# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## Appendix A  
### The Proof of Lemma 5  
The proof is similar to that of Lemma 4, and thus we use the same notation as in Lemma 4. The difference is that here, the generated signal is \(g_{DM,PP+DD}(t)=C_{PP,\delta_{0}}(t)+C_{DD,\delta_{0}}(t)\), instead of \(g_{DM,PP}(t)=C_{PP,\delta_{0}}(t)\).  
Consider the \(H_{0}\) hypothesis:  
\[\begin{array}{llll}H_{0}\ :&Z_{H_{0}}&=\sum\limits_{i=0}^{M-1}n(t_{i})n(t_{i}- 33T_{0})\{C_{PP,\delta_{0}}(t)+C_{DD,\delta_{0}}(t)\}\\ &&=\sum\limits_{i=0}^{M-1}n_{DM,\delta_{0}}(t)C_{PP,\delta_{0}}(t)+\sum\limits _{i=0}^{M-1}n_{DM,\delta_{0}}(t)C_{DD,\delta_{0}}(t)\end{array}\]  
Then we have the Gaussian distribution for \(H_{0}\) as \(Z_{H_{0}}\ \sim\ N\big{(}0,\ 2M\sigma_{n}{}^{4}\big{)}\).  
Suppose that the GPS L1C signal is present, and that the received signal and the generated signal are aligned in phase; the decision variable then becomes:  
\[\begin{array}{ll}H_{1}\ :\ \ Z_{H_{1}}&=\sum_{i=0}^{M-1}r_{DM,\hat{\omega}_{0}}(t_{ i})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD,\hat{\omega}_{0}}(t_{i})\}\\ &=\sum_{i=0}^{M-1}s_{DM,\hat{\omega}_{0}}(t_{i})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD, \hat{\omega}_{0}}(t_{i})\}\\ &+\sum_{i=0}^{M-1}s_{L1C}(t_{i})\cos(2\pi(f_{IF}+f_{D})t_{i})\ n(t_{i}-\delta_{0})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD,\hat{\omega}_{0}}(t_{i})\}\\ &+\sum_{i=0}^{M-1}s_{L1C}(t_{i}-\delta_{0})\cos(2\pi(f_{IF}+f_{D})(t_{i}-\delta _{0})\ n(t_{i})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD,\hat{\omega}_{0}}(t_{i})\}\\ &+\sum_{i=0}^{M-1}n_{DM,\hat{\omega}_{0}}(t_{i})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD, \hat{\omega}_{0}}(t_{i})\}.\end{array}\]  
The first term is given using Equation (16) as follows:  
\[\begin{array}{ll}Z_{H_{1,1}}&=\sum_{i=0}^{M-1}s_{DM,\hat{\omega}_{0}}(t_{ i})\ \{C_{PP,\hat{\omega}_{0}}(t_{i})+C_{DD,\hat{\omega}_{0}}(t_{i})\}\approx\ 3d_{0} \sum_{i=0}^{M-1}C_{PP,\hat{\omega}_{0}}(t_{i})C_{PP,\hat{\omega}_{0}}(t_{i})+ d_{0}\sum_{i=0}^{M-1}C_{DD,\hat{\omega}_{0}}(t_{i})C_{DD,\hat{\omega}_{0}}(t_{i})\\ &=4d_{0}M=c\cos(2\pi f_{D}\delta_{0})M.\end{array}\]  
The second term \(Z_{H_{1,2}}\) has the Gaussian distribution with mean zero and variance \(\sigma^{2}_{Z_{H_{1,2}}}\), which is given as follows using the similar procedure in the proof of Lemma 4:  
\[\sigma^{2}_{Z_{H_{1,2}}}=\sum_{i=0}^{M-1}\ \{S^{2}_{L1C}(t_{i})\text{cos}^{2}(2 \pi(f_{IF}+f_{D})t_{i})\}\{C^{2}_{PP,\hat{\omega}_{0}}(t_{i})+C^{2}_{DD,\hat{ \omega}_{0}}(t_{i})\}\}\sigma^{2}_{n}\approx 2 cM\sigma^{2}_{n}\]  
Similarly, we obtain that \(E\Big{[}Z_{H_{1,3}}\Big{]}=0,\sigma^{2}_{Z_{H_{1,3}}}\approx 2 cM\sigma^{2}_{n}\). Also \(E\Big{[}Z_{H_{1,4}}\Big{]}=0,\ \sigma^{2}_{Z_{H_{1,4}}}=M\sigma^{4}_{n}\). In summary, the decision variable has a Gaussian distribution, as follows:  
\[Z_{H_{1}}\sim N\Big{(}\text{c}\cdot\cos(2\pi f_{D}\delta_{0})M,\ \ \Big{(}2\sigma^{4}_{n}+4c\sigma^{2}_{n}\Big{)}M\Big{)}.\]  
Thus the SNR is obtained as follows:  
\[SNR_{DM}=\frac{E\big{[}Z_{H_{1}}\big{]}^{2}}{\sigma^{2}_{H_{1}}}=\frac{(c\cdot \cos(2\pi f_{D}\delta_{0})M)^{2}}{2(\sigma^{4}_{n}+2c\sigma^{2}_{n})M}=\frac{ \frac{1}{2}\text{:}c^{2}\text{cos}^{2}(2\pi f_{D}\delta_{0})M}{\sigma^{4}_{n} +2c\sigma^{2}_{n}}\]