# Optimal Order of Time-Domain Adaptive Filter for Anti-Jamming Navigation Receiver
## 3 Analysis of Filter Order  
The purpose of the study is to adaptively design the shortest anti-jamming filter according to interference characteristics, which should meet specifications requirements in the meantime. Since the algorithm parameters cannot be set directly according to the interference, the window function changing with interferences can be used to design a filter to achieve interference suppression.  
The method by window function is a time-domain design method for digital filters. By adopting window functions with different finite time widths to cut the infinite length sequence, the finite length sequence is obtained, and the corresponding FIR filter is realized. The most basic window function is a rectangular window [35]. By weighting the amplitude of the rectangular window function, more window functions with different performances can be obtained, such as triangular window, Hanning window, Hamming window, Blackman window, and Kaiser window. Kaiser window can adjust the width of the main lobe and the attenuation of the side lobe by restricting filter length, which means it has good adaptability to the interference of different power and bandwidth [36]. Other window functions cannot control the filter performance as flexibly as the Kaiser window. Therefore, the Kaiser window is most suitable for calculating the optimal filter order.  
The Kaiser window comprises Bessel functions and contains complex and variable window parameters \(\alpha\), but it can achieve the steepest window function transition zone with the same performance. The Kaiser window function can be expressed as [37].  
\[\omega(n)=\frac{I_{0}\Big{(}\alpha\sqrt{1-\big{(}1-\frac{2n}{N-1}\big{)}^{2}} \Big{)}}{I_{0}(\alpha)},n=0,1,\ldots,N-1, \tag{14}\]The Fourier transform of Kaiser window function can be derived to be [37].  
\[W(\omega)=\left\{\begin{array}{ll}\frac{2\tau\sinh\left[a\sqrt{1-(\omega/\omega_ {a})^{2}}\right]}{-I_{0}|a|a\sqrt{1-(\omega/\omega_{a})^{2}}}&|\omega|\leq \omega_{a}\\ \frac{2\tau\sin\left[a\sqrt{(\omega/\omega_{a})^{2}-1}\right]}{I_{0}|a|\alpha \sqrt{(\omega/\omega_{a})^{2}-1}}&|\omega|>\omega\end{array}\right., \tag{15}\]  
where, \(\alpha=\omega_{a}\tau,f_{a}=\omega/2\pi\), and \(I_{0}(x)\) is the zero-order first-class modified Bessel function, which is defined as:  
\[I_{0}(x)=1+\sum_{k=1}^{\infty}\left[\frac{(x/2)^{k}}{k!}\right]^{2}, \tag{16}\]  
A typical Fourier transform of a Kaiser window (for \(\alpha=8\), \(\tau=2\)) is shown in Figure 4. The normalized window width is the precise transition zone of the Kaiser window, which is normalized by the product of the window width and the transition zone.  
\[D=\Delta f\cdot(2\tau), \tag{17}\]  
where \(\Delta f\) is the transition bandwidth and \(2\tau\) is the window span, which are defined as follows.  
\[\Delta f =f_{s}-f_{p}, \tag{18}\] \[2\tau =\frac{N-1}{fs}, \tag{19}\]  
where \(f_{s}\) is the stop-band cut-off frequency, and \(f_{p}\) is the pass-band cut-off frequency.  
\(\alpha\) is a parameter for adjusting the shape of the Kaiser window. The larger \(\alpha\), the narrower the window width and the increase of the main lobe width. Both \(\alpha\) and the normalized window width \(D\) can all be expressed by the minimum stop-band attenuation [38].  
\[\alpha=\left\{\begin{array}{ll}0.1102(As-8.7)&,&As>50\\ 0.5842(As-21)^{0.4}+0.07886(As-21),&21<As<50\\ 0&,&As<21\end{array}\right., \tag{20}\]  
\[D=\left\{\begin{array}{ll}(As-7.95)/14.36&As>21\\ 0.9222&else\end{array}\right., \tag{21}\]  
Figure 4: A typical Fourier transform of Kaiser window.  
The optimal filter order of the filter designed by the Kaiser window can be obtained by Formula (19) and Formula (21).  
\[N=\frac{As-7.95}{14.36(\Delta f/f_{s})} \tag{22}\]