# Adaptive Satellite Navigation Anti-Interference Algorithm Based on Inverse Cosine Function
## 2 LMS Algorithm Interference Suppression Principle  
The fundamental concept of an adaptive filter for interference suppression is depicted in Figure 1, which illustrates the core principle behind its operation. \(x(n)\) is the received signal of the receiver, including the satellite navigation signal, narrowband interference signal, and noise signal, \(d(n)\) denotes the delay of \(x(n)\) which is used as the reference signal of the adaptive filter, \(y(n)\) is the output of the adaptive filter after processing, \(e(n)\) denotes that the difference between the output signal and the reference signal is the error signal, and the error signal automatically adjusts the tap weight \(w(n)\) of the adaptive filter according to the adaptive algorithm [30]. The input satellite navigation signals and noise signals are time domain broadband signals with small sample time delay correlation, but the correlation of the narrowband interference sample value is much stronger than that of the broadband signal, and the current signal value can be estimated using the past signal value [31]. Depending on the correlation, the estimated signal of narrowband interference can be generated through an adaptive filter, canceling it out from the input signal to achieve narrowband interference suppression [32].  
The receiver receives the satellite navigation signal and down-converts it to obtain the intermediate frequency signal (IF signal), which is digitally sampled by the analog-to-digital convert (ADC) to obtain the input signal as follows:  
\[x(n)=s(n)+j(n)+n(n) \tag{1}\]  
where \(s(n)\) is the received satellite navigation sign, \(j(n)\) is the narrowband interference sign and \(n(n)\) is the noise sign. The reference signal \(d(n)\) is the delay of the input signal \(x(n)\). \(x(n)\) and \(d(n)\) will cancel the signal with correlation and the signal without correlation will be unaffected, where:  
\[e(n)=d(n)-y(n) \tag{2}\]  
The LMS algorithm replaces the mean square error (MSE) \(E(\left|e(n)\right|^{2})\) by a single sample of data \(\left|e(n)\right|^{2}\)[33], and the gradient estimate during the iteration needs to satisfy the following [34]:  
\[\begin{array}{l}\dot{\
abla}(n)=\partial|e(n)|^{2}/\partial W(n)\\ =\partial\left\{|x(n)|^{2}+W^{H}(n)X(n)W(n)\right.\\ \left.-2\text{Re}[x(n)X^{T}(n)W(n)]\right\}/\partial W(n)\\ =2W^{H}(n)X(n)W(n)-2x(n)X(n)\\ =-2e(n)X(n)\end{array} \tag{3}\]  
Therefore, the filter weight coefficient vector iterative formula of the LMS algorithm is as follows [34]:  
\[W(n+1)=W(n)+2\mu e(n)X(n) \tag{4}\]  
where \(W(n)\) is the filter coefficient vector, \(X(n)\) is the input vector, \(e(n)\) is the error signal, and \(\mu\) is the key to determine the convergence and error level of LMS algorithm, which can reduce the time required for convergence and limit the error in a small range [35].  
The main implementation process of the LMS algorithm is as follows:  
\[y(n)=\mathbf{W}(n)^{H}\mathbf{X}(n) \tag{5}\]  
\[e(n)=d(n)-y(n) \tag{6}\]  
\[W(n+1)=\mathbf{W}(n)+2\mu(n)e(n)\mathbf{X}(n) \tag{7}\]  
where \(\mathbf{X}(n)\) is the input signal at moment \(n\), \(\mathbf{W}(n)\) is the weight coefficient of the filter at moment \(n\), and \(\mu(n)\) is the iteration step size factor. The condition that the LMS algorithm must satisfy to achieve the convergence effect is: \(0<\mu<1/\lambda_{m}\), and \(\lambda_{m}\) is the maximum eigenvalue of the filter input signal autocorrelation matrix.  
Figure 1: Basic principle of adaptive filter. \(x(n)\) is a navigation signal contaminated with narrowband interference. \(y(n)\) is the output signal of the filter. \(d(n)\) is the desired signal. \(e(n)\) is error signal.