## 4 Performance Evaluation
### Accuracy of the Code Phase Measurement  
In order to verify that the grid sizes gained from EWPRE bring the optimal code phase measuring accuracy for the corresponding \(C/N_{0}\) in the presence of AGWN, Monte Carlo simulation is carried out to evaluate the measurement accuracy of open loop tracking method. In the simulation, the measurement is directly extracted from the grid. The mean measurement error of the open loop tracking is computed to testify that the EWPRE can exactly indicate the difference of the measuring accuracy among different grid sizes. The expression of the mean measurement error is  
\[E_{m}=\frac{1}{N_{t}}\sum_{n=1}^{N_{t}}|\tau_{t,n}-\tau_{t,n}| \tag{36}\]  
where \(N_{t}\) is the total number of the tests which is 5000 in the simulation. \(\tau_{t,n}\) is the real code phase of the \(nth\) test and \(\tau_{t,n}\) is that derived from the open loop tracking. Then, the code phase measurement error standard deviations of open loop tracking with the optimal grid size and other grid sizes are calculated for different \(C/N_{0}\). The simulation parameters are shown in Table 2. The front-end filtering, quantization and other non-idealities are not considered for analytical simplicity. The Doppler frequency error is assumed to be zero since it almost has nothing to do with the code phase measurement accuracy.  
Taking the \(C/N_{0}\) of 10 dBHz, 15 dBHz, 20 dBHz and 25 dBHz as examples, the simulation results of the mean code phase measurement error of open loop tracking method with the code search grid sizes listed in Table 2 is shown in Figure 7. It is clear that the shape of the mean error curves in the figure are more or less the same as those in Figure 4 and the grid sizes with the minimum mean measurement error are the same as those with minimum EWPRE. For other \(C/N_{0}\), the simulation results demonstrate that the grid size calculated from EWPRE brings the best accuracy as well. As a result, it verified that the grid size derived from EWPRE is optimal.  
Figure 8 illustrates the simulation result of the code phase measurement error standard deviation for the open loop tracking method with the optimal code search grid size that is calculated in Section 3.1. The standard deviations of the code phase measurement error of open loop tracking with the grid sizes of 0.2 chip, 0.1 chip, 0.05 chip and 0.02 chip are drawn for comparison. It confirmed that the optimal code search grid sizes show higher measurement accuracy than the other grid sizes in the condition of the same \(C/N_{0}\). Therefore, by changing the grid size to the optimal ones according to \(C/N_{0}\), the accuracy of the code phase measurement can be greatly improved. In addition, the EMLD based open loop tracking  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Signal & GPS L5 pilot signal \\ Signal \(C/N_{0}\) & 10 dBHz\(-\)35 dBHz \\ Signal \(C/N_{0}\) step & 1 dB \\ Signal sampling rate & 62 MHz \\ Code search grid size & 0.01 chip\(-\)0.4 chip \\ Code search grid size step & 0.01 chip \\ Coherent integration length & 300 ms \\ Doppler frequency error & 0 Hz \\ Code phase & Uniformly distributed in the range of 2 adjacent chip \\ \hline \hline \end{tabular}
\end{table}
Table 2: The parameters of the code phase accuracy simulation.  
Figure 7: The mean code phase measurement error.  
method with the grid size of 0.1 chip is simulated as well. The error standard deviation of the code phase measurements computed by the EMLD is also drawn in the figure. It demonstrates that the accuracy of the method is more or less the same as that directly extracted from the grid with optimal size when \(C/N_{0}\) is high. However, it will be quickly degraded when \(C/N_{0}\) is lower than 20 dBHz which is much poorer than the measurement directly extracted from the grid.