## 3 The Acquisition Method Based on PIC
### Design of the Hard Reconstruction Module  
As shown in Figure 5, the hard decision module and the reconstruction module serve as the fundamental pillars of the aforementioned PIC-based acquisition algorithm. The reconstructed signal directly affects the effectiveness of interference cancellation. In this paper, we propose a method to estimate the amplitude of strong signals based on the acquisition results, which can augment the efficacy of PIC in eliminating interference signals.  
First, the hard decision module necessitates estimating the amplitude of the received signal. So, we discuss the the correlation between the reconstructed signal sequence and the received signal. The reconstructed sequence is delineated as  
\[x(k)=Re_{coef}\times x(k), \tag{5}\]  
where \(x(k)=b_{n}c_{n}\) is the non-data-modulated spread spectrum sequence, and \(Re_{coef}\) is the predicted coefficient of the reconstructed sequences.  
Figure 6: Block diagram of the coherent accumulation acquisition algorithm.  
The power of the reconstructed signal is  
\[\hat{P_{Re}}=Re_{coef}^{2}\times\sum\limits_{k=0}^{N-1}x^{2}(k). \tag{6}\]  
The objective of the reconstruction module is to minimize the disparity between the estimated parameter \(Re_{coef}\) and the received signal amplitude \(A_{n}\). Thus, we direct our focus towards the aforementioned parameters. The results of the acquisition algorithm manifest as a correlation between the received signal and the localized PRN code, which is defined as  
\[z(n,m)=\sum\limits_{k=0}^{N-1}x(k)y_{in}(k-n,m). \tag{7}\]  
In the event of a successful acquisition, there are values \(N\) and \(M\) such that \(z(N,M)\) achieves the maximum value denoted as \(z_{max}\). Precisely, \(z_{max}\) indicates the position where the local PRN code sequence exhibits the highest correlation value with the received signal sequence:  
\[\begin{split} z_{max}&=\sum\limits_{k=0}^{N-1}x(k)y _{in}(k)\\ &=\sum\limits_{k=0}^{N-1}x(k)[A_{n}x(k)+n_{0}]\\ &=\sum\limits_{k=0}^{N-1}A_{n}x^{2}(k)+\sum\limits_{k=0}^{N-1}x( k)n_{0}.\end{split} \tag{8}\]  
The power information of the reconstructed signal is further derived from the above equations:  
\[A_{n}=\frac{z_{max}-\sum\limits_{k=0}^{N-1}x(k)n_{0}}{\sum\limits_{k=0}^{N-1} x^{2}(k)}. \tag{9}\]  
Since Gaussian white noise is theoretically uncorrelated with the local PRN code sequence:  
\[\sum\limits_{k=0}^{N-1}n_{0}\times x(k)=N\times E[n_{0}\times x(k)]=N\times E[ n_{0}]\times E[x(k)]=0. \tag{10}\]  
Combining the above derivations, \(Re_{coef}\) is expressed as  
\[Re_{coef}=\frac{z_{max}}{\sum\limits_{k=0}^{N-1}x^{2}(k)}. \tag{11}\]  
This paper further introduces coherent accumulation to augment the correlation peaks of the received signal with the local PRN code. The magnitude of the correlation peaks is directly proportional to the accumulation of frames. Therefore, the above equation can be further modified as  
\[Re_{coef}=\frac{z_{max}}{Acc_{num}\times\sum\limits_{k=0}^{N-1}x^{2}(k)}, \tag{12}\]  
where \(Acc_{num}\) indicates the quantity of frames accumulated.  
During our preliminary research, we discovered that the duration of a complete up-link DSSS signal packet, including the non-data-modulated section, data segment,and other components, is less than one second [1]. During the interval of signal continuity, the amplitude of the signal remains relatively stable, thereby enabling us to apply the amplitude estimation result derived from this method to the entire data packet.  
Combining Equations (5) and (12), the reconstructed sequence used in this paper is  
\[x(\hat{k})=\frac{z_{max}}{Acc_{num}\times\sum\limits_{k=0}^{N-1}x^{2}(k)}\times x (k). \tag{13}\]