# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 5 Simulation Results
### BER Comparison between the MP-AIC and MP-A Schemes  
Figure 5 shows the BER of the non-clipped OTFS-BFDM system with different PWFs. In Section 2.2, we learned that the better the frequency-domain localization of non-rectangular PWFs, the lower the PAPR of the OTFS-BFDM system. When the OTFS-BFDM system adopts RRC/RC PWFs with \(\alpha=0\) or Dirichlet PWFs, their PAPR performance is consistent with that of the OTFS-Rect system. To verify that the BER performance of OTFS-BFDM system using non-rectangular PWFs is not inferior to that of OTFS-Rect system in high mobility scenarios, the relationship between PWFs and BER of the system can also be obtained through simulation. In Figure 5a, we adopt the MP [7] and linear minimum mean square error (LMMSE) [1] equalization to simulate the OTFS-BFDM system with different PWFs in three scenarios. It can be seen that when the OTFS-BFDM system adopts rectangular PWFs, RRC/RC PWFs (\(\alpha=0\)) and Dirichlet PWFs, respectively, the BERs of the system are almost the same regardless of the symbol mapping, channel equalization and maximum Doppler shift, which is consistent with the conclusion in [14]. It is worth noting that the BER of the linear equalizer LMMSE is higher than that of MP equalizer because it cannot obtain the time-frequency diversity of the channel. When the symbol modulation is 4-QAM and the maximum Doppler shift is 967.74 Hz, the BER of LMMSE equalizer and MP equalizer are \(1.563\times 10^{-4}\) and \(2.974\times 10^{-4}\), respectively, at \(SNR=15\) dB. Figure 5b depicts the BER performance of an OTFS-RRC system at different roll-off coefficients \(\alpha\), where the smaller the \(\alpha\), the smaller the BER of the system. When the symbol modulation is 4-QAM and the maximum Doppler shift is 967.74 Hz, the BERs of OTFS-RRC system with \(\alpha=0.1\) and \(\alpha=0.9\) are \(2.6\times 10^{-4}\) and \(7.681\times 10^{-3}\), respectively, at \(SNR=16\) dB. Therefore, the better the frequency-domain localization of non-rectangular PWFs, the better the BER of OTFS-BFDM system. In particular, the BER of OTFS-BFDM system using (\(\alpha=0\)) and Dirichlet PWFs is almost the same as that of OTFS-Rect system. Figure 6 shows the BER performance of the MP-A method for clipping noise suppression in [26]. Although the MP-A method considers the influence of clipping noise on MP algorithm, it only alleviates the clipping noise interference but does not cancel it, where MP-A takes clipping noise and channel noise as the whole noise input of MP algorithm. For the case of Figure 6a, the BER of OTFS-Rect system deteriorates obviously after clipping and increases with the clipping ratio \(\gamma\). When the MP-A method is used to suppress the clipping noise, the BER performance is slightly improved. On the channel with the maximum Doppler shift of 1935.48 Hz, when \(\gamma=1\) dB and \(SNR=16\) dB, the BER is \(3.717\times 10^{-3}\), while the BER in MP-A method is \(2.701\times 10^{-3}\). It can be concluded from the above that MP-A can alleviate the interference of clipping noise to MP decoding to a  
Figure 4: (**a**) PAPR distribution for the OTFS-Rect signal with different clipping ratio \(\gamma\); (**b**) PAPR distribution for the OTFS-RRC signal with different clipping ratio \(\gamma\) (\(\alpha=0.9\)). certain extent, but cannot completely cancel the interference. A similar conclusion can be drawn from Figure 6b. As shown in Figure 7a, the BER of the MP-AIC method decreases with the increase of iterations, because the residual clipping noise is gradually canceled out after several iterations. When the number of iterations increases from 0 to 2, the BERs of MP-AIC are \(5.814\times 10^{-3}\), \(1.809\times 10^{-3}\), and \(1.62\times 10^{-3}\), respectively, at \(\gamma=1\) dB and \(\text{SNR}=16\) dB. It can be seen that the BER at \(iter=2\) is about 72% lower than that at \(iter=0\) since clipping noise is not considered at \(iter=0\).