## 3 Acsa-Olta
### The Optimal Correlation Space Based on EWPRE
Only power can be utilized and false detection will occur when the correlated power of the adjacent grids is larger than the most accurate peak. Therefore, the probability that the error occurs is that the false peak has power larger than the most accurate peak. Based on the detection variable expressed in Equation (9), the correlation outputs of two grids in code dimension with the same Doppler frequency are considered. One of them is the most accurate peak, the other one is the false peak. There are two situations for the probability calculation. When the space between the girds of the two peaks in code dimension is larger than one code chip, the correlated powers of the most accurate peak and the false peak are independent from each other and only the most accurate peak contains signal power. If \(P_{r}\) and \(P_{f}\) are the correlated power of the most accurate peak and the false peak, according to Equation (9), \(P_{r}\) obeys non-central \(\chi^{2}\) (chi-square) distribution with two degrees of freedom and \(P_{f}\) obeys central \(\chi^{2}\) distribution with two degrees of freedom. The probability density functions (PDF) of them are  
\[f_{p}(x)=\frac{1}{2\sigma_{n}^{2}}exp(-\frac{x+\upalpha^{2}}{2\sigma_{n}^{2}})I _{0}(\frac{\upalpha}{\sigma_{n}^{2}}\sqrt{x}) \tag{14}\]  
\[f_{a}(x)=\frac{1}{2\sigma_{n}^{2}}exp(-\frac{x}{2\sigma_{n}^{2}}) \tag{15}\]  
where \(\upalpha\) is the non-central parameter and it is equal to \(\sqrt{P}\) which is the energy of the most accurate peak. \(I_{0}(\cdot)\) is the zero-order modified Bessel function of the first kind [26]. The joint PDF is  
\[f_{u}(P_{r},P_{f})=f_{p}(P_{r})\cdot f_{a}(P_{f}) \tag{16}\]  
The probability false detection occurs is that when \(P_{f}>P_{r}\) and it is expressed as  
\[ \begin{split}& F(P_{f}>P_{r})=\iint_{P_{f}>P_{r}}f_{u}(\upgamma, \upbeta)\upalpha\upbeta\\ =& exp(-\frac{\upalpha^{2}}{4\sigma_{n}^{2}})\cdot \int_{0}^{+\infty}\frac{1}{2\sigma_{n}^{2}}exp(-\frac{2\upbeta+\upalpha^{2}/ 2}{2\sigma_{n}^{2}})I_{0}(\frac{\upalpha/\sqrt{2}}{\sigma_{n}^{2}}\sqrt{2 \upbeta})\upalpha\upbeta\\ =&\frac{1}{2}exp(-\frac{\upalpha^{2}}{4\sigma_{n}^ {2}})\end{split}  \tag{17}\]  
When the space between the girds of the most accurate peak and the false peak in code dimension is smaller than one code chip, \(P_{r}\) and \(P_{f}\) are not independent from each other and both of them contain signal power. Let \(I_{r}\) and \(Q_{r}\) be the in-phase and quadrature branch correlation outputs of the most accurate peak and \(I_{f}\) and \(Q_{f}\) be the in-phase and quadrature branch correlation outputs of the false peak, supposing the Doppler frequency error is zero, their expressions are as below according to Equations (6) and (7)  
\[I_{r}=\sqrt{P_{r}}cos\Delta\upvarphi_{1}=\sqrt{P}R(\uptau-\uptau_{d})cos \Delta\upvarphi_{1}+n_{I,r} \tag{18}\]  
\[Q_{r}=\sqrt{P_{r}}sin\Delta\upvarphi_{1}=\sqrt{P}R(\uptau-\uptau_{d})sin \Delta\upvarphi_{1}+n_{Q,r} \tag{19}\]\[I_{f} = \sqrt{P_{f}}cos\Delta\varphi_{2}=\sqrt{P}R(\uptau-\uptau_{d}-\Delta \uptau_{b})cos\Delta\varphi_{2}+n_{I,f} \tag{20}\] \[Q_{f} = \sqrt{P_{f}}sin\Delta\varphi_{2}=\sqrt{P}R(\uptau-\uptau_{d}- \Delta\uptau_{b})sin\Delta\varphi_{2}+n_{Q,f} \tag{21}\]  
where \(\Delta\uptau_{b}\) is the code phase space between the two peaks and \(|\Delta\uptau_{b}|<1\ code\ chip\). \(\Delta\varphi_{1}\) and \(\Delta\varphi_{2}\) are the carrier phase residuals. \(n_{I,r}\), \(n_{Q,r}\), \(n_{I,f}\) and \(n_{Q,f}\) are the noise components. They are AWGN and the variances of them are all \(\upsigma_{n}^{2}\) according to Equation (8). Because of the orthogonal carrier phase relationship, \(n_{I,r}\) and \(n_{I,f}\) are independent of \(n_{Q,r}\) and \(n_{Q,f}\).