## 3 Acsa-Olta
### The Optimal Correlation Space Based on EWPRE
The covariance between \(n_{I,r}\) and \(n_{I,f}\) is  
\[\mathit{cov}(n_{I,r},n_{I,f};\Delta\uptau_{b}) = \frac{1}{N_{c}^{2}}E[n_{w}^{2}(t_{n})(1+cos(4\uptau(f_{IF}+f_{d}) t_{n}))]E[\sum_{n=0}^{N_{c}-1}c(t_{n}-\uptau_{d})c(t_{n}-\uptau_{d}-\Delta \uptau_{b})] \tag{22}\] \[= \upsigma_{n}^{2}\cdot R(\Delta\uptau_{b})\qquad(\Delta\uptau_{b }<1\ code\ chip)\]  
The covariance between \(n_{Q,r}\) and \(n_{Q,f}\) is  
\[\mathit{cov}(n_{Q,r},n_{Q,f};\Delta\uptau_{b}) = \frac{1}{N_{c}^{2}}E[n_{w}^{2}(t_{n})(1-cos(4\uptau(f_{IF}+f_{d}) t_{n}))]E[\sum_{n=0}^{N_{c}-1}c(t_{n}-\uptau_{d})c(t_{n}-\uptau_{d}-\Delta \uptau_{b})] \tag{23}\] \[= \upsigma_{n}^{2}\cdot R(\Delta\uptau_{b})\qquad(\Delta\uptau_{b }<1\ code\ chip)\]  
Thus, \(I_{r}\), \(I_{f}\), \(Q_{r}\) and \(Q_{f}\) are all Gaussian random variables. According to the definition of the variables \([I_{r},I_{f},Q_{r},Q_{f}]\), the covariance matrix \(\mathbf{K}\) is given by  
\[\mathbf{K}=\left[ \begin{array}{cccc}\upsigma_{n}^{2}&\upsigma_{n}^{2}\cdot R (\Delta\uptau_{b})&0&0\\ \upsigma_{n}^{2}\cdot R(\Delta\uptau_{b})&\upsigma_{n}^{2}&0&0\\ 0&0&\upsigma_{n}^{2}&\upsigma_{n}^{2}\cdot R(\Delta\uptau_{b})\\ 0&0&\upsigma_{n}^{2}\cdot R(\Delta\uptau_{b})&\upsigma_{n}^{2}\end{array} \right] \tag{24}\]  
Supposing \(\mathbf{P}_{r,f}=[I_{r},I_{f},Q_{r},Q_{f}]^{T}\), the joint PDF is  
\[f(I_{r},I_{f},Q_{r},Q_{f};\Delta\uptau_{b})=\frac{1}{(2\uptau)^{2}\sqrt{| \mathbf{K}|}}exp(-\frac{1}{2}(\mathbf{P}_{r,f}-\upmu)\mathbf{K}^{-1}(\mathbf{ P}_{r,f}-\upmu)^{T}) \tag{25}\]  
where \(\upmu=[\upmu_{I_{r}},\upmu_{I_{f}},\upmu_{Q_{r}},\upmu_{Q_{f}}]^{T}\) and it is the mean value vector of \(\mathbf{P}_{r,f}\). By computing the inverse matrix of \(\mathbf{K}\), the expression of the joint PDF is  
\[f(I_{r},I_{f}, Q_{r},Q_{f};\Delta\uptau_{b})=\frac{1}{(2\uppi)^{2}\cdot\upsigma_{n }^{4}(1-R^{2}(\Delta\uptau_{b}))}exp(-\frac{1}{2\upsigma_{n}^{2}(1-R^{2}( \Delta\uptau_{b}))} \tag{26}\] \[[(I_{r}-\upmu_{r,I})^{2}+(I_{f}-\upmu_{f,I})^{2}-2R(\Delta\uptau_{ b})(I_{r}-\upmu_{r,I})(I_{f}-\upmu_{f,I})\] \[+(Q_{r}-\upmu_{r,Q})^{2}+(Q_{f}-\upmu_{f,Q})^{2}-2R(\Delta\uptau_{ b})(Q_{r}-up\mu_{r,Q})(Q_{f}-\upmu_{f,Q})]\]  
Submitting \(I_{r}=\sqrt{P_{r}}cos(\Delta\varphi_{1})\), \(Q_{r}=\sqrt{P_{r}}sin(\Delta\varphi_{1})\), \(I_{f}=\sqrt{P_{f}}cos(\Delta\varphi_{2})\) and \(Q_{f}=\sqrt{P_{f}}sin(\Delta\varphi_{2})\) into Equation (26), after some algebra manipulations, the expression of the joint PDF of \(P_{r}\) and \(P_{f}\) is \[f(P_{r},P_{f};\Delta\tau_{b})=\frac{1}{(2\uppi)^{2}\cdot\sigma_{n}^{4 }(1-R^{2}(\Delta\uptau_{b}))}exp(-\frac{1}{2\sigma_{n}^{2}}(P_{r}+P_{f}+\upmu_{ I_{r}}^{2}+\upmu_{I_{f}}^{2}+\upmu_{Q_{r}}^{2}+\upmu_{Q_{f}}^{2} \tag{27}\] \[-2R(\Delta\uptau_{b})(\upmu_{I_{r}}\upmu_{I_{f}}+\upmu_{Q_{r}} \upmu_{Q_{f}})))\times\int_{0}^{2\uppi}\int_{0}^{2\uppi}exp(\frac{\sqrt{P_{r} \cdot P_{f}}}{\sigma_{n}^{2}}R(\Delta\uptau_{b})cos(\Delta\upvarphi_{1}- \Delta\upvarphi_{2})\] \[+\frac{\sqrt{P_{r}}}{\sigma_{n}^{2}}(\upmu_{f,I}R(\Delta\uptau_{ b})-\upmu_{r,I})cos(\Delta\upvarphi_{1})+\frac{\sqrt{P_{f}}}{\sigma_{n}^{2}}( \upmu_{r,I}R(\Delta\uptau_{b})-\upmu_{f,I})cos(\Delta\upvarphi_{2})\] \[+\frac{\sqrt{P_{r}}}{\sigma_{n}^{2}}(\upmu_{f,Q}R(\Delta\uptau_{ b})-\upmu_{r,Q})sin(\Delta\upvarphi_{1})+\frac{\sqrt{P_{f}}}{\sigma_{n}^{2}}( \upmu_{r,Q}R(\Delta\uptau_{b})-\upmu_{f,Q})sin(\Delta\upvarphi_{2}))\mathrm{ d}\Delta\upvarphi_{1}\mathrm{d}\Delta\upvarphi_{2}\]  
When \(\Delta\uptau_{b}\) varies from 0 to 1 chip, the probability that false detection occurs is  
\[F(P_{f}>P_{r};\Delta\uptau_{b})=\int_{0}^{+\infty}[\int_{0}^{P_{f}}f(P_{r},P _{f})\mathrm{d}P_{r}]\mathrm{d}P_{f} \tag{28}\]  
Taking the coherent integration length of 300 ms as an example, the false detection probability for different \(\Delta\uptau_{b}\) and different carrier-to-noise density ratio (\(C/N_{0}\)) is shown in Figure 2. It is obvious the false peak detection probability is related to the space between the most accurate peak and the false peak.