## 3 Acsa-Olta
### The Optimal Correlation Space Based on EWPRE
Since more signal power is contained in the false peak when the space is small, the probability increases as the space between the two peaks decreases. While, when the space is too small, the probability become lower because of the significant positive correlation between the powers of the two peaks according to Equations (22) and (23). Hence, when the grid size is small, the false peak detection will probably occur especially at the point with the largest false detection probability. However, large grid size contains large inherent measurement error. In addition, when the correlation space exceeds 1 code chip, the false peak detection probability is extremely small with respect to those within the code chip duration so that it is not considered in the following analysis. In consideration that open-loop tracking extracts code phase measurement from that of the grid where the detected peak is, the results further validate that the search grids with the size too large or two small will both degrades the measurement accuracy. \[f(I_{r},I_{f}, Q_{r},Q_{f};\Delta\uptau_{b})=\frac{1}{(2\uppi)^{2}\cdot\upsigma_{n }^{4}(1-R^{2}(\Delta\uptau_{b}))}exp(-\frac{1}{2\upsigma_{n}^{2}(1-R^{2}( \Delta\uptau_{b}))} \tag{26}\]0 False detection probability of different correlation space and \(C/N_{0}\). Furthermore, it also shows that the false detection probability becomes high as (\(C/N_{0}\)) decreases. When \(C/N_{0}\) is high, the most accurate peak with the maximum signal power is easily detected from the false peak. While, when \(C/N_{0}\) is low, both the two peaks are greatly disturbed by noise so that they are difficult to be distinguished and false detection will probably occur. In addition, the code phase spaces with the largest false detection probability are various among different \(C/N_{0}\) due to the positive correlation between the two peaks besides the signal power. The lower the \(C/N_{0}\) is, the larger the corresponding space with the highest false detection probability is. Therefore, the signal power should be considered in the choice of code search grid size. Considering that the false peak detection probabilities can indicate the possibility of the occurrence of the measurement errors induced by the false detection at different points, the weight coefficients are designed based on the probability. The weight coefficient for the \(nth\) grid relative to grid with the most accurate measurement is expressed as below  
\[W_{n}(\tau_{b})=\left\{ \begin{aligned} &\frac{1}{T_{p}}\int_{0}^{\frac{ \tau_{b}}{2}}\mathrm{d}x+\frac{1}{T_{p}}\int_{\frac{\tau_{b}}{2}}^{T_{p}}(1-F (P_{f}>P_{r};x))\mathrm{d}x\\ &\frac{1}{T_{p}}\int_{\frac{2n-1}{2}\tau_{b}}^{\frac{2n+1}{2} \tau_{b}}F(P_{f}>P_{r};x)\mathrm{d}x\\ \end{aligned} \right.\qquad n=0 \tag{29}\]  
where \(n=0\) corresponds to the grid with the most accurate measurement. \(T_{p}\) is the code chip duration. The weighed coefficients are the integration of the probability that the detection result is in the false grid or in the right grid. Accordingly, it evaluates the likelihood of the measurement errors corresponding to the detection of the correct or false peak. Based on the weight coefficients, the EWPRE is  
\[S(\tau_{b})=W_{0}(\tau_{b})\cdot c_{me}(\tau_{b})+\sum_{n=1}^{T_{p}/\tau_{b}}[ W_{n}(\tau_{b})\cdot e_{n}(\tau_{b})] \tag{30}\]  
It is the weighed sum of the code phase measurement errors when the detection results vary from the grid of the most accurate peak to that one code chip away. The measurement error induced by the false detection in other grids and that when the most accurate peak is detected are both involved and weighed to accumulate together. With the help of the weight coefficients, when the false detection probability is large, the measurement error induced by the false peak detection will be dominant in the weighted error. On the contrary, the measurement error within the grid will be more significant. As a result, it synthesizes the effects of the uncertainty in the search grid and false peak detection on the code phase measurement accuracy.