# Adaptive Satellite Navigation Anti-Interference Algorithm Based on Inverse Cosine Function
## 4 Simulation Analysis  
### Performance Verification of The Improved Variable Step Size Algorithm  
To showcase the benefits of the ICVS-LMS algorithm in iterative convergence times, steady-state error control, and tracking capability, as well as the impacts of parameters \(\alpha\), \(\beta\), \(\gamma\), and \(m\) on the properties of the algorithm, the algorithm is simulated in MATLAB R2017b software under the following conditions [20]: let the order of the adaptive filter \(M=2\), and the unknown system is a transverse FIR structure. The tapping coefficients are \(\mathbf{W}^{*}=[0.8,0.5]^{T}\). At the 500 th iteration, the system is time-varying, and the tap coefficients change abruptly to \(\mathbf{W}^{*}=[0.4,0.2]^{T}\). Assuming that the reference input signal \(\mathbf{X}(n)\) and the interference noise \(v(n)\) are both Gaussian white noise with zero mean, the variance of the reference input signal \(\mathbf{X}(n)\) is 1 and the variance of the interference noise \(v(n)\) is 0.04. The number of sampling points of the system is set to 1000, and 200 independent simulations are performed for each algorithm to achieve the statistical average and the learning curve against different signal-to-noise ratio (SNR) backgrounds. Figure 4 illustrates the convergence curves of the four algorithms. Looking at the figure, it is evident that when faced with three different noise backgrounds (SNR values of 20, 30, and 40, respectively), the VVS-LMS algorithm requires the highest number of iterations to converge. On the other hand, the proposed ICVS-LMS algorithm demonstrates the fewest iterations during the initial stage of convergence while maintaining a relatively low steady-state error, which can effectively address the challenges of accelerating convergence and handling large errors encountered in the LMS algorithm. The steady-state error is effectively suppressed at a low level by the proposed ICVS-LMS algorithm under three noise backgrounds, and the benefits of lower iterative convergence times and low error of the proposed ICVS-LMS algorithm become more obvious as the SNR continues to increase. To underline the superiority of the proposed ICVS-LMS algorithm, we compare the convergence performance of the four algorithms across different signal-to-noise ratio (SNR) levels by examining their respective iteration counts, as detailed in Table 1. When the SNR is set at 20 dB, the ICVS-LMS algorithm achieves convergence within just 30 iterations, outperforming the LVS-LMS algorithm, which requires 170 iterations, and significantly surpassing both the VVS-LMS and NDVS-LMS algorithms, both of which demand more than 350 iterations. At an increased SNR of 30 dB, the ICVS-LMS algorithm attains convergence in 60 iterations, while the LVS-LMS algorithm necessitates 200 iterations. Moreover, the VVS-LMS and NDVS-LMS algorithms exhibit considerably higher iteration counts at 400 and 380 iterations, respectively. As the SNR escalates to 40 dB, the ICVS-LMS algorithm achieves convergence within 100 iterations, once again surpassing the LVS-LMS algorithm's requirement of 260 iterations. Additionally, the VVS-LMS and NDVS-LMS algorithms maintain convergence behavior above 400 iterations. To validate the traceability and convergence capability of the proposed ICVS-LMS algorithm, as well as its comparison against existing algorithms, we tested the algorithm when dealing with time-varying system occurrences. When the algorithm iterative process reaches 500 iterations, the unknown system occurs in a time-varying manner, and Figure 5 shows the algorithm's ability to cope with the time-varying system under different SNR. Figure 4: Comparison of algorithm under different SNR values: (**a**) SNR = 20 dB; (**b**) SNR = 30 dB; (**c**) SNR = 40 dB.