# OTFS-Based Handover Triggering in UAV Networks
## 4 OTFS Modulation/Demodulation
### OTFS Modulation  
The first step in the OTFS modulation is to convert the DD matrix \(s_{DD}[l,k]\in\mathbb{C}^{M\times N}\) containing quadrature amplitude modulation (QAM) Tx symbols into its corresponding  
Figure 2: Baseband block diagram of the proposed DD channel gain-based handover triggering using OTFS modulation.  
FT matrix \(s_{FT}[m,n]\in\mathbb{C}^{M\times N}\) using ISFFT operation. ISFFT is a two-dimensional Fourier transform, where FFT is applied on the delay space of \(s_{DD}[l,k]\), while IFFT is applied on the Doppler space of it [12]:  
\[s_{FT}[m,n]=\sum_{d=0}^{M-1}\sum_{k=0}^{N-1}s_{DD}[l,k]e^{2\pi(\frac{\alpha_{ \mathrm{I}}}{N}-\frac{\alpha_{\mathrm{I}}}{M})}, \tag{5}\]  
In matrix notation [13],  
\[\mathbf{S}_{FT}=\mathbf{F}_{M}\mathbf{S}_{DD}\mathbf{F}_{N}^{\dagger}, \tag{6}\]  
where both \(\{\mathbf{S}_{DD},\mathbf{S}_{FT}\}\in\mathbb{C}^{M\times N}\) indicate the complex DD and FT matrices of \(s\). \(\mathbf{F}_{N}\) and \(\mathbf{F}_{M}\) are the discrete Fourier transform (DFT) matrices of sizes \(N\) and \(M\), respectively, where \(\mathbf{F}_{M}(m,l)=e^{-\frac{\beta_{\mathrm{I}}m}{M}}\), \(\mathbf{F}_{N}(n,k)=e^{\frac{\beta_{\mathrm{I}}mck}{N}}\), and the superscript \(\dagger\) indicates the Hermitian transpose. Then, Heisenberg transform using windowing function \(b_{T\mathrm{X}}(t)\) is used to convert \(s_{FT}[m,n]\) into the continuous DT signal \(x_{DT}(t)\):  
\[x_{DT}(t)=\sum_{m=0}^{M-1}\sum_{m=0}^{N-1}s_{FT}[m,n]b_{T\mathrm{X}}(t-nT)e^{ 2\pi m\Delta/(t-nT)}, \tag{7}\]  
In the matrix notation, (7) is equivalent to performing column-wise IFFT on \(\mathbf{S}_{FT}\) with pulse shaping as follows:  
\[\mathbf{X}_{DT}=\mathbf{B}_{T\mathrm{X}}\mathbf{F}_{M}^{\dagger}\mathbf{S}_{ FT}\overset{\eqref{eq:f_T}}{\Rightarrow}\ \mathbf{B}_{T\mathrm{X}}\mathbf{S}_{DD}\mathbf{F}_{N}^{\dagger} \tag{8}\]  
where \(\overset{\eqref{eq:f_T}}{\Rightarrow}\) means by substituting (6), \(\mathbf{X}_{DT}\in\mathbb{C}^{M\times N}\) is a matrix containing \(N\)-OFDM symbols in its column, and \(\mathbf{B}_{T\mathrm{X}}=\mathrm{diag}\Big{[}b_{T\mathrm{X}}(0),\ b_{T\mathrm{X}} \Big{(}\frac{T}{M}\Big{)},\ \dots,b_{T\mathrm{X}}\Big{(}\frac{nT}{M}\Big{)},\ \dots,b_{T\mathrm{X}}\Big{(}\frac{(M-1)T}{M}\Big{)} \Big{]}\in\mathbb{C}^{M\times M}\) is an \(M\times M\) diagonal matrix containing the sampled \(b_{T\mathrm{X}}\) in its diagonal. For rectangular pulse, i.e., \(\mathbf{B}_{T\mathrm{X}}=\mathbf{I}_{M}\), where \(\mathbf{I}_{M}\) is the identity matrix, \(\mathbf{X}_{DT}=\mathbf{S}_{DD}\mathbf{F}_{N}^{\dagger}\), which is called the IZT on \(\mathbf{S}_{DD}\)[13; 14]. Thus, the whole OTFS modulation, including both ISSFT and Heisenberg transform with windowing, is just equivalent to performing IZT on the DD data matrix. By applying vectorization on (8), the transmitted column-wise of \(\mathbf{x}_{DT}\in\mathbb{C}^{MN\times 1}\) becomes the following:  
\[\mathbf{x}_{DT}=\mathrm{vec}(\mathbf{X}_{DT})=\Big{(}\mathbf{F}_{N}^{\dagger} \bigotimes\mathbf{B}_{T\mathrm{X}}\Big{)}\mathbf{s}_{DD}, \tag{9}\]  
where \(\ \mathbf{s}_{DD}=\mathrm{vec}(\mathbf{S}_{DD})\)\(\in\mathbb{C}^{MN\times 1}\) is the column-wise vectorization of \(\mathbf{S}_{DD}\), and \(\bigotimes\) means the Hadamard product.