## 3 Acsa-Olta
### The Measurement Calculation Strategy  
In order to reduce the correlators needed and keep the accuracy at the same time, early-minus-late discriminator (EMLD) is introduced [8]. However, no previous study has analyzed the accuracy of the method. In this section, the accuracy of the measurement derived from the discriminator is evaluated. Based on the accuracy comparison between the code phase calculated with the EMLD method and that directly extracted from the local reference signal of the detected grid, together with the computational cost assessment results, the measurement calculation strategy is developed. The EMLD based code phase estimation begins after the peak detection. The correlation outputs of the two code search grids adjoining the peak are used to construct the discriminator. If the two grids are represented by \(E\) and \(L\) corresponding to that with the code phase earlier or later than the peak respectively, the in-phase and quadrature correlation outputs of the two grids are expressed as below  
\[ \begin{split}& E_{I}=\sqrt{P}R(\uptau-k\uptau_{b}-\uptau_{b})cos( \Delta\varphi)+n_{I,E}\\ & E_{Q}=\sqrt{P}R(\uptau-k\uptau_{b}-\uptau_{b})sin(\Delta \varphi)+n_{Q,E}\\ & L_{I}=\sqrt{P}R(\uptau-k\uptau_{b}+\uptau_{b})cos(\Delta \varphi)+n_{I,L}\\ & L_{Q}=\sqrt{P}R(\uptau-k\uptau_{b}+\uptau_{b})sin(\Delta \varphi)+n_{Q,L}\end{split}  \tag{32}\]  
Figure 5: Optimal code search grid size of different \(C/N_{0}\) with 300 ms coherent integration. where \(n_{I,E}\), \(n_{Q,E}\), \(n_{I,L}\) and \(n_{Q,L}\) are the noise components in the correlation results. \(k\) is the grid index of the detected peak. The final code phase estimation result \(\uptau_{f}\) is  
\[\uptau_{f}=k\uptau_{b}+\frac{1}{1-2\uptau_{b}}\frac{(E_{I}^{2}+E_{Q}^{2})-(L_ {I}^{2}+L_{Q}^{2})}{(E_{I}^{2}+E_{Q}^{2})+(L_{I}^{2}+L_{Q}^{2})} \tag{33}\]  
In this way, regardless of the noise, the discriminator can completely compensate the uncertainty of the code phase in the detected grid [8]. If the most accurate peak is detected, the code phase estimation is totally unbiased. However, noise will distort the code phase measurement so that the error standard deviation is utilized for the evaluation. Assuming that it is perfectly normalized discriminator and the front-end bandwidth is infinite, the variance of the measurement calculated from the discriminator is as below (details of derivation are shown in the appendix)  
\[\upsigma_{EML}^{2}=\frac{\uptau_{b}}{2\cdot C/N_{0}\cdot T_{c}}(1+\frac{1}{C /N_{0}\cdot T_{c}(1-\uptau_{b})}) \tag{34}\]  
At the same time, for the code phase that directly extracted from the grid parameters, when the most accurate peak is detected, the variance of the measurement is  
\[\upsigma_{c}^{2}=\int_{-\uptau_{d}/2}^{\uptau_{d}/2}c_{e}^{2}\cdot f(c_{e}) dc_{e}=\uptau_{b}^{2}/12 \tag{35}\]  
In the following, we use the abbreviations below to represent the open loop tracking with the two different code phase measurement calculation methods. * DIS: The open-loop tracking with EMLD dependent code phase estimation method
* DET: The open-loop tracking that directly calculate the code phase from the grid parameters. Assuming that the peak detection is correct, the error standard deviations of measurements calculated by DIS and DET with different grid sizes are drawn in Figure 6 in which the coherent integration length is 300 ms. It shows that, when \(C/N_{0}\) is high, the accuracy of DIS is much better than that of DET with the same grid size. Furthermore, although the accuracy of DET with optimal grid size is better than the discriminator based method with the grid size of 0.1 chip when \(C/N_{0}\) is high, the improvement is a little and it needs a huge number of correlators. However, as the \(C/N_{0}\) decreases, the accuracy of DIS is destroyed and it becomes worse than DET. Moreover, if false peak detection occurs, the accuracy of DIS will be further distorted. Thus, the EMLD method cannot be utilized all the time as that in [19] and a \(C/N_{0}\) threshold is needed to switch the code phase calculation methods.