# 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
 \tag{20}\]  
To eliminate the influence of the initial phase difference \(\varphi_{0}\), the squares of \(I_{p}\) and \(I_{q}\) are summed together:  
\[I_{p}^{2}+Q_{p}^{2}=\frac{P_{X}}{2}T^{2}\,\text{sinc}^{2}(\pi\Delta fT)R^{2}( \Delta\tau)+N\sim\chi^{2} \tag{21}\]  
\(N\) is the noise term, and the result of the addition follows a non-central chi-square distribution with 2 degrees of freedom. When the correlation peak value exceeds the threshold, it is considered that the correct correlation peak has been found. The capture probability can be expressed as:  
\[P=\int_{\sqrt{th}}^{+\infty}\frac{z}{\sigma^{2}}e^{-\frac{T^{2}P_{X}/2+z^{2}}{ 2\sigma^{2}}}I_{0}\Big{[}\frac{\sqrt{P_{X}/2}Tz}{\sigma^{2}}\Big{]}dz \tag{22}\]  
where \(I_{0}\) is the zeroth-order Bessel function of the first. The correlation integral judgment module based on a priori information utilizes other short-code frequency signals broadcast on the same satellite to acquire long-code pilot signals. Then, it passes the acquired results to the digital transmission channel. The principle and implementation structure of the algorithm is shown in Figure 8. The algorithm is based on the premise that the frequency signal is broadcast on the same satellite, the two signals are almost identical in terms of propagation delay, and the modulated frequencies are adjacent, as well as containing the same on-board reference clock and its propagation path to the receiver antenna, resulting in a strong correlation between the two signals, which can be used to acquire the B1I signal, and then assist in the acquisition of the B1C signal after resolving the phase ambiguity. The overall effectiveness of the algorithm is higher than the direct capture of B1C, and it avoids introducing multi-peak interference in the acquisition session. The SC-PMF-FFT algorithm involves the configuration of many parameters, which are summarized here for the B1 and 5G signals, respectively, as well as the default values, as shown in Table 9.  
Some of the parameters in the table have constraint relationships with each other, which are discussed as follows:  
1. _SC_PMF_FFT_X_ determines the FFT calculation resolution = downsampling rate \(\div\)\(SC\_PMF\_FFT\_X\) \(\div\)\(SC\_PMF\_FFT\_N\). 2. _SC_PMF_FFT_D_ = (downsampling rate \(\times\) coherent integration time) \(\div\)\(SC\_PMF\_FFT\_X\), but with \(SC\_PMF\_FFT\_D\)\(\leq\)\(SC\_PMF\_FFT\_N\). 3. _CNT_CAL_PER_RAM_ = operating clock \(\div\) (downsampling rate \(\times\) coherent integration time ), with the result rounded down. 4. _NUM_RAM_SWAP_ = downsampling rate \(\times\) coherent integration length \(\div\)\((NUM\_CORRELATOR\times CNT\_CAL\_PER\_RAM)\), the result is rounded upwards, the reasonable choice of \(NUM\_CORRELATOR\) and \(CNT\_CAL\_PER_RAM\), there will be no phase secondary alignment problems. Otherwise, there may be two related peaks. Otherwise, two correlation peaks may occur. The SC-PMF-FFT workflow is discussed in detail: Firstly, the local pseudocode goes through the \(NUM\_CORRELATOR\) stage flow, and the relative data slides through \(NUM\_CORRELATOR\) phase-detection points. The data and pseudocode are fed to \(NUM\_CORRELATOR\) correlators simultaneously. These parallel correlators work simultaneously under the control of integration pulses, outputting one partial coherent integration result per \(PMF\_FFT\_X\) clock, and each correlator outputs \(PMF\_FFT\_D\) coherent integration results in one complete operation. In one complete operation, all correlators output a total of \(PMF\_FFT\_D\)\(\times\)\(NUM\_CORRELATOR\) coherent integration results; these results  
Figure 8: Obtaining the code phase ambiguity point of B1C from the a priori information of the B1I signal.