# Acquisition of Weak BDS Signal in the Lunar Environment
## 2 Materials and Methods
### Signal Acquisition Algorithms  
Traditional acquisition search algorithms that rely on code correlation and carrier correlation typically fall into one of the following categories: serial search, parallel frequency space search, and parallel code phase search [34; 35]. Parallel code phase search operates in parallel on the code phase and is faster than the other two algorithms. Therefore, all the algorithms used in this study are based on this method. The flowchart of the parallel code phase search algorithm is illustrated in Figure 5.  
\(Z(\hat{\tau},\hat{f}_{d})\) denotes the correlation function, where \(\hat{\tau}\) and \(\hat{f}_{d}\) are the estimates of the code phase and Doppler frequency, respectively. Since performing correlation operations on a single C/A code period (1 ms for BDS B1I signal) is not sufficient in a weak signal environment, the integration time must be increased to increase the SNR. An extra correlation operation is carried out on the output value \(Z(\hat{\tau},\hat{f}_{d})\) in Figure 5. Three techniques are used for this purpose and can be classified as follows: (1) coherent integration (COI), (2) non-coherent integration (NCOI), and (3) differential integration (DI). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Intermediate frequency, \(f_{IF}\) (MHz) & 12.276 \\ Sampling frequency, \(f_{s}\) (MHz) & 49.104 \\ Doppler shift, \(f_{d}\) (KHz) & \([-30,30]\) \\ Code Phase, \(\tau\) (chip) & [1; 2046] \\ \(C/N_{0}\) (dB-HZ) & [10; 35] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the BDS B1I IF signal generator. The detection function, \(T\), of the coherent integration method has the following relationship:  
\[T(\hat{\tau},\hat{f}_{d})=|\sum_{m=1}^{M}Z_{m}(\hat{\tau},\hat{f}_{d})|^{2}, \tag{13}\]  
where \(M\) represents the number of performed correlations, i.e., \(M\) millisecond data length for the correlation operation. However, a long coherent integration leads to an increase in the number of fast Fourier transforms (FFIs); thus, the coherent integration expression can also be expressed as [36]:  
\[T(\hat{\tau},\hat{f}_{d}) = |\sum_{m=1}^{M}X_{m}Y^{*}|^{2} \tag{14}\] \[= |Y^{*}\sum_{m=1}^{M}FFT(x_{m})|^{2}\] \[= |Y^{*}FFT(\sum_{m=1}^{M}x_{m})|^{2},in\]  
where \(X_{m}\) represents the FFT of the received signal unit \(x_{m}\). \(Y^{*}\) is obtained by the complex conjugate of the C/A code signal replicated locally through the FFT. It means that the received signal is split into \(M\) blocks of processing units before carrier correlation is performed and then directly summed. The following operations requiring long code coherent integration are performed using the method shown in Equation (14). Due to the presence of a data bit sign transition in the received signal, coherent integration does not grow the coherence gain indefinitely and can even lead to worse results. The non-coherent integration method can be expressed in the following form:  
\[T(\hat{\tau},\hat{f}_{d})=\sum_{m=1}^{M}|Z_{m}(\hat{\tau},\hat{f}_{d})|^{2}, \tag{15}\]  
where the meanings of the symbols in this expression are the same as in Equation (13). Although this method is insensitive to the data bit sign transition, the problem of square loss occurs due to the square operation. In fact, this operation increases the signal power while amplifying the amount of noise at the output. Moreover, combining coherent and non-coherent integration is a powerful approach that can significantly improve the sensitivity of the receiver. Differential integration is usually combined with coherent integration and is called coherent differential integration (CODI). The output noise will be lost as a result of the  
Figure 5: Block diagram of the parallel code phase search algorithm. differential operation because the noise of the correlated outputs are almost independent of each other.