# An Unambiguous Delay-And-Multiply Acquisition Scheme for GPS L1C Signals
## 3 Results
### New DM Acquisition Schemes for GPS LIC Signal and the Unambiguous Property
#### 3.1.2 The Proposed 33-chip Delay DM Acquisition Scheme  
This section provides the two proposed DM acquisition schemes. Figure 5a shows the first 33-chip delay (\(\delta_{0}=33T_{0}\)) DM acquisition scheme. Figure 4: Auto correlation function (ACF) of; (**a**) \(\sqrt{3}SC_{P}(t)\); (**b**) \(SC_{D}(t)\). Figure 5: The proposed 33-chip delay DM acquisition scheme; (**a**) Block diagram; (**b**) The decision variable. The following Lemma shows the unambiguous property of the proposed 33-chip delay DM acquisition scheme in Figure 5a. **Lemma 1**. : _Consider the GPS LIC DM signal \(s_{DM,\lambda_{0}}(t)\) in Equation (10) \((\delta=\delta_{0}=33T_{0})\), and also the generated signal:_  
\[g_{DM,PP}(t)=C_{PP,\lambda_{0}}(t)=C_{\rm P}(t)C_{\rm P}(t-33T_{0})\]  
_then the CCF of \(s_{DM,\lambda_{0}}(t+\tau)\) and \(g_{DM,PP}(t)\) during the coherent integration time \(T_{c}\) = 10 ms is defined as the decision variable \(d_{DM,PP}(\tau)=\int_{0}^{T_{c}}s_{DM,\lambda_{0}}(t+\tau)g_{DM,PP}(t)dt\) in Figure 5a. The CCF can be obtained as:_  
\[d_{DM,PP}(\tau)=3d_{0}(1-|\tau|)(u(\tau+1)-u(\tau-1)) \tag{14}\]  
_where \(d_{0}=\pm\frac{\varsigma}{4}\cos(2\pi f_{D}\delta_{0})\) and \(u(\tau)\) is the unit step function._  
**Proof. ** Considering the optimal time delay \(\sigma_{0}=33T_{0}\), we obtain:  
\[SC_{P}(t)\cdot SC_{P}(t-\delta_{0})=SC_{P}^{2}(t)=1,\,SC_{D}(t)\cdot SC_{D}(t -\delta_{0})=SC_{D}^{2}(t)=1,\,\text{and}\]  
\[ \begin{array}{l}\int_{0}^{T_{c}}\,C_{DP,\lambda_{0}}(t+\tau)C_{PP,\lambda_{ 0}}(t)dt\\ \approx 0,\,\int_{0}^{T_{c}}\,C_{PD,\lambda_{0}}(t+\tau)C_{PP,\lambda_{0}}(t)dt \approx 0,\,\int_{0}^{T_{c}}\,C_{DD,\lambda_{0}}(t+\tau)C_{PP,\lambda_{0}}(t)dt \approx 0.\end{array} \]  
\[ \begin{array}{l}d_{DM,PP}(\tau)\\ =\int_{0}^{T_{c}}\,s_{DM,\lambda_{0}}(t+\tau)g_{DM,PP}(t)dt\\ =\int_{0}^{T_{c}}\,d[3C_{PP,\lambda_{0}}(t)+\tau)\pm\sqrt{3}\,C_{DP,\lambda_{0 }}(t+\tau)SC_{D}(t+\tau)SC_{P}(t+\tau-\delta_{0})\\ \pm\sqrt{3}C_{PD,\lambda_{0}}(t)SC_{P}(t+\tau)\,SC_{D}(t+\tau-\delta_{0})+C_{ DD,\lambda_{0}}(t+\tau)]C_{PP,\lambda_{0}}(t)d\\ \approx 3d_{0}\int_{0}^{T_{c}}C_{PP,\lambda_{0}}(t)+C_{PP,\lambda_{0}}(t)dt \end{array} \]  
Since \(C_{PP,\lambda_{0}}(t)\) is a pseudo random noise (PRN) signal, the \(d_{DM,PP}(\tau)\) is the \(3d_{0}\) times of ACF of \(C_{PP,\lambda_{0}}(t)\), which has the form of Equation (14), and is shown as Figure 5b. \(\square\)  
Figure 6a shows the second 33-chip delay (\(\delta_{0}=33T_{0}\)) DM acquisition scheme. Figure 6: The proposed 33-chip delay DM acquisition scheme; **(a)** Block diagram; **(b)** The decision variable. The following Lemma shows another 33-chip delay DM acquisition scheme as in Figure 6a. **Lemma 2**. : _Consider the GPS LLC DM signal \(s_{DM,33T_{0}}(t)\) in Equation (10) \((\delta=\delta_{0}=33T_{0})\) and also the generated signal:_  
\[\mathcal{S}_{DM,PP+DD}(t)=C_{PP,\delta_{0}}(t)+C_{DD,\delta_{0}}(t)=\ \ C_{P}(t)C_{P}(t-33T_{0})+C_{D}(t)C_{D}(t-33T_{0}). \tag{15}\]  
_then the CCF of \(s_{DM,\delta_{0}}(t+\tau)\) and \(g_{DM,PP+DD}(t)\) during the coherent integration time \(T_{c}=10\) ms is defined as the decision variable \(d_{DM,PP+DD}(\tau)=\int_{0}^{T}s_{DM,\delta_{0}}(t+\tau)g_{DM,PP+DD}(t)dt\) in Figure 6a. The CCF can be obtained as:_  
\[d_{DM,PP+DD}(\tau)=4d_{0}(1-|\tau|)(u(\tau+1)-u(\tau-1)) \tag{16}\]  
_where \(d_{0}=\pm\frac{\xi}{4}cos(2\pi f_{D}\delta_{0})\) and \(u(\tau)\) is the unit step function._  
**Proof. ** Using the similar manipulation as in the Proof of Lemma 1, we obtain that:  
\[