# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 3 Proposed MP-AIC Scheme of Clipping Noise
### The Procedure of the MP-AIC Method  
In the original method of iterative clipping noise cancellation, the residual clipping noise is gradually canceled and the BER tends to converge. However, the initial probability of modulation alphabet used in MP decoding is still equal after each clipping noise cancellation. In the MP-AIC method, we use the variance of the residual signal and channel noise together as input to the MP algorithm and use the probability of modulation alphabet after MP decoding in the current iteration as the initial probability of MP decoding in the next iteration to accelerate the convergence rate of MP decoding. Therefore, the proposed  
Figure 3: The iterative cancellation of clipping noise for OTFS-BFDM signals on the receiver. MP-AIC method can further improve the BER of the clipped system and accelerate the BER convergence. Assume that \(\mathbf{H}_{eq}\) has only \(S\) non-zero elements in each row and column. Each observation node \(\tilde{d}_{a}\) is connected to the set of variable nodes \(\{d_{a},a\in S\}\), and each variable node \(d_{b}\) is connected to the set of observation nodes \(\{\tilde{d}_{b},b\in S\}\). The maximum a posteriori (MAP) decision rule for (13) is given by:  
\[\mathbf{\hat{d}}=\operatorname*{arg\,max}_{\mathbf{d}\in\mathbb{A}^{L\times 1}}\, \Pr(\mathbf{d}|\mathbf{\tilde{d}},\mathbf{H}_{eq}), \tag{29}\]  
where \(\mathbb{A}\) is a modulation alphabet of size \(Q\). Using the Bayesian criterion, the MAP joint detection output is:  
\[\hat{d}_{a}=\operatorname*{arg\,max}_{a_{j}\in\mathbb{A}}\prod_{b\in S}\,\Pr( \tilde{d}_{b}|d_{a}=a_{j},\mathbf{H}_{eq}). \tag{30}\]  
In the MP algorithm, the message passed from a variable node \(d_{a}\) to the observation nodes \(\tilde{d}_{b}\) is the probability mass function (pmf) of the alphabet \(P_{ab}=\{p_{ab}(a_{j})|a_{i}\in\mathbb{A}\}\). The procedure of the MP-AIC method is expressed as follows:  
1. Input: clipping attenuation factor \(\beta\), Doppler tap \(k_{p}\), delay tap \(l_{p}\), path gain \(h_{p}\), iteration number \(N_{iter}\) (Iterative Clipping Noise Cancellation ), iteration number \(\tau_{\text{max}}\) (Iterative MP decoding), Gaussian white noise variance \(\sigma_{n}^{2}\), the damping factor \(\Delta=0.6\), \(\varepsilon=0.01\). 2. Initialization: overall noise variance \(\sigma_{All(0)}^{2}=\sigma_{All}^{2}\) (Equation (24)), the initial probability of constellation symbols \(p_{ab}^{(0)}=1/Q\) and the observation signal \(\mathbf{\hat{d}}_{(0)}=\mathbf{\hat{d}}\) (Equation (25)) in the zero-th iteration. 3. Calculate the total noise variance and observation signal of the \(k\)-th iteration by \(\sigma_{All(k)}^{2}=\beta^{-2}(\sigma_{\hat{s}(k)}^{2}+\sigma_{n}^{2})\) (Equations (20), (24), and (28)). \(\mathbf{\hat{d}}_{(k)}=\beta^{-1}\mathbf{\hat{s}}_{(k)}\) (Equations (25) and (27)). 4. Message from \(\tilde{d}_{b}\) to \(x_{a}\): The mean \(\mu_{ba}^{(i)}\) and variance \((\sigma_{ba}^{(i)})^{2}\) of the interference term \(I_{ba}\) are passed as messages from \(\tilde{d}_{b}\) to \(x_{a}\): \[I_{ba}=\sum_{c\in S,c\
eq a}d_{c}H_{b,c}+\xi_{b}.\] (31) The mean and variance of \(I_{ba}\) are expressed, respectively, as: \[\mu_{ba}^{(i)}=\sum_{\begin{subarray}{c}c\in S,c\
eq a}\sum_{j=1}^{Q}p_{cb}^{ (i)}\,(a_{j})a_{j}H_{b,c},\] (32) and: \[\left(\sigma_{ba}^{(i)}\right)^{2}=\sum_{ \begin{subarray}{c}c\in S\\ c\
eq a\end{subarray} }\sum_{j=1}^{Q}p_{cb}^{(i)}\,(a_{j})|a_{j}|^{2}|H_{b,c}| ^{2}-\sum_{ \begin{subarray}{c}c\in S\\ c\
eq a\end{subarray} }\left|\sum_{j=1}^{Q}p_{cb}^{(i)}\,(a_{j})a_{j}H_{b,c} \right|^{2}+\sigma_{all(k)}^{2},\] (33) where \(k\) and \(i\) represent iterations of IC and MP, respectively. 5.