# A Fast Acquisition Algorithm for Hybrid Signals of 5G and BeiDou B1
## 2 Proposed Method
### Receiver Acquisition Algorithm Design
#### 2.2.2 Receiver Hybrid Coherent Acquisition Algorithm Design  
In the signal processing session, the acquisition process is the first stage of the hybrid signal processing, aiming to determine the presence of the detected signal and to obtain a rough estimate of the residual carrier frequency and code delay of the received signal, providing the necessary parameters and conditions for the initialization of the subsequent tracking stages. The acquisition's accuracy also directly impacts the tracking performance [34], such as the loop convergence time. On the other hand, the acquisition process is one of the more time-consuming phases of baseband processing [35]. Therefore, hybrid signal reception systems must have high accuracy and fast acquisition in a weak-signal environment. In order to achieve high accuracy and fast acquisition in weak signals, we propose an SC-PMF-FFT (strong correlation partial matched filters fast Fourier transformation) acquisition algorithm based on convolutional blocks with a priori information, starting from search domain evaluation and detection. The structure of the implementation of this algorithm is shown in Figure 7. The SC-PMF-FFT algorithm consists of the signal preprocessing module, the block convolution-based code phase deblurring fast-matching filtering module, and the a priori information-based correlation integral judgment module. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **BeiDou B1 (MHz)** & **5G n50 (MHz)** \\ \hline RF & 1561.098\(\sim\)1575.42 & 1427\(\sim\)1517 \\ Baseband IF & 20.46 & 120 \\ Local Oscillator Frequency & 1540.638 & 1592 \\ Baseband BandWidth & 4.092 & 90 \\ Baseband BandWidth/2 & 2.046 & 45 \\ RF Range & 1559.052\(\sim\)1577.466 & 1427\(\sim\)1517 \\ Baseband IF Range & 18.414\(\sim\)22.506 & 75\(\sim\)165 \\ ADC Sampling Rate & 20.46 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Hybrid signal frequency allocation. where \(f_{s}\) is the sampling frequency, and \(f_{ds}\) is the downsampling rate. Performing coherent integration on \(D_{I\_ds}(n)\) and \(D_{Q\_ds}(n)\):  
\[ \begin{array}{l}I_{i}(n)=\sum_{k=0}^{PMF\_FFT\_X}D_{I\_ds}(k+n\times PMF\_FFT\_X )C(PMF\_FFT\_D-1-k)\\ Q_{i}(n)=\sum_{k=0}^{PMF\_FFT\_X}D_{Q\_ds}(k+n\times PMF\_FFT\_X)C(PMF\_FFT\_D-1-k) \end{array}  \tag{13}\]  
where \(PMF\_FFT\_X\) is the number of data points for partial coherent integration, \(PMF\_FFT\_D\) is the number of blocks, and \(C\) function is the local pseudocode sequence. Performing FFT processing on \(I_{i}\) and \(Q_{i}\), the code phase points can be obtained at the correlation peaks:  
\[\theta_{C}=\frac{NUM\_PN}{f_{s}\times T_{c}}\times n \tag{14}\]  
where \(\theta_{C}\) is the pseudocode phase point. A pseudocode period consists of \(NUM\_PN\) code chips. \(f_{s}\) is the sampling rate, \(T_{c}\) is the duration of coherent integration, and \(n\) corresponds to the number of FFT computations at the correlation peak. Similarly, the carrier Doppler value can be obtained as:  
\[f_{d}=\left\{ \begin{array}{cc}m\times\frac{f_{ds}}{PMF\_FFT\_X}\times PMF\_FFT \_N},&m<\frac{PMF\_FFT\_X}{2}\\ (m-PMF\_FFT\_N)\times\frac{PMF\_FFT\_X}{PMF\_FFT\_X},&m>\frac{PMF\_FFT\_X}{2} \end{array} \right. \tag{15}\]  
where \(f_{d}\) refers to the carrier Doppler value, \(f_{ds}\) is the downsampling rate, \(PMF\_FFT\_X\) is the number of data points for partial coherent integration (accumulation), and \(PMF\_FFT\_D\) is the number of points for FFT computation. The obtained \(\theta_{C}\) and \(f_{d}\) are output and passed on to the B1C carrier compensation module and the B1C pilot code phase fixed delay module. The formula for generating the local pilot code in B1C is as follows:  
\[ \begin{array}{l}W(k;w)=L(k)\oplus L((k+w)\text{ mod }N),k=0,1,2, \cdots,N-1\\ L(k)=\left\{\begin{array}{l}0,k=0\\ 1,k\
eq 0,\text{ exists a positive integer }x\text{ such that }k=x^{2}\text{ mod }N\\ 0,\text{ others}\end{array}\right.\end{array}  \tag{16}\]  
where \(L\) is a Legendre sequence, and \(w\) is the phase parameter.