## 3 Acsa-Olta
### The Optimal Correlation Space Based on EWPRE  
Based on the generalized open-loop signal tracking method described earlier, the accuracy of the code phase measurement heavily relies on the code search grid size. If the grid size is too large, the uncertainty in the grid will distort the accuracy. On the contrary, if the grid size is too small, the accuracy may be greatly destroyed by the false peak detection. Since the length of the PRN code chip in distance is really large such as GPS L1 CA signal whose code chip is as long as 300 m, the grid size should be much smaller than the code chip duration to guarantee the accuracy. If the peak with the most accurate measurement is called most accurate peak, the correlation outputs beside the most accurate peak contain partial signal power. Thus, the powers of these correlation outputs are easier to be larger than the most accurate peak and false peak detection will probably occur. It is obvious that the smaller the grid is, the more signal power the adjacent grids contains and the more likely the false detection occurs. Therefore, the balance between the accuracy degradation induced by the uncertainty in the grid and false peak detection should be gained in the choice of the code search grid size. In order to obtain the optimal search grid size for the code phase measurement accuracy improvement, we propose the equivalent weight pseudo range error (EWPRE) to evaluate the measurement error and gain the balance. Both the errors brought by the uncertainty in the grid and false peak detection are involved so as to synthesize the effects of the two factors on the measurement accuracy. In the design of EWPRE, we firstly evaluate the code phase measurement errors when the most accurate peak or false peak is detected. For the measurements that are directly extracted from the parameters of the search grid, as it is a biased estimation with the jitter limited within the grid, the mean error is utilized for the error assessment. When the most accurate peak in the signal image is detected, the code phase measurement error is derived from the uncertainty of the real code phase in the search grid. In this case, the real signal code phase obeys the uniform distribution in the search grid expressed as below [25]  
\[f(c_{e})=1/\tau_{b},-\tau_{b}/2\leq c_{e}\leq\tau_{b}/2 \tag{11}\]  
Thus, the mean measurement error is  
\[c_{me}(\tau_{b})=\int_{-\tau_{b}/2}^{\tau_{b}/2}|c_{e}|\cdot f(c_{e})dc_{e}= \tau_{b}/4 \tag{12}\]  
According to the equation, it is clear that, when the most accurate peak is detected, the measurement error is proportional to the grid size and wide search grid will induce large code phase measuring error. When the false peak is detected, besides the error induced by the uncertainty in the search grid, the deviation of the search grid against that with the most accurate measurement is involved in the measuring error. If the \(nth\) grid relative to the grid with the most accurate measurement is falsely detected, the mean measurement error is  
\[e_{n}(\tau_{b})=\int_{n\tau_{b}-\tau_{b}/2}^{n\tau_{b}+\tau_{b}/2}|x|\cdot \frac{1}{\tau_{b}}\mathrm{d}x=n\tau_{b} \tag{13}\]  
It is obvious that the measurement error brought by false peak detection is much larger than that in the grid. Hereafter, we will develop the weight coefficients of the measurement errors when the most accurate peak or false peak is detected in the EWPRE. In order to synthetically evaluate the effects of the two kinds of errors on the measurement, the weight coefficients should be the probability that the errorsoccur. Usually, in open loop tracking, the peak with the largest power is regarded as the most accurate peak and the measurement will be extracted from the grid of that peak. Although prior information has been used to judge whether the peak is the most accurate one, it cannot distinguish the false peak whose corresponding code search grid is close to the most accurate peak since measurements extracted from these grids are all probable.