# Jamming Detection and Suppression Technique for OTFS Systems in an AWGN Channel
## 3 Jamming Detection and Suppression in the Frequency Domain
### Overlapping and Windowing  
In the anti-jamming OTFS receiver, the received signal is divided into two consecutive overlapping frames and multiplied by a smoothing window to reduce amplitude discontinuities at their boundaries. The selection of the window function is a critical factor that demands thorough consideration. The Hamming window, Hanning window, and Blackman window are commonly used window functions that help suppress spectral leakage and improve frequency response performance. Since the received signal may suffer distortion after being filtered by the window function, SNR degradation can occur, negatively impacting BER performance. In this subsection, we evaluate the SNR loss caused  
Figure 3: Partial-band jamming in time–frequency domain and delay-Doppler domain.  
by the windowing process and use it as a criterion for selecting the most appropriate window function.  
The filtered signal in the frequency domain can be expressed as  
\[r_{win}(t)=r(t)*h_{win}(t), \tag{11}\]  
where \(h_{win}(t)\) is the window function and the symbol \(*\) denotes the convolution operator. The SNR gain caused by windowing the signal can be expressed as  
\[SNR_{gain}=\frac{\left[\sum\limits_{L_{win}}h_{win}(nT)\right]^{2}}{\sum \limits_{L_{win}}h_{win}^{2}(nT)}, \tag{12}\]  
where \(T\) is the sampling interval and \(L_{win}\) is the length of window function. Table 1 presents the SNR gain caused by different window functions, where a negative gain indicates that the windowing process leads to SNR degradation. Additionally, the Hamming window function results in the least SNR degradation.  
In order to compensate the SNR loss caused by the window filtering, an overlap window technique is introduced. The overlap ratio parameter must be carefully selected to achieve a better trade-off between SNR loss and computational cost. The relationship between the overlap ratio \(\alpha\) and the SNR gain achieved by adding an overlap window can be expressed as  
\[SNR_{gain,overlap}=\frac{\left[\sum\limits_{n=1}^{nN_{FFT}}[h_{win}(nT)+h_{win }(nT+(1-\alpha)N_{FFT}T)]\right]^{2}}{\alpha N_{FFT}\sum\limits_{n=1}^{nN_{ FFT}}[h_{win}(nT)+h_{win}(nT+(1-\alpha)N_{FFT}T)]^{2}}. \tag{13}\]  
Figure 4 illustrates the relationship between the overlap ratio \(\alpha\) and the SNR gain achieved by applying an overlap window with different window functions, including the Hamming, Hanning, and Blackman windows. It can be observed that the overlapping windowing technique effectively mitigates SNR degradation. However, this technique comes at the cost of increased computational complexity, which grows proportionally with the overlap ratio. Simulation results also show that the Hamming window function experiences less SNR degradation compared to the Hanning and Blackman window functions over a wide range of overlap ratios. Additionally, the minimum SNR degradation occurs when \(\alpha\approx 0.5\). As a result, the Hamming window function is employed, and the overlap ratio \(\alpha\) is set to 0.5 for the subsequent analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Hamming** & **Hanning** & **Blackman** \\ \hline SNR gain (dB) & \(-1.351\) & \(-1.769\) & \(-2.381\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: SNR gain caused by different window functions.