# Adaptive Satellite Navigation Anti-Interference Algorithm Based on Inverse Cosine Function
## 3 Improved Variable Step Size LMS Algorithm  
### Improved Inverse Cosine Variable Step Size LMS Algorithm  
Achieving convergence in a short time while maintaining a low level of steady-state error are two crucial evaluation metrics of the algorithm and, in order to achieve both, experts and scholars have studied and proposed various new algorithms [36]. When the variable step size LMS algorithm has not yet reached the convergence state, it will choose an outsize step size value to shorten the time spent on convergence. When the algorithm enters the convergence period, the step size value gradually becomes smaller, maintaining a low level of error value and avoiding missing the best point. The calculation formula of step length is a key part used to enhance the application effect of the algorithm [37], and the new step length calculation formula is constructed through functions, which are widely chosen as sigmoid functions, hyperbolic tangent functions, exponential functions, inverse tangent functions, etc. In the pursuit of enhancing conventional algorithms, one frequently encountered path for improvement involves the utilization of the sigmoid function. This pathway holds broader applicability and often yields favorable outcomes. To underscore the merits of the algorithm introduced in this manuscript, we have opted to juxtapose it with three prominent algorithms from pertinent references. These selections have been chosen to highlight the algorithm's advantages in comparison to both traditional and currently popular algorithms. Innovatively, reference [38] employed the logarithmic function to construct the step size formula of the algorithm and obtained the improved LVS-LMS algorithm with strong stability. The new step size calculation formula is as follows:  
\[\mu(n)=b\log(a|e(n)|^{m}) \tag{8}\]  
Reference [39] skillfully used the Versiera function to obtain the step size calculation method and obtained the VVS-LMS algorithm with superior performance. The new step size formula is as follows:  
\[\mu(n)=a[1-\frac{1}{be^{2}(n)+1}] \tag{9}\]  
Reference [40] integrated the normal distribution curve into the construction of the step size formula in order to improve the LMS algorithm, and the NDVS-LMS algorithm with stable performance is obtained. The improved step size formula is as follows:  
\[\mu(n)=c(1-\exp(-a|e(n)|^{b})) \tag{10}\]  
In this paper, according to the LMS algorithm step factor adjustment rules, it is possible to use the inverse cosine function to make the size of the step size change according to the different convergence states of the algorithm. The expression of the inverse cosine function is as follows:  
\[y=\arccos(x) \tag{11}\]  
The function is defined over the interval (\(-1\), \(1\)), exhibiting properties of an odd function characterized by a monotonically decreasing trend within this range. The corresponding graphical representation is depicted as the dashed line in Figure 2. Upon manipulation of the independent variable in Equation (11), the resultant function takes the following form:  
\[y=\left|\arccos(x)-\frac{\pi}{2}\right| \tag{12}\]The plotted function resulting from Equation (12) is represented by the solid line in Figure 2. The visual analysis of the graph reveals a conformity to the algorithm's step size adjustment mechanism, as evidenced by the gradual decline in slope with the increase in the independent variable. This trend signifies that during the initial convergence stages, the step size is larger, becoming smaller as convergence nears completion, and exhibiting gradual adjustments when the error approaches zero.