# 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
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **BeiDou B1** & **5G n50** \\ \hline Operating Clock & 120 MHz & 120 MHz \\ Downsampling Rate & 30 MHz & 90 MHz \\ Coherent Integration Time & 1 ms & 5 ms \\ Number of Partial Coherent Integration (\(SC\_PMF\_FFT\_X\)) & 600 & 3000 \\ Number of Chunks (\(SC\_PMF\_FFT\_D\)) & 20 & 100 \\ Number of Points of FFT (\(SC\_PMF\_FFT\_N\)) & 128 & 128 \\ Number of Parallel Correlators (\(NUM\_CORRELATOR\)) & 60 & 300 \\ Number of Parallel Correlators within Each Correlation Integration (\(CNT\_CAL\_PER\_RAM\)) & 5 & 1 \\ Number of RAM Switches of the Data within the Correlation Integration (\(NUM\_RAM\_SWAP\)) & 40 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: SC-PMF-FFT algorithm parameters. are stored in the Bing Bang cache according to a particular organization, the read and write operations of the Bing Bang cache do not conflict, and the data read in are read out after the complementary zero operation for the FFT operation of \(PMF\_FFT\_N\) points. The FFT results and synchronization signals are sent to the peak detection circuit at the back. Depending on the timing margin, the above operation process can be performed \(CNT\_CAL\_PER\_RAM\) times within each coherent integration duration, that is, within one coherent integration duration, \(CNT\_CAL\_PER\_RAM\times NUM\_CORRELATOR\) phase-detection points can be searched, and to thoroughly search all phase points of the data within one coherent integration duration \(NUM\_RAM\_SWAP\) sub-RAM switching operation is required. Figure 9 shows the timing diagram of the workflow described above. Below, we present an analysis of the computational expenses associated with the receiver's operation employing the SC-PMF-FFT capture algorithm during the capture phase. The computation cost is quantified as the aggregate of multiplier and adder computations performed by the receiver throughout the capture process. The computational burden imposed by the SC-FFT capture algorithm mainly manifests in three distinct phases: downsampling, coherent integration, and FFT. In the downsampling process, considering the inherent multiplication operations involved in sampling, an analysis with Equation (12) indicates \(\frac{2f_{\text{ds}}}{f_{s}}\) multiplications and \(\frac{2f_{\text{ds}}}{f_{s}}\) additions were performed. In coherent integration, analysis with Equation (12) indicates \(2\times PMF\_FFT\_X^{2}\times PMF\_FFT\_D\) multiplications and \(2\times PMF\_FFT\_X^{2}\times PMF\_FFT\_D\) additions were performed. Where \(PMF\_FFT\_X\) is the number of data points for partial coherent integration and \(PMF\_FFT\_D\) is the number of blocks. For a radix-2 FFT, the number of complex multiplication operations \(\alpha_{1}\) and complex addition operations \(\beta_{1}\) can be expressed as follows:  
\[\left\{ \begin{array}{c}\alpha_{1}=N_{FFT}/2\log_{2}(N_{FFT})\\ \beta_{1}=N_{FFT}\log_{2}(N_{FFT})\end{array} \right. \tag{23}\]  
One complex multiplication operation is equivalent to four real multiplications and two additions, while one complex addition operation is equivalent to two real additions. Figure 9: (**a**) One parallel correlator operation timing. (**b**) Timing of operations within a correlation integration time. (**c**) Timing of one complete full-phase search process. Thus, the number of multiplication operations \(\alpha_{2}\) and addition operations \(\beta_{2}\) can be expressed as follows:  
\[\left\{ \begin{array}{l}\alpha_{2}=2N_{FFT}\log_{2}(N_{FFT})\\ \beta_{2}=3N_{FFT}\log_{2}(N_{FFT})\end{array} \right. \tag{24}\]  
The analysis shows that the number of multiplication operations is \(2\times PMF_{-}FFT_{-}D\times PMF_{-}FFT_{-}L\times\log_{2}PMF_{-}FFT_{-}L\) and the number of addition operations is \(3\times PMF_{-}FFT_{-}D\)\(\times PMF_{-}FFT_{-}L\times\log_{2}PMF_{-}FFT_{-}L\) in the FFT operation. Where \(PMF_{-}FFT_{-}L\) is the number of points for FFT computation. Table 10 is the computation cost analysis of SC-PMF-FFT algorithm.