# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 3 Proposed MP-AIC Scheme of Clipping Noise  
### Statistical Properties of Clipping Noise  
Based on the central limit theory, \(s(\ell)\) can be approximated as a complex Gaussian process with sufficiently large discrete samples, which has zero mean and variance \(\sigma^{2}\). When a soft limiter is used to clip \(s(\ell)\), the clipped signal can be written as:  
\[s^{\prime}(\ell)= \begin{cases}Ae^{i\phi(t)},|s(\ell)|>A\\ s(\ell),|s(\ell)|\leq A\end{cases}  \tag{17}\]  
where \(\phi(t)\) represents the phase of \(s(\ell)\) and \(A\) is the clipping level. As mentioned in [26], the signal after clipping is referred to as a clipped signal. According to the time-domain characteristics of the clipped signal, the clipped signal can be described as the sum of the original signal and the clipping distortion:  
\[s^{\prime}(\ell)=\beta s(\ell)+\delta(\ell),0\leq\ell\leq OL-1, \tag{18}\]  
In (18), \(\delta(\ell)\) is the clipping distortion uncorrelated with \(s(\ell)\), and the clipping attenuation factor \(\beta\in(0,1)\) is defined as:  
\[\beta=1-e^{-\gamma}+\frac{\sqrt{\gamma\gamma}}{2}erfc(\sqrt{\gamma}), \tag{19}\]  
where \(\gamma=\frac{A^{2}}{\sigma^{2}}\) is the clipping ratio, representing the ratio of the clipping threshold power to the average power of the clipped signal. The variance of the clipping noise can be obtained as [28]:  
Figure 2: (a) The CCDF of PAPR of OTFS-BFDM systems with different Doppler bins (\(N\)); (b) the CCDF of PAPR of OTFS-BFDM systems with different PWFs. The variance of clipping distortion can be divided into two cases: one is Nyquist rate sampling (oversampling factor \(O=1\), the filtering is not required) and the other is oversampling. For the Nyquist rate case, the variance of the clipping noise can be obtained as [26]:  
\[E\Big{[}\big{|}\delta(l)\big{|}^{2}\Big{]}=\sigma_{\delta}^{2}=(1-e^{-r}-\beta^{ 2})\sigma^{2}. \tag{20}\]  
For the oversampling case, the clipping distortion of the out-of-band part is filtered out. The average power of distortion term can no longer be obtained by simply adding or subtracting based on in-band power, but by integrating the power spectral density (PSD) of the clipping distortion. The PSD of the clipping distortion can be obtained as [29]:  
\[Sf_{\delta}(v)=FFT(R_{\delta}(u)), \tag{21}\]  
where \(R_{\delta}(u)=E[\delta_{\ell}\cdot\delta_{\ell+u}]=\sum\limits_{n=1}^{OL}C_{ \ell}\Big{[}\frac{R_{\delta}(u)}{R_{\delta}(0)}\Big{]}^{2\ell+1}\), \(R_{\delta}(u)\) and \(R_{\delta}(u)\) are autocorrelation functions of distortion term and original signal, respectively, and FFT represents the fast Fourier transform. The coefficient \(C_{\ell}\) can be seen in [29], which depends on the clipping ratio \(\gamma\) only. Then, the variance of the clipping distortion can be obtained by integrating \(Sf_{\delta}(v)\) within the bandwidth, that is, take norm-1 for vector \(\mathbf{S}f_{\delta}\) and then multiply it by \(\Delta f\). Referring to (11), the clipped OTFS-BFDM signal after downsampling can be given as:  
\[\mathbf{\hat{s}}=\mathbf{H}(\beta\mathbf{s}+\mathbf{\delta})+\mathbf{w}, \tag{22}\]  
where \(\delta=[\delta(0),\delta(1),\cdots,\delta(L-1)]^{T}\) and \(\mathbf{w}=[w(0),w(1),\cdots,w(L-1)]^{T}\). At the receiver, the signal \(\mathbf{\hat{s}}\) is first de-attenuated by multiplying \(\beta^{-1}\). Then, the estimated signal \(\mathbf{\hat{d}}\) in the DD domain can be obtained through by (13), as follows:  
\[\mathbf{\hat{d}} =\mathbf{\Lambda}^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{H}(\mathbf{F}_{N} ^{-1}\otimes\mathbf{I}_{M})\mathbf{\Lambda}\mathbf{d}\] \[+\underbrace{\beta^{-1}\mathbf{\Lambda}^{-1}(\mathbf{F}_{N}\otimes\mathbf{I} _{M})\mathbf{H}\delta}_{\Xi}+\underbrace{\beta^{-1}\mathbf{\Lambda}^{-1}(\mathbf{F}_{N} \otimes\mathbf{I}_{M})\mathbf{w}}_{\hat{\mathbf{\psi}}}, \tag{23}\]  
where \(\Xi\) is the clipping noise and \(\mathbf{\psi}\) is the channel noise. Based on the linear properties of the mean and variance, \(\Xi\sim\mathcal{CN}(0,\beta^{-2}\sigma_{n}^{2}\mathbf{1})\) can be obtained from [27].