# A Message Passing-Assisted Iterative Noise Cancellation Method for Clipped OTFS-BFDM Systems
## 2 OTFS-BFDM System
### OTFS-BFDM System Model
#### 2.1.2 Receiver  
Since BFDM satisfies the PR condition \(\mathbf{A}\mathbf{B}=\mathbf{I}_{L}\), once the modulation matrix \(\mathbf{A}\) is determined, the demodulation matrix \(\mathbf{B}\) can be derived as follows:  
\[\mathbf{B}=diag\{[\mathbf{F}_{M},\mathbf{F}_{M},\cdots,\mathbf{F}_{M}]_{N\times 1}\}\times \underbrace{circ\{\{\{\mathbf{Q}_{0},\mathbf{Q}_{1},\cdots,\mathbf{Q}_{N-1}\}_{M\times 1}\}}} _{\mathbf{Q}}, \tag{8}\]  
where \(0\leq i\leq N-1\) and \(\mathbf{Q}_{i}=diag\{q(iM),q(iM+1),\cdots,q(iM+M-1)\}\). The PWF \(q(\ell)\) of synthesis filter bank can be extracted from the first-row matrix element of \(\mathbf{Q}\). Similar to (5), the synthesis filter coefficient matrix \(\mathbf{Q}\) can be reinterpreted as:  
\[\mathbf{Q}=(\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\mathbf{\Lambda}^{-1}(\mathbf{F}_{N}\otimes \mathbf{I}_{M}), \tag{9}\]  
Assume that the OTFS-BFDM signal undergoes a linear time-varying (LTV) channel with \(P\) discrete propagation paths. Let \(h_{p}\), \(l_{p}\) and \(k_{p}\) denote the complex path gain, delay tap, and Doppler tap associated with the \(p\)-th path, respectively. Considering the sparsity of channel representation, it is convenient to express the response \(h(\tau,v)\) as [27]:  
\[h(\tau,v)=\sum_{p=1}^{p}h_{p}\delta(t-\frac{l_{p}}{M\Delta f})\delta(v-\frac{ k_{p}}{NT}), \tag{10}\]  
where \(\delta(\cdot)\) is the Dirac delta function.  
Adding a cyclic prefix (CP) of sufficient length to \(\mathbf{s}\) before transmission and discarding the CP at the receiver, the received time-domain signal can be presented by:  
\[\mathbf{\bar{s}}=\mathbf{H}\mathbf{s}+\mathbf{w}, \tag{11}\]  
where \(\mathbf{H}\) is the \(L\times L\) matrix  
\[\mathbf{H}=\sum_{p=1}^{p}h_{p}\mathbf{\Pi}^{t}\mathbf{\rho}\mathbf{\bar{k}}^{k_{p}}. \tag{12}\]In (12), \(\mathbf{\Pi}=circ\{[0,1,0,\cdots,0]_{L\times 1}\}\) and \(\mathbf{\Delta}=diag\{1,e^{j2\pi/L},\cdots,e^{j2\pi(L-1)/L}\}\). The Gaussian noise \(\mathbf{w}\in\mathbb{C}^{L\times 1}\) is independently and identically distributed (i.i.d) with \(w_{i}\sim\mathcal{CN}(0,\sigma_{n}^{2})\), \(i=0,\cdots,L-1\).  
On the receiver, the signal \(\mathbf{\hat{s}}\) can be mapped to \(\mathbf{\hat{d}}\) in DD domain by BFDM demodulation and symplectic finite Fourier (SFFT) transform, which is expressed as:  
\[\mathbf{\hat{d}}= (\mathbf{F}_{N}\otimes\mathbf{F}_{M}^{-1})\mathbf{B}\mathbf{\hat{s}}\] \[= \underbrace{\mathbf{\Lambda}^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{H} (\mathbf{F}_{N}^{-1}\otimes\mathbf{I}_{M})\mathbf{\Delta}}_{\mathbf{H}_{eq}}\mathbf{d}+ \underbrace{\mathbf{\Lambda}^{-1}(\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{w}}_{\mathbf{w}}, \tag{13}\]  
where \(\mathbf{H}_{eq}\in\mathbb{C}^{L\times L}\) is the ECM of the OTFS-BFDM system, \(\mathbf{F}_{N}\otimes\mathbf{F}_{M}^{-1}\) is the symplectic finite Fourier transform (SFFT) transform, and \(\mathbf{\tilde{w}}\in\mathbb{C}^{L\times 1}\) represents the channel noise vector.  
When OTFS-BFDM adopts the shorter rectangular PWF, \(\mathbf{\Lambda}\) evolves into an \(L\times L\) identity matrix. The OTFS-BFDM input-output relationship is re-expressed as:  
\[\mathbf{\hat{d}} =(\mathbf{F}_{N}\otimes\mathbf{F}_{M}^{-1})\mathbf{B}\mathbf{\hat{s}}\] \[=(\underbrace{\mathbf{F}_{N}\otimes\mathbf{I}_{M})\mathbf{H}(\mathbf{F}_{N}^{-1} \otimes\mathbf{I}_{M})}_{\mathbf{H}_{eq}^{Rec}}\mathbf{d}+(\mathbf{F}_{N}\otimes\mathbf{I}_{M}) \mathbf{w}. \tag{14}\]  
In (14), \(\mathbf{H}_{eq}^{Rec}\) is the ECM of the OTFS system, given in [7] to be sparsely connected. Because \(\mathbf{\Lambda}^{-1}\) is a diagonal matrix, \(\mathbf{H}_{eq}\) in (13) is also sparsely connected.