## 4 Performance Analysis  
In this section, the widely used Add White Gaussian Noise (AWGN) channel model and Monte Carlo method are adopted. We numerically evaluate the simulation results based on \(P_{cd}\) and BER to demonstrate the effectiveness of the proposed algorithm for the acquisition module. Moreover, we compare the performance of our proposed algorithm with the traditional correlation-based acquisition algorithm [26]. This simulation solely focuses on the analysis of interference caused by AWGN and MAI within an ideal Doppler environment. For the transmitters, each user utilizes a 2048-bit m sequence as the spreading code, which is generated by an 11-bit linear shift register. Subsequently, the spread spectrum signal is shaped and filtered using a root ascending cosine filter (\(\alpha=0.25\)). After aliasing, the signals from the \(K\) users are received by the receiver via the AWGN channel. The system encounters multi-path delays, where the \(K\) users do not arrive at the on-board receiver simultaneously, yet possess the same transmit frequency. In the acquisition module, the improved algorithm is used to eliminate the effect of MAI on multi-user receivers. In this analysis, we investigate the impact of the quantity of multi-user parallel communications on the acquisition process. Figure 7 illustrates the results, where an ISR of 20 dB is maintained and the number of users (\(K\)) ranges from 1 to 16. The interference-free scenario is used as a benchmark, represented by a dotted red dashed line. The traditional correlation-based acquisition algorithm is depicted by colored dashed lines. Notably, the \(P_{cd}\) of individual user acquisition is found to be very low, indicating that it fails to meet the requirements of satellite communication due to the presence of MAI. The proposed PIC-based acquisition algorithm, represented by colored solid lines, significantly enhances the \(P_{cd}\) for weak signals across the entire SNR range. Furthermore, no significant degradation in performance is observed for different values of \(K\), with only minor variations evident in the mid-SNR range. In the high-SNR region, all curves achieve \(P_{cd}\) of 94%. In simpler terms, we discuss the impact of parameter ISR on the performance of the PIC-based acquisition algorithm. In Figure 8, we fixed the parameter \(K\) to 6 and varied the ISR from 5 to 30. When the ISR is below 20 dB, the performance of the algorithm is almost  
Figure 7: ISR = 20 dB, \(P_{cd}\) comparison of the proposed acquisition algorithm over AWGN channel. comparable to the theoretical limit. However, as the ISR increases, the performance of the algorithm gradually decreases across the entire SNR range. When the ISR exceeds 25 dB, even the improved algorithm struggles to meet the demands of satellite communication. On the other hand, we also analyze the effect of the parameters \(K\) (number of users) and ISR on the BER of the receiver, assuming successful acquisition. Similar to Figures 7 and 8, Figures 9 and 10 respectively analyze the effects of parameters \(K\) and ISR on the BER. As shown in Figure 9, the ISR is fixed at 20 dB, and \(K\) is varied from 1 to 16. The BER theoretical curve serves as a benchmark, appearing as a dotted red dashed line. The outcomes of the improved and traditional algorithms are categorized into two groups: the improved algorithm is illustrated by colored solid lines, while the traditional algorithm is depicted by colored dashed lines. Curves with identical parameters are distinguished by matching colors and markers. Notably, when \(K\) equals 7, the system becomes greatly influenced by MAI, resulting in a degradation of 1.3 dB in the BER performance compared to the absence of interference. Subsequently, regardless of the increase or decrease in the number of users, the BER performance exhibits improvement. The introduction of the single-stage PIC algorithm elevates the BER of all curves by 0.8 dB. Figure 8: \(K=6\), \(P_{cd}\) of the proposed acquisition algorithm over AWGN channel.