# Optimal Order of Time-Domain Adaptive Filter for Anti-Jamming Navigation Receiver
## 4 Optimal Filter Order  
### Demand Analysis of Anti-Jamming Filter  
Interference suppression performance is a significant indicator of anti-jamming satellite navigation receivers [39]. CNR is usually used to access the interference suppression ratio and loss of signals resulting from the anti-jamming filter [40]. The interference suppression performance of time-domain adaptive anti-jamming filters differ from different interference characteristic of power and interference bandwidth, and its ultimate anti-jamming performance can use the maximum estimated CNR as the equivalent. The adaptive algorithm correlates signal characteristics in the time domain and forms a null in the frequency band where the interference signal is located [41]. A deeper null means a more excellent interference rejection ratio, and a narrower transition bandwidth indicates less loss of the navigation signal. JSR (Jamming-to-Signal Ratio) and interference bandwidth are two characteristic parameters of interference signals, and these two parameters can be estimated effectively by existing corresponding algorithms in the navigation receiver [42; 43]. In this paper, they are assumed to be known parameters. The difference between interference signals' peak power and spread spectrum signals' level is defined as the jam-to-signal ratio, and the bandwidth of the narrow-band interference can be expressed as. \[B_{J}=\eta\cdot(2F_{c}), \tag{23}\]  
where \(\eta\) is the interference relative bandwidth, \(2F_{c}\) is the signal bandwidth, and \(F_{c}\) is the frequency of pseudo random noise (PRN) code. While the interference relative bandwidth is less than 20%, the interference is a narrow-band interference signal [5]. The requirements of filters for suppressing various narrow-band interferences are analyzed. The simulation results shown in Figure 5 illustrate the frequency spectrum of navigation signals before and after anti-jamming, with different JSR and interference bandwidths. In order to show the bandwidth relationship between interference and receiver, the negative frequency is retained in Figure 5. There is no symmetry in the spectrum since all received signals are complex signals. When the filter length is increased sufficiently, the anti-jamming performance can be effectively improved. However, while the filter order has become overlarge, the room for improvement in interference suppression performance is gradually compressed. In Figure 5a-c are the spectrograms before and after the anti-jamming module of different length filters: When the filter length increases from 9 to 19, the null is deepened, and the transition band is narrowed, which means that the interference suppression is enhanced while the signal loss is reduced. When the filter length increases from 19 to 29, the performance improvement is not as apparent as before. That is, e improvement of interference suppression performance is reduced. The higher the interference signal power, the worse the interference suppression performance of the same length filter and the lower the ultimate performance the finite-length filter can achieve. Figure 5c,d are the before and after anti-interference spectrum of the filter order under interferences of different power. While the interference power increases, there is no apparent change in the null depth, but the transition band is widened. The increase in navigation signal loss leads to a decrease in anti-jamming performance. As the interference relative bandwidth increases, the loss of navigation signals increases, resulting in a more obvious reduction in the anti-jamming performance. Therefore, the ultimate performance of the adaptive filter is reduced gradually. Pictures (c), (e), (f) are the spectrum diagrams before and after the anti-jamming filter under different bandwidth interference. The increase of interference bandwidth will not significantly impact the transition band and null of the filter frequency response.