# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## 3 Results
### Noise Characteristics of the Proposed DM and Conventional Acquisition Method  
#### 3.2.1 Noise Characteristics of the Proposed DM Acquisition Method  
In this section, we show the SNR of the decision variables for the proposed DM acquisition methods, which are shown in Figures 5a and 6a.  
**Lemma 4**: _Consider the proposed DM acquisition scheme in Figure 5a, where the generated signal is \(g_{DM,PP}(t)=C_{PP,\delta_{0}}(t)=C_{P}(t)C_{P}(t-33T_{0})\). The SNR of the decision variable in the presence of GPS LIC signal can be approximated as Equation (19):_  
\[SNR_{DM,PP}=\frac{\frac{9}{16}\cdot c^{2}\{\cos(2\pi f_{D}\delta_{0})\}^{2}M}{ \sigma_{n}^{4}+2c\cdot\sigma_{n}^{2}} \tag{19}\]  
_where \(\delta_{0}=33T_{0}\) is the optimal delay in DM signal, \(M\) is the number of samples according to the appropriate sampling frequency and the coherent integration period \(T_{c}\), and \(\sigma_{n}^{2}\) is the variance of the noise \(n(t)\) in Equation (4)._  
**Proof.** Denote \(H_{0}\) as the hypothesis in the absence of GPS LIC signal, and \(H_{1}\) in the presence of the signal, and \(Z_{H_{0}}\) and \(Z_{H_{1}}\) as the decision variables in the case of \(H_{0}\) and \(H_{1}\), respectively. Suppose that the optimal time delay is \(\delta_{0}=33T_{0}\), and the generated signal is \(g_{DM,PP}(t)=C_{PP,\delta_{0}}(t)=C_{P}(t)C_{P}(t-33T_{0})\), then we obtain the decision variables for \(H_{0}\) and \(H_{1}\) as follows:  
\[H_{0}\ :Z_{H_{0}}=\sum_{i=0}^{M-1}n(t_{i})n(t_{i}-33T_{0})C_{PP,\delta_{0}}(t)\]  
Denote \(n_{DM,\delta_{0}}(t)=n(t)n(t-33T_{0})\), then it is known that \(E\big{[}n_{DM,\delta_{0}}\big{]}=0,\ \sigma_{n_{DM,\delta_{0}}}^{2}=\sigma_{n}^{4}\)[19].  
The signal \(n_{DM,\delta_{0}}(t)\) is actually a random process, and thus \(n_{DM,\delta_{0}}(t_{i})\), \(i=0\), \(1,2,\cdots\), \(M-1\), are random variables that are independent and identically distributed. Thus by the central limit theorem, \(Z_{H_{0}}\) has the Gaussian distribution with mean zero and variance, \(M\sigma_{n}\)\({}^{4}\), that is:  
\[H_{0}:\quad Z_{H_{0}}\ \sim\ N\Big{(}0,\ M\sigma_{n}\ ^{4}\Big{)}\]  
Suppose that the GPS LIC signal is present, and that the received signal and the generated signal are aligned in phase, then the decision variable becomes:  
\[\begin{array}{l}H_{1}\ :\ Z_{H_{1}}=\sum\limits_{i=0}^{M-1}r_{DM,\delta_{0}}(t_ {i})C_{PP,\delta_{0}}(t_{i})\\ =\sum\limits_{i=0}^{M-1}s_{DM,\delta_{0}}(t_{i})\ C_{PP,\delta_{0}}(t_{i})+ \sum\limits_{i=0}^{M-1}S_{1\mathrm{C}}(t_{i})\cos(2\pi(f_{\mathrm{IF}}+f_{D})t _{i})\ n(t_{i}-\delta_{0})C_{PP,\delta_{0}}(t_{i})\\ +\sum\limits_{i=0}^{M-1}S_{L\mathrm{IC}}(t_{i}-\delta_{0})\cos(2\pi(f_{ \mathrm{IF}}+f_{D})(t_{i}-\delta_{0})\ n(t_{i})C_{PP,\delta_{0}}(t_{i})\\ +\sum\limits_{i=0}^{M-1}n_{DM,\delta_{0}}(t_{i})C_{PP,\delta_{0}}(t_{i})\end{array} \tag{20}\]  
**Lemma 6**.: _Consider the conventional acquisition scheme in Figure 7, and assume that we know the Doppler frequency, and thus we use \(f_{D}=0\). Note that the generated signal is \(g_{conv,i}(t)\). We consider three cases of generated signals, which are:_  
\[\begin{array}{l}g_{conv,P}(t)=C_{P}(t)SC_{P}(t)\cos(2\pi f_{IF}t)\\ g_{conv,\sqrt{3}p+D}(t)=\left\{\sqrt{3}C_{P}(t)SC_{P}(t)+C_{D}(t)SC_{D}(t) \right\}\cos(2\pi f_{IF}t)\\ g_{conv,P+D}(t)=\left\{C_{P}(t)SC_{P}(t)+C_{D}(t)SC_{D}(t)\right\}\cos(2\pi f_ {IF}t)\end{array}\]  
The SNR of the decision variable in the presence of the GPS L1C signal can be approximated for the three cases as (22):  
\[SNR_{conv,P}=\frac{\frac{3}{4}c\cdot M}{\sigma_{n}^{2}},SNR_{conv,\sqrt{3}p+ D}=\frac{\frac{1}{4}c\cdot M}{\sigma_{n}^{2}},\qquad SNR_{conv,P+D}=\frac{ \frac{1}{2}c\cdot M}{\sigma_{n}^{2}}\text{ or }0 \tag{22}\]  
_where, \(M\) is the number of samples according to the sampling frequency and the coherent integration period, and \(\sigma_{n}^{2}\) is the variance of the noise \(n(t)\) in Equation (4)._  
The proof of Lemma 6 has similar procedure as that of Lemma 4, and thus is omitted here and given in Appendix B.