## 3 Methodology and Solution
### Adaptive Bandwidth Adjustment for Notch Filter  
As previously discussed, a notch filter is an adjustable gap in the frequency domain. That gap center frequency and bandwidth are controlled by two parameters: \(f_{c}\) and \(r\). The proposed interference detector has two main capabilities: center frequency estimation and interference power calculation. Detected center frequency can directly be utilized to adjust the center of the gap of the notch filter on narrowband interference. However, filter bandwidth adjusting is more complicated and demands a careful study about the SNR based on the \(r\) parameter after filtering. It should be considered that the filter also removes a part of the useful received signal. Clearly, interference cancelation for powerful interference demands a wider gap or a greater attenuation from the notch in the frequency domain, but there is a tradeoff between the amount of interference power reduction and the removed bandwidth from the signal. To study the effects of bandwidth removal on the system's performance, a study based on Figure 9 is done to calculate the optimum \(r\) parameter based on the input JSR. The target of this process is to generate a general modulated telecommunication signal with pulse shaping. After the signal generator, a CWI interference with variable power is added to  
Figure 8: Notch filters with different \(r\) parameters. the desired signal. In the receiver, a notch filter tries to remove the interference. Then, the system performance is measured with the SNR measurement between the transmitted signal \(S_{t}\) and final signal after filtering \(S_{f}\)[34]:  
\[SNR=\ \frac{var(S_{t})}{var\Big{(}S_{t}-S_{f}\Big{)}}, \tag{13}\]  
where \(var(S_{t})=\ \frac{\sum_{i=1}^{K}(S_{t}|k|-u)^{2}}{K}\) denotes the variance of \(S_{t}\) and \(u\) is the mean value of \(S_{t}\). The \(r\) parameter must be adjusted in a way that the interference is removed completely from the desired signal with a minimum amount of useful signal being removed from the original signal. In other words, the notch filter must remove the interference completely with the minimum available bandwidth to preserve the quality of service. CWI decreases the received SNR and notch filter tries to remove the interference power by generating a gap at the position of interference and decrease its power. Decreasing interference power increases SNR. \(r\) parameter adjusts the notch bandwidth and it can vary between \(0<r<1\). Selecting the notch bandwidth (or \(r\) parameter) affects the SNR improvement:  
* If notch bandwidth is small, interference power is not completely removed. * If notch bandwidth is larger than enough, the interference power is removed but the extra useful spectrum of signal is also removed. Both mentioned cases cause that the SNR does not achieve its maximum. The optimum \(r\) adjusts the notch bandwidth in the way that interference power is mitigated and the minimum bandwidth of the spectrum is removed or in another words, we must adjust the notch gap in the way that the maximum SNR after filtering is obtained:  
\[r_{optimum}=\ Arg\ \max_{r}(SNR)\,\ 0<r<1. \tag{14}\]  
Equation (14) expresses that the value of \(r\) that maximizes the received SNR after filtering is the optimum selection. Utilizing optimum \(r\) will maximize the SNR value in Figure 9.  
Figure 10a displays the output SNR of the notch filter based on the notch bandwidth (\(r\) parameter) for different CWI powers (JSR parameter). It can be seen that, for a special JSR (interference power) value, the output SNR peaks at a special \(r\) parameter value that is optimal for that JSR. Based on these figures, when the JSR is large, there is a powerful peak on top of the desired spectrum and a notch filter with a larger gap is required to completely remove all interference power. On the other side, when the JSR is small, a notch filter with a smaller gap must be utilized to preserve the spectrum quality as much as possible.