# Adaptive Damping Log-Domain Message-Passing Algorithm for FTN-OTFS in V2X Communications
## 2 System Model
### Receiver  
Following passage through the time-varying channel, the received signal \(r(t)\) is given by  
\[\begin{split} r(t)&=\int_{-\infty}^{\infty}\int_{- \infty}^{\infty}h(\tau,\
u)e^{j2\pi\
u(t-\tau)}s(t-\tau)\mathrm{d}\tau\ \mathrm{d}\
u+n(t)\\ &=\sum_{p=1}^{p}h_{p}s\big{(}t-\tau_{p}\big{)}e^{j2\pi\
u_{p}(t- \tau_{p})}+n(t),\end{split} \tag{8}\]  
where \(n(t)\) is the the complex-valued additive white Gaussian noise (AWGN) with a mean of zero and variance \(N_{0}\).  
The received signal \(r(t)\) is then passed through an FTN matched filter \(p^{*}(-t)\). The matched-filtered output can be expressed as  
\[\begin{split} y(t)=& r(t)\star p^{*}(-t)\\ =&\left[\sum_{p=0}^{p-1}h_{p}e^{j2\pi\frac{(k_{p}+ \kappa_{p})(t-p\beta T_{0})}{MN\beta T_{0}}}s\big{(}t-l_{p}\beta T_{0}\big{)}+ n(t)\right]\star p^{*}(-t)\\ =&\sum_{p=0}^{p-1}\sum_{n=0}^{MN-1}h_{p}e^{j2\pi \frac{(k_{p}+\kappa_{p})(t-p\beta T_{0})}{MN\beta T_{0}}}s_{n}g\big{(}t-\big{(} n+l_{p}\big{)}\beta T_{0}\big{)}+\eta(t),\end{split} \tag{9}\]  
where \(\star\) denotes the convolution operation. Specifically, we define \(g(t)\triangleq p(t)\star p^{*}(-t)\) and \(\eta(t)\triangleq n(t)\star p^{*}(-t)\)[24]. Then, \(y(t)\) is sampled with interval \(\beta T_{0}\) and the discrete received signal is obtained as  
\[\mathbf{y}_{\mathrm{T}}=\mathbf{Hs}+\boldsymbol{\eta}, \tag{10}\]  
From (9), the entry in the \(k\)th-row and \(m\)th-column of effective channel matrix \(\mathbf{H}\in\mathbb{C}^{MN\times MN}\) can be written as  
\[\mathbf{H}(k,m)=\sum_{p=0}^{p-1}h_{p}e^{j2\pi\frac{(k_{p}+\kappa_{p})(k-l_{p} )}{MN}}g\big{(}\big{(}k-m-l_{p}\big{)}\beta T_{0}\big{)}. \tag{11}\]  
Next, \(\mathbf{y}_{\mathrm{T}}\) is rearranged into an \(M\times N\) matrix and SEFDM demodulation is performed. The demodulated received signal is given by  
\[\mathbf{Y}_{\mathrm{T}}=\mathbf{F}_{\alpha}\,\mathrm{vec}^{-1}(\mathbf{y}_{ \mathrm{T}}). \tag{12}\]  
Then, OTFS demodulation is applied to \(\mathbf{Y}_{\mathrm{T}}\) to obtain the DD domain received signal \(\mathbf{Y}_{\mathrm{DD}}\). Specifically, an \(M\)-point FFT operation is first applied to each column of \(\mathbf{Y}_{\mathrm{T}}\). Subsequently, the SFFT is performed; this involves an \(M\)-point IFFT applied to the columns, followed by an \(N\)-point FFT applied to the rows. Therefore, \(\mathbf{Y}\)DD can be expressed as  
\[\mathbf{Y}_{\mathrm{DD}}=\mathbf{F}_{M}^{H}\mathbf{F}_{M}\mathbf{Y}_{ \mathrm{T}}\mathbf{F}_{N}=\mathbf{F}_{\alpha}\,\mathrm{vec}^{-1}(\mathbf{y}_{ \mathrm{T}})\mathbf{F}_{N}. \tag{13}\]where \(\mathbf{Y}_{\text{DD}}\in\mathbb{C}^{M\times N}\), and whose vectorized form can be expressed as  
\[\mathbf{y}_{\text{DD}}=\text{vec}(\mathbf{Y}_{\text{DD}})=(\mathbf{F}_{N}\otimes \mathbf{F}_{\alpha})\mathbf{y}_{\text{T}}. \tag{14}\]  
Finally, the relationship between the transmitted vector \(\mathbf{x}_{\text{DD}}\) and the received vector \(\mathbf{y}_{\text{DD}}\) is given by  
\[\mathbf{y}_{\text{DD}} =(\mathbf{F}_{N}\otimes\mathbf{F}_{\alpha})\mathbf{H}\Big{(} \mathbf{F}_{N}^{\text{H}}\otimes\mathbf{F}_{\alpha}^{-1}\Big{)}\mathbf{x}_{ \text{DD}}+(\mathbf{F}_{N}\otimes\mathbf{F}_{\alpha})\boldsymbol{\eta} \tag{15}\] \[=\mathbf{H}_{\text{DD}}\mathbf{x}_{\text{DD}}+\boldsymbol{\eta}_ {\text{DD}}.\]  
where \(\mathbf{x}_{\text{DD}}\), \(\mathbf{y}_{\text{DD}}\), and \(\boldsymbol{\eta}_{\text{DD}}\) are vectors of dimension \(NM\times 1\). The DD domain channel matrix is defined as \(\mathbf{H}_{\text{DD}}=(\mathbf{F}_{N}\otimes\mathbf{F}_{\alpha})\mathbf{H} \Big{(}\mathbf{F}_{N}^{\text{H}}\otimes\mathbf{F}_{\alpha}^{-1}\Big{)}\), and \(\mathbf{H}_{\text{DD}}\in\mathbb{C}^{MN\times MN}\).