# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## 3 Results
### Noise Characteristics of the Proposed DM and Conventional Acquisition Method
#### 3.2.3 Comparison of the SNR Performance between DM and Conventional Methods  
This section summarizes the SNR ratio in the presence of L1C signal for the DM and conventional methods from the results of Lemmas 4-6. Table 1 compares the performance with each other. For the two DM methods, the SNR of \(g_{DM,PP}\) is greater than that of \(g_{DM,PP+DD}\), while the mean value, which is the correlation peak, of \(g_{DM,PP+DD}\) is greater than that of \(g_{DM,PP}\). In this case the use of \(g_{DM,PP}\) provides greater probability of detection than that of \(g_{DM,PP+DD}\). Thus, we can conclude that the DM method of \(g(t)=g_{DM,PP}(t)\) is preferable to obtain better performance such as the probability of detection. In similar way, \(g(t)=g_{conv,P}(t)\) is the best among the three candidates in the conventional method.  
As mentioned in Section 3.1.1, the optimal delay \(\delta_{0}=33T_{0}\) is long and thus we cannot ignore the effect of the Doppler frequency such that \(0.5\leq\lvert\cos(2\pi f_{D}\delta_{0})\rvert\leq 1\), for \(-5167\) Hz \(\leq f_{D}\leq 5167\) Hz, which is the maximum Doppler frequency when a GPS receiver moves at the speed of 198 km/h. The Doppler frequency is slowly varying and thus the cosine value in Table 1 can be treated as a constant for the acquisition process. The following Corollary shows the ratio of SNR between the DM method and the conventional method.  
Figure 7: The conventional acquisition scheme.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{4}{c}{**Delay-And-Multiply Method**} & \multicolumn{2}{c}{**Conventional Method**} \\ \hline \(g(t)\) & \(g_{\text{DM},PP}\) & \(g_{\text{DM},PP+DD}\) & \(g_{\text{comp},P}\) & \(g_{\text{comp},\sqrt{P}+D}\) & \(g_{\text{comp},P+D}\) \\ \hline \(Z_{H_{b}}\) & \(N(0,\ M\sigma_{a}^{2})\) & \(N(0,\ 2M\sigma_{a}^{2})\) & \(N(0,\ \frac{M}{2}\sigma_{a}^{2})\) & \(N(0,\ 2M\sigma_{a}^{2})\) & \(N(0,\ M\sigma_{a}^{2})\) \\ \hline \(Z_{H_{1}}\) & \(N\Big{(}\frac{N}{2}\cos(2\pi f_{D}b_{3})M,\ \left(\sigma_{a}^{2}+2\omega_{a}^{2} \right)M\Big{)}\) & \(N\Big{(}\zeta\cos(2\pi f_{D}b_{3})M,\ \left(2\sigma_{a}^{2}+4\omega_{a}^{2} \right)M\Big{)}\) & \(N\Big{(}\pm\sqrt{\frac{N}{2}}M,\ \frac{1}{2}\sigma_{a}^{2}M\Big{)}\) & \(N\Big{(}\pm\sqrt{\frac{N}{2}}M,\ 2\sigma_{a}^{2}M\Big{)}\) or \(N\Big{(}\pm\sqrt{\frac{N}{2}}M,\ 2\sigma_{a}^{2}M\Big{)}\) & \(N\Big{(}\pm\sqrt{\frac{N}{2}}M,\ 2\sigma_{a}^{2}M\Big{)}\) & \(N\Big{(}\pm\sqrt{\frac{N}{2}}M,\ 2\sigma_{a}^{2}M\Big{)}\) & \(N\Big{(}0,\ \sigma_{a}^{2}M\Big{)}\) \\ \hline SNR for \(Z_{H_{1}}\) & \(\frac{\frac{N}{2}\cdot\zeta^{\text{DM}}(2\pi f_{D}b_{3})M}{\frac{M}{2}+2\omega _{a}^{2}}\) & \(\frac{1}{2}\cdot\zeta^{\text{DM}}(2\pi f_{D}b_{3})M\) & \(\frac{1}{2}\cdot\zeta^{\text{DM}}\) & \(\frac{1}{2}\cdot\zeta^{\text{DM}}\) or \(\frac{\zeta\text{DM}}{\sigma_{a}^{2}}\) & \(\frac{1}{2}\cdot\zeta^{\text{DM}}\) or \(0\) \\ \hline preference & \(\mathbf{o}\) & \(\mathbf{o}\) & & \(\mathbf{o}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of the performance between the DM method and the conventional method.  
**Corollary 1**.: _Consider the SNR of the decision variable for the DM acquisition method \(SNR_{DM,PP}\) and the SNR of the decision variable for the conventional acquisition method \(SNR_{conv,P}\). Then, \(SNR_{DM,PP}\) is always less than \(SNR_{conv,P}\), and the SNR ratio is given as Equation (23):_  
\[\frac{SNR_{DM,PP}}{SNR_{conv,P}}=\frac{\frac{3}{4}c\text{-}\cos^{2}(2\pi f_{D} \delta_{0})}{\sigma_{n}^{2}+2c}. \tag{23}\]