# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 3 Performance Analysis
### Ber  
When CSI is well known at the receiver, it can be seen from Equation (6) that the conditional pairwise error probability (CPEP) of misjudging \(\mathbf{x}\) as \(\hat{\mathbf{x}}\) is as follows [28]:  
\[\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}}|\mathbf{H}_{eff})=Q(\sqrt{\frac{ \left|\left|\mathbf{H}_{eff}(\hat{\mathbf{x}}-\mathbf{x})\right|\right|^{2}} {2N_{0,\text{DD}}}}) \tag{13}\]  
where \(Q(\cdot)\) is the Gaussian \(Q\) function. In order to obtain the non-conditional probability, it is necessary to calculate the expected \(\mathbf{H}_{eff}\). To facilitate the solution, the input-output expression is rewritten into a form related to the channel gain \(\mathbf{h}\):  
\[\mathbf{y}=\mathbf{\Phi}(\mathbf{x})\mathbf{h}+\mathbf{\widetilde{n}} \tag{14}\]  
where \(\mathbf{\Phi}(\mathbf{x})=[\mathbf{\Phi}_{1}\mathbf{x},\mathbf{\Phi}_{2} \mathbf{x},\cdots,\mathbf{\Phi}_{p}\mathbf{x}]\in\mathbb{C}^{MN\times P}\), \(\mathbf{\Phi}_{i}=(\mathbf{F}_{N}\otimes\mathbf{G}_{rx})\mathbf{\Pi}^{i} \mathbf{\Delta}^{(k_{i})}(\mathbf{F}_{N}^{H}\otimes\mathbf{G}_{tx})\in\mathbb{ C}^{MN\times MN}\), \(\mathbf{h}=[h_{1},h_{2},\cdots,h_{p}]^{T}\in\mathbb{C}^{P\times 1}\), and \(h_{i}\sim C(0,\sigma^{2})\). The CPEP can be expressed as:  
\[\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}}|\mathbf{h})=Q(\sqrt{\frac{\left| \left(\mathbf{\Phi}(\hat{\mathbf{x}})-\mathbf{\Phi}(\mathbf{x})\right) \mathbf{h}\right|^{2}}{2N_{0,\text{DD}}}})=Q(\sqrt{\frac{\mathbf{\alpha}}{2N_{ 0,\text{DD}}}}) \tag{15}\]  
where \(\mathbf{\alpha}=\mathbf{h}^{H}((\mathbf{\Phi}(\hat{\mathbf{x}})-\mathbf{\Phi }(\mathbf{x}))^{H}((\mathbf{\Phi}(\hat{\mathbf{x}})-\mathbf{\Phi}(\mathbf{x} ))\mathbf{h}=\mathbf{h}^{H}\Theta\mathbf{h}\). Then, substitute the approximation of the \(Q\) function into (15):  
\[\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}}|\mathbf{h})\approx\frac{1}{12}e^{- \frac{\mathbf{\alpha}}{2N_{0,\text{DD}}}}+\frac{1}{4}e^{-\frac{\mathbf{\alpha} }{2N_{0,\text{DD}}}} \tag{16}\]  
The unconditional pairwise error probability (UPEP) is calculated as follows [28]:  
\[\begin{split}\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}})& =E_{\mathbf{h}}(\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}}| \mathbf{h}))\\ &=\frac{1/12}{\det(\mathbf{I}_{P}+\frac{1}{4N_{0,\text{DD}}} \mathbf{K}_{P}\mathbf{\Theta})}+\frac{1/4}{\det(\mathbf{I}_{P}+\frac{1}{3N_{0, \text{DD}}}\mathbf{K}_{P}\mathbf{\Theta})}\end{split} \tag{17}\]  
where \(\mathbf{K}_{P}=E(\mathbf{h}\mathbf{h}^{H})=\sigma^{2}\mathbf{I}_{P}\). It is obvious that \(\mathbf{\Theta}\) is the Hermitian matrix and can be decomposed into \(\mathbf{\Theta}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{H}\), where \(\mathbf{\Lambda}=diag(\lambda_{1},\lambda_{2},\cdots\lambda_{P})\), and \(\lambda_{i}\) is the eigenvalue of \(\mathbf{\Theta}\). Hence, \(\det(\mathbf{\Theta})=\prod_{i=1}^{P}\lambda_{i}\), and we can simplify (17) as:  
\[\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}})=\frac{1/12}{\prod_{i=1}^{P}\left(1 +\frac{\sigma^{2}}{4N_{0,\text{DD}}}\lambda_{i}\right)}+\frac{1/4}{\prod_{i=1} ^{P}\left(1+\frac{\sigma^{2}}{3N_{0,\text{DD}}}\lambda_{i}\right)} \tag{18}\]Finally, the BER can be expressed as:  
\[P_{b}=\frac{1}{pgn_{\mathbf{x}}}\sum_{\mathbf{x}}\sum_{\hat{\mathbf{x}}\
eq\hat{ \mathbf{x}}}\Pr(\mathbf{x}\rightarrow\hat{\mathbf{x}})\,e(\mathbf{x},\hat{ \mathbf{x}}) \tag{19}\]  
where \(n_{\mathbf{x}}=\left(\left(M_{c}\right)^{k/2}n/k\right)^{\delta}\) is the number of potential cases of signal \(\mathbf{x}\), and \(e(\mathbf{x},\hat{\mathbf{x}})\) denotes the number of error bits when \(\mathbf{x}\) is detected as \(\hat{\mathbf{x}}\). Additionally, the error bits from misjudging includes the modulation symbol only, the index selection only, and both.