# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## Appendix B  
### The Proof of Lemma 6  
Consider the received signal \(r(t)\) of the GPS L1C signal in Equation (4), and the three cases of generated signal \(g_{conv,i}(t)\). We use the same notation and similar procedure as in the proof of Lemma 4, and the Doppler frequency \(f_{D}=0\) from the assumption:  
\[\begin{array}{ll}(i)\ g_{conv,P}(t)=C_{P}(t)SC_{P}(t)\cos(2\pi f_{IF}t).\\ H_{0}:\ \ Z_{H_{0}}=\sum_{i=0}^{M-1}n(t_{i})C_{P}(t_{i})SC_{P}(t_{i})\cos(2\pi f_{IF}t _{i})\ \ \ E[Z_{H_{0}}]=0,\\ \sigma^{2}_{Z_{H_{0}}}=\sum_{i=0}^{M-1}\ \sigma^{2}_{n}(C_{P}(t_{i}))^{2}(SC_{P}(t_{i}))^{ 2}\text{cos}^{2}(2\pi f_{IF}t_{i})\approx\ \sigma^{2}_{n}\sum_{i=0}^{M-1}1\text{:}1\text{:} \frac{1}{2}=\frac{1}{2}\sigma^{2}_{n}M,\end{array}\]  
that is, \(Z_{H_{0}}\ \sim\ N\Big{(}0,\ \frac{1}{2}\sigma^{2}_{n}M\Big{)}\).  
\[\begin{array}{ll}H_{1}\ :\ Z_{H_{1}}&=\sum_{i=0}^{M-1}S_{L1C}(t_{i})\cos(2\pi f _{IF}t_{i})\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\texttexttexttexttexttexttexttexttexttexttexttext \\[\approx\sum_{i=0}^{M-1}\ \left\{\sqrt{\frac{3}{2}}c\cdot C_{OL}(t_{i})C_{P}^{2}(t_{ i})SC_{P}^{2}(t_{i})\cos^{2}(2\pi f_{IF}t_{i})\right\}+Z_{H_{0}}\]  
\[=\sum_{i=0}^{M-1}\ \left\{\sqrt{\frac{3}{2}}c\cdot\pm 1\cdot 1\cdot 1\cdot\frac{1}{2} \right\}+Z_{H_{0}}\ \ =\pm\sqrt{\frac{3}{8}}M+Z_{H_{0}}.\]  
that is, \(Z_{H_{1}}\ \sim\ N(\pm\sqrt{\frac{3}{8}}M,\ \frac{1}{2}\sigma_{n}^{2}M)\).  
Thus, the SNR is obtained as follows:  
\[SNR_{Conv,P}=\frac{E\left[Z_{H_{1}}\right]^{2}}{\sigma_{H_{1}}^{2}}=\frac{ \frac{3}{4}M}{\sigma_{n}^{2}}.\]  
(ii) \(g_{conv,\sqrt{3}P+D}(\mathrm{t})=\left(\sqrt{3}C_{P}(t)SC_{P}(t)+C_{D}(t)SC_{ D}(t)\right)\cos(2\pi f_{IF}t)\).  
For the \(H_{0}\) hypothesis:  
\[\begin{array}{ll}H_{0}:&Z_{H_{0}}=\sum_{i=0}^{M-1}n(t_{i})\Big{(}\sqrt{3}C_ {P}(t)SC_{P}(t)+C_{D}(t)SC_{D}(t)\Big{)}\cos(2\pi f_{IF}t_{i})\ \ \ E[Z_{H_{0}}]=0,\\ &\sigma_{Z_{H_{0}}}^{2}=\sum_{i=0}^{M-1}\ \sigma_{n}^{2}\Big{(}\sqrt{3}C_{P}(t)SC _{P}(t)+C_{D}(t)SC_{D}(t)\Big{)}^{2}\cos^{2}(2\pi f_{IF}t_{i})\\ &\approx\sigma_{n}^{2}\sum_{i=0}^{M-1}\left\{\Big{(}\sqrt{3}C_{P}(t)SC_{P}(t) \Big{)}^{2}+\left(C_{D}(t)SC_{D}(t)\right)^{2}\right\}\cdot\frac{1}{2}=(3+1) \cdot\frac{1}{2}M\sigma_{n}^{2}=2M\sigma_{n}^{2},\end{array}\]  
we obtain that \(Z_{H_{0}}\ \sim\ N\big{(}0,\ 2\sigma_{n}^{2}M\big{)}\).  
\[H_{1}:\ Z_{H_{1}}=\sum_{i=0}^{M-1}S_{L1C}(t_{i})\cos(2\pi(f_{IF}t_{i})t_{i}) \{\sqrt{3}C_{P}(t_{i})SC_{P}(t_{i})+C_{D}(t_{i})SC_{D}(t_{i})\}\cos(2\pi f_{IF }t_{i})+Z_{H_{0}}\]  
\[\begin{array}{ll}E[Z_{H_{1}}]=\sum_{i=0}^{M-1}\left\{\sqrt{\frac{3}{2}}C_{OL }(t_{i})C_{P}(t_{i})SC_{P}(t_{i})+\sqrt{\frac{3}{2}}D(t_{i})C_{D}(t_{i})SC_{D}( t_{i})\right\}\{\sqrt{3}C_{P}(t_{i})SC_{P}(t_{i})+C_{D}(t_{i})SC_{D}(t_{i})\} \cos^{2}(2\pi f_{IF}t_{i})\\ &\approx\sqrt{\frac{3}{2}}\sum_{i=0}^{M-1}\{S_{L1C}(t_{i})C_{P}^{2}(t_{i})SC_{ P}^{2}(t_{i})+D(t_{i})C_{D}^{2}(t_{i})SC_{D}^{2}(t_{i})\}\frac{1}{2}=\sqrt{\frac{3}{2} \pm 1\}\cdot\frac{1}{2}M\\ \sigma_{Z_{H_{1}}}^{2}=\sigma_{Z_{H_{0}}}^{2}=2\sigma_{n}^{2}M\end{array}\]  
that is, \(Z_{H_{1}}\ \sim\ N(\pm\sqrt{\frac{5}{2}}M,\ 2\sigma_{n}^{2}M)\) or \(N\Big{(}\pm\sqrt{2c}M,\ 2\sigma_{n}^{2}M\Big{)}\)  
Thus, the SNR is obtained as follows:  
\[SNR_{conv,\sqrt{3}P+D}=\frac{E\left[Z_{H_{1}}\right]^{2}}{\sigma_{H_{1}}^{2}}= \frac{\frac{1}{4}c\cdot M}{\sigma_{n}^{2}}\ \text{or}\ \frac{c\cdot M}{\sigma_{n}^{2}}\]  
(iii) \(g_{conv,P+D}(\mathrm{t})=\left(C_{P}(t)SC_{P}(t)+C_{D}(t)SC_{D}(t)\right)\cos(2 \pi f_{IF}t)\).  
By the similar procedure as in (ii), we obtain that:  
\[Z_{H_{0}}\ \sim\ N(0,\ \sigma_{n}^{2}M),\ \text{and}\ \ Z_{H_{1}}\ \sim\ N(\pm\sqrt{\frac{c}{2}}M,\ \sigma_{n}^{2}M)\ \ \text{or}\ \ N(0,\ 2\sigma_{n}^{2}M)\]  
Thus, the SNR is obtained as follows:  
\[SNR_{conv,P+D}=\frac{E\left[Z_{H_{1}}\right]^{2}}{\sigma_{H_{1}}^{2}}=\frac{ \frac{1}{2}c\cdot M}{\sigma_{n}^{2}}\ \text{or}\ 0\]