only the results of five satellite signals among 10 satellite signals are given in order to improve the readability. From
Figure 10b, it is shown that the correlator outputs for each satellite signals are well synchronized in time. The noise power of the correlator output with respect to the number of overlapped signals is given in
Figure 11. It can be confirmed from the figure that the measured power is similar to the theoretical value that the noise power decreases to
when
signals are overlapped. By the simulation results using Spirent simulator output, it is verified that the noise of the correlator output decreases by signal overlapping algorithm.
Since the estimated rotation frequency of the vehicle is used to calculate the time delay to overlap multiple satellite signals, the effect of the estimation error of the rotation frequency on the overlapped signal is analyzed in this section. Using the same Spirent simulator’s output described above, for the case that error of the estimated frequency is −50–50%, the correlator output of the overlapped signal and the output of the rotation discriminator are presented in
Figure 12 and
Figure 13, respectively. From
Figure 13, it can be seen that when the estimation error of the rotation frequency is −50%, the response of the discriminator is slightly lowered only in the region where the input error angle is larger than 30 degrees, but it has little effect on the output of the discriminator in the other regions. Additional experiments were performed using the output signal of the software-based signal generator for signal strengths of 34 dB–Hz and 29 dB–Hz, and the outputs of the discriminator are presented in
Figure 14a,b, respectively, which are similar to the results in
Figure 13.
3.2. Proposed Enhanced RLL Algorithm
In this section, simulations were performed to evaluate the performance of the proposed algorithm according to the signal strength. In order to change the signal strength and the rotation frequency of the vehicle over time, a satellite signal was generated using a software-based signal generator developed by Chungnam National University and stored as a computer file. In this simulation, the beam width of the antenna was set to 180 degree as in the previous case of using Spirent simulator and 12 GPS satellite signals were simulated. The rotation scenario of the vehicle was set to start from non-rotating state as shown in
Figure 15, rotate at 5 Hz after 2 s, and rotate at 6 Hz after 10 s. In this simulation, the rotation frequencies of 5 and 6 Hz are chosen to show the simulation results clearly in figures but the rotation frequency itself does not likely to affect the performance of the proposed algorithm. In order to verify the roll angle estimation performance of the proposed algorithm, in this paper, we used a software receiver to track the code and carrier of the signal stored in the computer file and generate the correlator outputs. In addition, the correlator outputs were synchronized by using the time delay calculated in the same manner as in
Section 3.1, and were overlapped on each other, and then the roll angle and the rotation frequency were estimated using the RLL algorithm. In this simulation, the difference from
Section 3.1 is that the rotation frequency of the vehicle for calculating the time delay was estimated from the output of RLL instead of using the known true rotation frequency, and the signal strength to compensate the rotation discriminator was applied as the known true value, not the estimated value.
In the simulation study, a wide range of signal strengths with a step of 1 dB are used as the input signal strength but only interested signal strengths of C/N0 34 dB–Hz to 28 dB–Hz are described in
Figure 16,
Figure 17,
Figure 18 and
Figure 19 and
Table 2. Among the results, the estimated roll angle error and the estimated rotation frequency for the four signal strengths are graphically presented in
Figure 16,
Figure 17,
Figure 18 and
Figure 19. The average and standard deviation of the roll angle estimation error and the rotation frequency for each signal strength, the convergence time of the rotation frequency are presented in
Table 1. In the Algorithm column of the table, Proposed RLL-C and Proposed RLL-C/O means that “only Compensation algorithm is used” and “Compensation with Overlapping algorithm with 12 signals is used” respectively. Here, four cases were simulated considering the overlap of the satellite signals and the compensation of the rotation discriminator response for each signal strength, but the results are presented only for the three modes because there was little performance change when only the satellite signals were overlapped without compensating the discriminator response. The determination of the failure of the convergence of RLL algorithm is based on the case that the estimated roll angle error exceeds 180 degrees and does not converge to 0 degree when the rotational frequency changes. When the convergence failure is determined, it can be confirmed from the graphs that estimated rotation frequency does not converge to 6 Hz. Mean and standard deviation in
Table 2 were measured after the estimated rotation frequency had converged and these values were the results from one simulation.
When the signal strength was 34 dB–Hz, the estimated roll angle was converged after about 10 s after the rotation frequency was changed in the case of using one satellite signal without compensating the discriminator response (previous RLL). When the discriminator response was compensated (proposed RLL-C), the roll angle was estimated continuously, and the rotation estimation error was reduced when the discriminator response was compensated and 12 signals were overlapped (Proposed RLL-C/O) as shown in
Figure 16. In this case, the estimation error became larger in Proposed RLL-C. This is because the open loop gain becomes larger as the discriminator response is compensated, and the noise bandwidth of the closed loop becomes larger in turn.
In case the signal strength was 31 dB–Hz, the roll angle could not be estimated in Previous RLL. However, the roll angle was estimated continuously in Proposed RLL-C, and the rotation estimation error was reduced in proposed RLL-C/O as shown in
Figure 17.
When the signal strength was 29 dB–Hz, the roll angle could not be estimated in Previous RLL or in Proposed RLL-C. However, the roll angle was estimated after about 5 s after the rotation frequency was changed in Proposed RLL-C/O as shown in
Figure 18.
In case the signal strength was 28 dB–Hz, the roll angle was estimated after about 10 s after the rotation frequency was changed only in Proposed RLL-C/O as shown in
Figure 19. However, since the mean and standard deviation of the roll angle error were high, it is considered to be the limit of the signal strength that can be tracked by the proposed algorithm.
In this simulation, the marginal point where RLL stably tracks the roll angle was determined by the roll angle error and the convergence time. In Previous RLL the phase error was high and the convergence time was long at the signal strength of 32 dB–Hz, so 33 dB–Hz was determined as a marginal tracking point. In Proposed RLL-C the roll angle error was small and convergence time was short at the signal strength of 31 dB–Hz, so this strength was considered as a marginal point. In Proposed RLL-C/O the roll angle error was high and the convergence time was long at the signal strength of 28 dB–Hz, so 29 dB–Hz was considered as a marginal point. Overall, by using the proposed enhanced RLL algorithm, the marginal point for stable signal tracking is improved by 4 dB from 33 dB–Hz to 29 dB–Hz.
The improvement by signal overlapping algorithm is theoretically 10.8 dB (
). Actually, however, the performance improvement was about 4 dB because the noise characteristic was degraded by increasing the response of the rotation discriminator. When the signal strength was 29 dB–Hz, an additional gain of 8.3 was applied for the discriminator compensation as shown in
Figure 7, and the noise power increased about 7.7 dB as is described below due to the additional gain. Therefore, the performance improvement of 3.1 dB was expected.
The noise bandwidth (
) and the 2nd order RLL tracking loop is shown as follows [
10]:
where,
and
denote the natural frequency and the damping ratio. If additional gain due to the discriminator compensation is
, then
and
become
and
respectively, and thus the modified noise bandwidth (
is given as follows:
Therefore, the modified noise bandwidth is , which means that the noise power increased by 7.7 dB ().
Through the simulation, the necessity for the compensation of the discriminator response was shown and it was verified that the roll angle estimation error was reduced and the roll angle could be estimated even at the lower signal strength by overlapping multiple signals.
3.3. Effect of the C/N0 Estimation Error
In this section, the effect of the C/N0 estimation error on the tracking performance of roll angle and rotation frequency is simulated. The effect of the rotation frequency error was not analyzed because the overlapping algorithm used the RLL algorithm output as the estimated rotation frequency in
Section 3.2. Therefore, the simulations only for the effect of the C/N0 estimation error were performed in this section. The simulation was performed for the case that the signal strength is 29 dB–Hz as C/N0 with proposed enhanced RLL (compensated + overlapped) which is the marginal point as described in
Section 3.2. The C/N0 error was assumed to be 1 dB–Hz and 2 dB–Hz with Gaussian random distribution in reference to the Pini et al. [
11]. The simulation results are shown in
Figure 20 and mean and standard deviation of the tracking error are summarized in
Table 3. In this table, None, 1 dB–Hz, and 2 dB–Hz in C/N0 error column mean that no C/N0 error was applied, 1 dB–Hz and 2 dB–Hz of C/N0 error was applied, respectively.
In
Table 3, the mean and standard deviation for the case 1 were the result from one simulation, and those for the case 2 and 3 were the averaged values from ten simulations with Gaussian random error. Through these simulation results, it is shown that the C/N0 estimation error increases the standard deviation of the tracking error, but the estimated roll angle and rotation frequency are still converged to the true value.