# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 2 OTFS-BFDM System
### OTFS-BFDM System Model
#### 2.1.1 Transmitter  
On the DD plane, all symbols on each transmission frame can be composed as an \(MN\times 1\) vector represented by:  
\[\mathbf{d}=[\mathbf{d}_{0}^{T},\mathbf{d}_{1}^{T},\cdots,\mathbf{d}_{N-1}^{T}]^{T}, \tag{1}\]  
where \(\mathbf{d}_{i}=[d[0,i],d[1,i],\cdots,d[M-1,i]]^{T}\), \(0\leq i\leq N-1\).  
The QAM symbol vector \(\mathbf{d}\) in the DD domain can be mapped to \(\mathbf{D}\) in the TF domain by the inverse symplectic finite Fourier transform (ISFFT) transform, as follows:  
\[\mathbf{D}=(\mathbf{F}_{N}^{-1}\otimes\mathbf{F}_{M})\mathbf{d}. \tag{2}\]  
Since the BFDM modulation follows the cyclic convolutional filter bank (CCFB) structure, the BFDM modulation matrix can be obtained as:  
\[\mathbf{A}=\underbrace{circ\{[\mathbf{P}_{0},\mathbf{P}_{N-1},\cdots,\mathbf{P}_{1}]_{M\times 1 }\}}_{\mathbf{P}}\times diag\{[\mathbf{F}_{M}^{-1},\mathbf{F}_{M}^{-1},\cdots,\mathbf{F}_{M}^{- 1}]_{N\times 1}\} \tag{3}\]  
by filtering each sub-carrier through the polyphase network.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \((*)^{-1}/(*)^{T}\) & The inverse/transpose of a matrix \\ \(\otimes\) & The kronecker product \\ \(\mathbf{I}_{M}\) & A \(M\)-dimensional identity matrix \\ \(\mathbf{F}_{N}\) & A \(N\)-point DFT matrix \(\left\{\frac{1}{\sqrt{N}}e^{2\pi\beta|M/N}\right\}_{k,l=0}^{N-1}\) \\ \(\mathbf{F}_{N}^{-1}\) & A \(N\)-point IDFT matrix \(\left\{\frac{1}{\sqrt{N}}e^{-2\pi\beta|M/N}\right\}_{k,l=0}^{N-1}\) \\ \(diag\{*\}\) & Return a diagonal matrix \\ \(circ\{*\}\) & Return a circular matrix \\ \hline \hline \end{tabular}
\end{table}
Table 1: Mathematical notation.  
Figure 1: OTFS-BFDM system model.  
In (3), \(\mathbf{P}\) is an \(L\times L\) block-circulant analysis filter coefficient matrix of the BFDM modulation, where \(p(\ell)\) is the PWF of the analysis filter bank with length \(L=MN\). The matrix \(\mathbf{P}_{i}\) consists of \(p(\ell)\) and can be expressed as:  
\[\mathbf{P}_{i}=diag\{p(iM),p(iM+1),\cdots,p(iM+M-1)\}, \tag{4}\]  
where \(0\leq i\leq N-1\). In addition, according to the properties of the block-circulant matrix, \(\mathbf{P}\) can be rewritten as:  
\[\mathbf{P}=(\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\mathbf{\Lambda}(\mathbf{F}_{N}\otimes\mathbf{I}_ {M}), \tag{5}\]  
where \(\mathbf{\Lambda}=diag(\mathbf{\Lambda}_{0},\mathbf{\Lambda}_{1},\cdots,\mathbf{\Lambda}_{N-1})\), \(\mathbf{\Lambda}_{n}\in\mathbb{C}^{M\times M}\) can be calculated by:  
\[\begin{cases}\mathbf{\Phi}(i,j)=[\mathbf{P}_{0}(i,j),\mathbf{P}_{1}(i,j),\cdots,\mathbf{P}_{N -1}(i,j)]^{T}\\ \left[\mathbf{\Lambda}_{0}(i,j),\mathbf{\Lambda}_{1}(i,j),\cdots,\mathbf{\Lambda}_{N-1}(i,j)\right]=\sqrt{N}\mathbf{F}_{N}\mathbf{\Phi}(i,j)\\ 0\leq i,j\leq M-1.\end{cases} \tag{6}\]  
Referring to (4) and (6), since \(\mathbf{P}_{n}(i,j)\) is a diagonal matrix and acts as a vector element of \(\mathbf{\Phi}(i,j)\), \(\mathbf{\Phi}(i,j)\
eq 0\) only when \(i=j\). Moreover, the \(N\)-DFT transform of \(\mathbf{\Phi}(i,j)\) yields \(\mathbf{\Lambda}_{n}(i,j)\). Therefore, the elements of \(\mathbf{\Lambda}\) in the main diagonal are not zero, i.e., \(\mathbf{\Lambda}\) is a diagonal matrix.  
The time domain signal \(\mathbf{s}\) on the physical channel is obtained by performing BFDM modulation on \(\mathbf{D}\) as follows:  
\[\begin{split}\mathbf{s}&=\mathbf{A}\mathbf{D}=\mathbf{P}(\mathbf{I}_{N }\otimes\mathbf{F}_{M}^{-1})(\mathbf{F}_{N}^{-1}\otimes\mathbf{F}_{M})\mathbf{d}\\ &=\mathbf{P}(\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\mathbf{d}.\end{split} \tag{7}\]