# Optimal Order of Time-Domain Adaptive Filter for Anti-Jamming Navigation Receiver
## 2 Mathematics Model
### Time Domain Adaptive Anti-Jamming Filter Model  
The time-domain adaptive anti-jamming technology can achieve good interference suppression performance for narrow-band interference, and it can be embedded in navigation receiver DSP independently with a small processor [33]. The classical least-mean-square (LMS) algorithm is used most often in adaptive filters because of its characteristics of small calculation and simple implementation, which uses MMSE as the optimal criterion and the steepest descent of the gradient to minimize the mean square of the error signal [34].  
However, increasing the filter length cannot guarantee the improvement of anti-jamming performance for a classical LMS filter. The iteration step limits the effect of filter order on anti-jamming performance. In terms of the general relationship, as the order increases, the filter transition band is compressed, the null is deepened to obtain better interference suppression performance, and the room for improvement of higher performance will gradually be limited in the meantime. Therefore, the estimated CNR increases rapidly with the filter order and then tends to be stable with no significant increase. However, the steady-state error of the interference suppression performance is also affected by the iteration step, the convergence condition of which is inversely proportional to the filter order. Therefore, with the same iteration step, the interference suppression performance of adaptive filters does not increase monotonically with the filter order, but there are inevitable fluctuations, as shown in Figure 2.  
In this paper, the LMS algorithm of filter vectors' zero-padding iteration can be used to alleviate the insufficient convergence of the adaptive algorithm, facilitating studying the relationship between the anti-interference performance and filter order, the structure of which is shown in Figure 3.  
Figure 3: LMS adaptive filter of filter vectors’ zero padding iteration.  
Figure 2: Estimated CNR after anti-jamming filter of LMS algorithm.  
The error signal between the filter output signal and the input signal is defined as \(e(n)\),which can be expressed as [21].  
\[e[n]=x[n]-y[n] \tag{8}\]  
The adaptive algorithm adjusts the filter coefficients \(\mathbf{W}_{M}\) according to the error signal and filter order, and the gradient vector of the mean square error can be expressed as [21].  
\[\
abla_{\mathbf{W}_{M}}E\Big{\{}\big{|}e[n]\big{|}^{2}\Big{\}}=2\mathbf{R_{X} }\mathbf{W}_{M}-2\mathbf{r}_{\text{Xd}}, \tag{9}\]  
Suppose that the initial value of \(\mathbf{W}_{M}\) is set to \(\mathbf{W}_{M,0}\), and the filter weight vector change iteratively along the negative gradient direction by the step \(\mu\), which can be expressed as [21].  
\[\mathbf{W}_{M}^{l+1}=\mathbf{W}_{M}^{l}-\mu\
abla_{\mathbf{W}_{M}}E\Big{\{} \big{|}e[n]\big{|}^{2}\Big{\}}, \tag{10}\]  
where \(l\) is the number of adaptive iterations, and the maximum number of iterations is \(L\). Different from the classical LMS algorithm, the initial value of the weight vector is obtained by the adaptive weight vector after the convergence of the filter of \(M-1\) orders.  
\[\mathbf{W}_{M,0}=\big{[}\begin{array}{cc}0&\mathbf{W}_{M-1,L}&0\end{array} \big{]}, \tag{11}\]  
where the initial number of the filter order is the minimum value of 2.  
Substituting Formula (9) into Formula (10) to get [21].  
\[\mathbf{W}_{M}^{l+1}=(\mathbf{I}-2\mu\mathbf{R_{X}})\mathbf{W}_{M}^{l}+2\mu \mathbf{r}_{\text{Xd}} \tag{12}\]  
In practical applications, the instantaneous mean square error is usually used instead of the mean square error, and finally the recursive formula of the LMS algorithm can be obtained [21].  
\[\mathbf{W}_{M}^{l+1}=\mathbf{W}_{M}^{l}+2\mu e^{*}[n]x[n] \tag{13}\]