# Jamming Detection and Suppression Technique for OTFS Systems in an AWGN Channel
## 3 Jamming Detection and Suppression in the Frequency Domain
### Smoothing Window Process  
In OTFS systems, jamming detection can be significantly impacted by abrupt variations in signal power. To mitigate this, smoothing window processing is applied to stabilize the spectral characteristics of the received signal, ensuring a more uniform sample for jamming detection. By suppressing power fluctuations, this method reduces the probability of false alarms in interference detection and improves the accuracy of subsequent signal processing.  
The smoothing window is applied in the frequency domain, averaging the power spectrum over a defined range. Specifically, for the frequency point \(k\), the amplitude of the signal at this point is denoted as \(R(k)\). The smoothing window processing involves selecting \(D\) frequency points on both sides of the target frequency point \(k\). For a total of \(2D+1\) frequency points \(|R(k-D)|,|R(k-D+1)|,\ldots,|R(k+D-1)|,|R(k+D)|\), their amplitudes are averaged to obtain \(\big{|}\hat{R}(k)\big{|}\). This value, \(\big{|}\hat{R}(k)\big{|}\), is then used as the smoothed amplitude of frequency point \(k\). When \(k-D\leq 0\) or \(k+D>N\), meaning that fewer than \(D\) frequency points are available on either side of \(p\), all available points are included in the averaging. The calculation is defined as  
\[|R(k)|=\left\{\begin{array}{ll}\sum\limits_{l=-k}^{D}R(k+l)/(k+1+D),&\quad k \leq D\\ \sum\limits_{l=-D}^{N-1-k}R(k+l)/(M-k+D),&\quad k\geq N-D\\ \sum\limits_{k=-D}^{D}R(k+l)/(2D+1),&\quad D<k<N-D\end{array}\right.. \tag{14}\]  
According to Equation (14), the smoothing window operation is defined as the weighted average of the magnitudes of all points within the window. Specifically, the first term on the right-hand side represents the case where the frequency point is within the first \(D\) positions, in which case the smoothing window remains fixed over the first \(D\) points without sliding. The second term corresponds to the case where the frequency point is within the last \(D\) positions, meaning the smoothing window remains fixed over the last \(D\) points without sliding. The third term describes the scenario where the frequency point is  
Figure 4: The relationship between the overlap ratio \(\alpha\) and the SNR gain achieved by adding an overlap window using different window functions.  
in an intermediate position, outside of the aforementioned two cases, which the smoothing window slides with the current frequency point as its center.  
Here, the length of \(D\) is defined by the smoothing factor \(\rho\), where \(D=\left\lceil\rho N\right\rceil\), and \(\left\lceil\rho N\right\rceil\) denotes the smallest integer value greater than or equal to \(\rho N\). The smoothing factor \(\rho\) represents the proportion of frequency points considered for weighted averaging within each smoothing window. Since the primary objective of smoothing is to mitigate high-power false alarms caused by non-jamming signals, the smoothing factor should be regarded as a trade-off between detection performance and false alarm probability. Figure 5 illustrates the probability density of the frequency-domain amplitudes of the received signal before and after smoothing. It is clearly observed that after processing with the smoothing window, a distinct boundary is formed between the amplitudes of the useful signal and the interference. This provides a stronger foundation for the detection process, facilitating more accurate differentiation between useful signals and interference, thereby improving the accuracy of jamming detection.