# OTFS-Based Handover Triggering in UAV Networks
## 4 OTFS Modulation/Demodulation
### OTFS Demodulation  
Based on (1), the received continuous DT signal, \(r_{DT}(t)\), at the front of the Rx side can be expressed as follows [12; 13; 14]:  
\[r_{DT}(t)=\sum_{d=1}^{\mathrm{Q}}\sum_{i_{q}=1}^{P_{q}}h_{i_{q}}e^{2\pi\beta_{ i_{q}}(t-\alpha_{i_{q}})}x_{DT}\Big{(}t-\alpha_{i_{q}}\Big{)}+g_{DT}(t) \tag{10}\]  
where \(g_{DT}(t)\sim\mathcal{N}\Big{(}0,\sigma_{\tilde{g}}^{2}\Big{)}\) is AWGN with zero mean and variance of \(\sigma_{\tilde{g}}^{2}\). In discrete from, \(r_{DT}(t)\) is sampled at \(t=z\frac{T}{M}\), \(0\leq z\leq MN-1\), producing \(r_{DT}[z]\):  
\[r_{DT}[z]=\sum_{q=1}^{\mathrm{Q}}\sum_{i_{q}=1}^{P_{q}}h_{i_{q}}e^{2\pi\beta_{ i_{q}}(z-\alpha_{i_{q}})}x_{DT}\Big{(}\Big{[}z-\alpha_{i_{q}}\Big{]}_{MN} \Big{)}+g_{DT}(z) \tag{11}\]  
where \([. ]_{\lambda}\) means mode \(\lambda\) operation. In matrix-vector notation, the vector of received symbols \(r_{DT}\in\mathbb{C}^{MN\times 1}\) can be expressed as the multiplication of the circulant DD channel matrix \(\mathbf{H}_{DD}\in\mathbb{C}^{MN\times MN}\) and the vector of transmitted symbols \(\mathbf{x}_{DT}\in\mathbb{C}^{MN\times 1}\)[26]:  
\[\mathbf{r}_{DT}=\mathbf{H}_{DD}\mathbf{x}_{DT}+g_{DT}, \tag{12}\]where \(\mathbf{x}_{DT}=\mathrm{vec}(\mathbf{X}_{DT})\), and \(\mathbf{g}_{DT}\in\mathbb{C}^{MN\times 1}\) is the AWGN vector. In this context, \(\mathbf{H}_{DD}=\sum_{q=1}^{Q}\Sigma_{i_{q}=1}^{P_{q}}I_{i_{q}}\mathbf{T}^{\mathbf{x} _{iq}}\mathbf{\Sigma}^{\mathbf{g}_{iq}}\) with \(\mathbf{T}^{\mathbf{x}_{iq}}\) is the permutation matrix of size \(MN\times MN\) related to delay spread \(\alpha_{i_{q}}\) and \(\mathbf{\Sigma}^{\mathbf{g}_{iq}}\) is a diagonal matrix of size \(MN\times MN\) containing the effect of Doppler shift \(\beta_{i_{q}}\), where \(\mathbf{T}\) is given as follows [26]:  
\[\mathbf{T}= \begin{bmatrix}0&\cdots&0&1\\ 1&\ddots&0&0\\ \vdots&\ddots&\ddots&\vdots\\ 0&\cdots&1&0\end{bmatrix} _{MN\times MN}, \tag{13}\]  
while \(\mathbf{\Sigma}\) is expressed as follows [26]:  
\[\mathbf{\Sigma}=\mathrm{diag}\Big{[}1,\ e^{2\pi},\ e^{4\pi},\ \ldots,\ e^{2\pi(MN-1)} \Big{]}, \tag{14}\]  
Thus, each channel path \(i_{q}\) introduces an \(\alpha_{i_{q}}\) cyclic shift to the transmitted vector \(\mathbf{x}_{DT}\) modeled by \(\mathbf{T}^{\mathbf{x}_{iq}}\),while its introduced Doppler shift \(\beta_{i_{q}}\) is modeled by \(\mathbf{\Sigma}^{\mathbf{\beta}_{iq}}\). The received \(r_{DT}(t)\) signal in (10) is first converted to the FT signal \(y(f,t)\) by applying the Wigner transform (inverse of the Heisenberg transform) using the received pulse \(b_{Rx}(t)\)[13]:  
\[y(f,t)=\int b_{Rx}^{*}\big{(}t^{\prime}-t)r_{DT}\big{(}t^{\prime}\big{)}e^{2 \pi f(t^{\prime}-t)}dt^{\prime} \tag{15}\]  
Then, it is sampled at \(f=m\Delta f\) and \(t=nT\) to produce \(y_{FT}[m,n]\) in FT domain [13]:  
\[y_{FT}[m,n]=\left.y(f,t)\right|_{f=m\Delta f,t=nT^{\prime}} \tag{16}\]  
Based on the detailed derivations given in [13], the relation between \(y_{FT}[m,n]\) and \(s_{FT}[m,n]\) including channel and noise effects can be written as follows [13]:  
\[y_{FT}[m,n]=H_{FT}[m,n]s_{FT}[m,n]+G_{FT}[m,n], \tag{17}\]  
where \(G_{FT}[m,n]\) is the sampled AWGN, and \(H_{FT}[m,n]\) is the channel response in the FT domain, expressed as follows [13]:  
\[H_{FT}[m,n]=\sum_{l=0}^{M-1}\sum_{k=0}^{N-1}h_{DD}[l,k]e^{j2\pi(\frac{M}{N}- \frac{m}{M})}, \tag{18}\]  
Simply, (18) is the ISFFT of \(h_{DD}[l,k]\) given in (2). In discrete form, the Wigner transform in (15) is equivalent to performing Fourier transform over the columns of the windowed received matrix \(\mathbf{R}_{DT}=\mathrm{vec}^{-1}(\mathbf{r}_{DT})\in\mathbb{C}^{M\times N}\), which results from un-vectorizing \(\mathbf{r}_{DT}\), giving the following:  
\[\mathbf{Y}_{FT}=\mathbf{F}_{M}\mathbf{B}_{Rx}\mathbf{R}_{DT}, \tag{19}\]  
where \(\mathbf{B}_{Rx}=\mathrm{diag}\Big{[}b_{Rx}(0),\ \ldots,\ b_{Rx}\Big{(}\frac{mT}{M} \Big{)},\ \ldots,\ b_{Rx}\Big{(}\frac{(M-1)T}{M}\Big{)}\Big{]}\in\mathbb{C}^{M\times M}\) is an \(M\times M\) diagonal matrix containing the sampled \(b_{Rx}\) in its diagonal.