# 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
The two segments of Legendre sequences, each with a length of 10,230, are truncated under the control of phase parameter \(w\). They are XORed to the desired satellite pilot code primary code sequence. After modulation with a 1.023 MHz subcarrier, the output is obtained. The code chips are sampled by a clock a downsampling rate \(f_{ds}\). These samples are sequentially stored Shift-FIFO circuit structures, each with a sampling depth of \((f_{ds}/1.023M)\times 1023\), resulting in a fixed 1 ms code phase delay in adjacent taps. The input B1C intermediate-frequency passband communication signal is:  
\[S_{X}(t)=\sqrt{2P_{X}}[s_{X1}(t)\cos(2\pi f_{X}t+\varphi_{0})-s_{X2}(t)\sin(2 \pi f_{X}t+\varphi_{0})] \tag{17}\]  
where \(P_{X}\) is the power of the intermediate-frequency signal, \(f_{X}\) is the intermediate frequency, \(\varphi_{0}\) is the initial phase of the intermediate-frequency signal, and \(S_{X}\) is the baseband signal. \(s_{X1}(t)\) refers to the real part of the baseband signal, \(s_{X2}(t)\) refers to the imaginary part of the baseband signal. \(S_{X}\) can be expressed as:  
\[ \begin{array}{l}s_{X}(t)=\frac{1}{2}D_{B1C\_data}(t)\cdot C_{B1C\_data}(t) \cdot\text{sign}(\sin(2\pi f_{sc\_B1C\_a}t))\\ +\sqrt{\frac{1}{11}}C_{B1C\_pilot}(t)\cdot\text{sign}(\sin(2\pi f_{sc\_B1C\_ b }t))\\ +j\sqrt{\frac{29}{44}}C_{B1C\_pilot}(t)\cdot\text{sign}(\sin(2\pi f_{sc\_B1C\_ a }t))\end{array}  \tag{18}\]where the first term represents the data component, the second term represents the pilot \(b\) component, and the third term represents the pilot \(a\) component. The output after digital downconversion (DDC) circuit processing is:  
\[ \begin{split} S_{I}&=\frac{\sqrt{2P_{X}}}{2}s_{X1}(t )[\cos(2\pi(f_{X}+f_{I})t+\varphi_{0})+\cos(2\pi\Delta ft+\varphi_{0})]\\ &-\frac{\sqrt{2P_{X}}}{2}s_{X2}(t)[\sin(2\pi(f_{X}+f_{I})t+ \varphi_{0})+\sin(2\pi\Delta ft+\varphi_{0})]\\ S_{Q}&=\frac{\sqrt{2P_{X}}}{2}s_{X1}(t)[\sin(2\pi(f _{X}+f_{I})t+\varphi_{0})-\sin(2\pi\Delta ft+\varphi_{0})]\\ &+\frac{\sqrt{2P_{X}}}{2}s_{X2}(t)[\cos(2\pi(f_{X}+f_{I})t+ \varphi_{0})-\cos(2\pi\Delta ft+\varphi_{0})]\end{split}  \tag{19}\]  
where \(f_{I}\) is the nominal intermediate frequency plus the B1C Doppler compensation, \(\Delta f\) is the difference between the actual intermediate frequency and \(f_{I}\). Therefore, \(\Delta f\) is an approximate value close to zero. The block convolution-based code phase deblurring fast-matching filtering module is an acquisition algorithm for CDMA modulation methods. By adjusting the parameters of the algorithm, a balance between algorithm accuracy and execution time can be obtained. However, the algorithm is essentially only a parallel frequency algorithm, and the search for the code phase is still performed serially. This design uses multiple sets of correlators working simultaneously to compensate for this drawback. These correlators simultaneously detect multiple code phase points, playing the role of parallel code phase search. Suppose the local code phase is aligned with the code phase of the input data. In that case, the algorithm will obtain a correlation peak acquired by the subsequent detection module. The number of FFT runs and frequency points corresponding to this peak will be obtained. The number of FFT runs indicates the code phase position, and the frequency point corresponding to the correlation peak in the FFT is the Doppler frequency. The peak detection module will also estimate the signal noise power. The specific formula is as follows:  
\[ \begin{split} I_{p}&=\frac{\sqrt{2P_{X}}}{2}T\, \text{sinc}(\pi\Delta fT)R(\Delta\tau)\sin(\pi\Delta fT+\varphi_{0})+N_{I}& N_{I}\sim N\Big{(}0,\sigma^{2}\Big{)}\\ Q_{p}&=\frac{\sqrt{2P_{X}}}{2}T\,\text{sinc}(\pi \Delta fT)R(\Delta\tau)\cos(\pi\Delta fT+\varphi_{0})+N_{Q}& N_{Q}\sim N \Big{(}0,\sigma^{2}\Big{)}\end{split}