# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 3 Proposed MP-AIC Scheme of Clipping Noise
### Iterative Clipping Noise Cancellation Scheme  
The main idea of the iterative clipping noise cancellation scheme is to cancel the clipping noise from the received signal by iteratively feeding the extracted residual signal back to the MP detector. As the number of iterations increases, the accuracy of MP decoding will be improved and the residual amount of clipping noise in the initial received signal is finally canceled.  
Figure 3 describes the iterative clipping noise cancellation with a maximum iteration number of \(\tau_{\max}\) in the clipped OTFS-BFDM system. The scheme includes the following steps:  
1. The received signal \(\mathbf{\hat{s}}\) is first de-attenuated by multiplying \(\beta^{-1}\). Then, BFDM demodulation and SFFT transformation are performed on the attenuated signal to obtain the observation signal \(\hat{\mathbf{d}}\). Finally, the decision signal \(\hat{\mathbf{d}}\) will be output when \(\hat{\mathbf{d}}\) is substituted into MP algorithm for equalization and decoding. According to (13) and (22), \(\hat{\mathbf{d}}\) can be derived as: \[\hat{\mathbf{d}}=(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{H}\mathbf{P}(\mathbf{F}_{N}^{-1} \otimes\mathbf{I}_{M})\mathbf{d}+\beta^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{H} \delta+\beta^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{w}.\] (25)
2. Let the decision output \(\hat{\mathbf{d}}\) passes through the BFDM modulator and ISEFT converter in turn to obtain a new OTFS-BFDM signal. The new OTFS-BFDM signal will be processed in two parallel ways. One way is to apply the same clipping and filtering as the transmitter to obtain a new clipped signal \(\tilde{\mathbf{s}}_{1}\). The other way is to multiply it by \(\beta\) to get the attenuated signal \(\tilde{\mathbf{s}}_{2}\). The constructed reference clipping distortion \(\delta^{\prime}\) can be generated by subtracting \(\tilde{\mathbf{s}}_{2}\) from \(\tilde{\mathbf{s}}_{1}\). According to (7) and (18), we get: \[\tilde{\mathbf{s}}_{1}=\beta\mathbf{P}(\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\tilde{\mathbf{d} }+\delta^{\prime},\tilde{\mathbf{s}}_{2}=\beta\mathbf{P}(\mathbf{F}_{N}^{-1}\otimes\mathbf{I} _{M})\tilde{\mathbf{d}}.\] (26) It is evident that \(\delta^{\prime}=\tilde{\mathbf{s}}_{1}-\tilde{\mathbf{s}}_{2}\). Note that the constructed reference clipping distortion \(\delta^{\prime}\) has to go through the channel convolution matrix \(\mathbf{H}\) before the next iteration, where \(\mathbf{H}\) is known due to the ideal channel estimation.
3. Remove the clipping distortion \(\mathbf{H}\delta^{\prime}\) from the received signal \(\tilde{\mathbf{s}}\) to obtain the revised received signal \(\tilde{\mathbf{s}}\). From Figure 3, the signal \(\tilde{\mathbf{s}}\) can be written as: \[\tilde{\mathbf{s}}=\beta\mathbf{H}\mathbf{P}(\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\mathbf{d}+ \mathbf{H}(\delta-\delta^{\prime})+\mathbf{w},\] (27) where \(\delta-\delta^{\prime}\) represents the difference in clipping distortion.
4. Replace \(\tilde{\mathbf{s}}\) with \(\tilde{\mathbf{s}}\) and return to step 1 for the next iteration. Until the number of iterations reaches \(\tau_{\max}\), the procedure terminates. In the first iteration, by attenuating and demodulating the signal \(\tilde{\mathbf{s}}\) in turn, the revised observation signal \(\tilde{\mathbf{d}}\) can be obtained as: \[\tilde{\mathbf{d}}=(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{H}\mathbf{P}(\mathbf{F}_{N}^{- 1}\otimes\mathbf{I}_{M})\mathbf{d}+\underbrace{\beta^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M}) \mathbf{Q}\mathbf{H}(\delta-\delta^{\prime})}_{Residual\_signal}+\beta^{-1}(\mathbf{F}_ {N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{w}.\] (28) In (28), \(\beta^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{Q}\mathbf{H}(\delta-\delta^{\prime})\) is the residual signal and will be further reduced in the next iteration.