# OTFS-Based Handover Triggering in UAV Networks
## Appendix B  
From [32], for any positive definite matrix \(A\in\mathbb{C}^{P\times P}\), \(\text{Tr}\Big{(}\mathbf{A}^{-1}\Big{)}\geq\frac{P^{2}}{\text{Tr}\{\mathbf{A }\}}\). Thus, from (51)  
\[MSE_{H,MMSE}\geq\frac{(QP)^{2}}{\text{Tr}\Big{\{}\Big{(}\boldsymbol{\varphi}_ {e}^{\dagger}\boldsymbol{\Omega}_{g0,DT}^{-1}\boldsymbol{\varphi}_{e}+ \boldsymbol{\Omega}_{h}^{-1}\Big{)}\Big{\}}}, \tag{102}\]  
where  
\[\text{Tr}\Big{\{}\Big{(}\boldsymbol{\varphi}_{e}^{\dagger}\boldsymbol{ \Omega}_{g0,DT}^{-1}\boldsymbol{\varphi}_{e}+\boldsymbol{\Omega}_{h}^{-1} \Big{)}\Big{\}}=\frac{\text{Tr}\big{\{}\boldsymbol{\varphi}_{e}^{\dagger} \boldsymbol{\varphi}_{e}\big{\}}}{\sigma_{e}^{2}\sum_{i=1}^{QP}\sigma_{hi}^{2} +\sigma_{g}^{2}}+\sum_{i=1}^{QP}\frac{1}{\sigma_{hi}^{2}}, \tag{103}\]  
From (46),  
\[\text{Tr}\Big{\{}\boldsymbol{\varphi}_{e}^{\dagger}\boldsymbol{\varphi}_{e} \Big{\}}=\text{Tr}\Big{\{}\sum_{i=1}^{QP}\big{(}\boldsymbol{\Theta}_{i} \boldsymbol{\mathbf{s}}_{DD}^{e}\big{)}^{\dagger}\big{(}\boldsymbol{\Theta}_ {i}\boldsymbol{\mathbf{s}}_{DD}^{e}\big{)}\Big{\}}=\text{Tr}\Big{\{}\sum_{i=1} ^{QP}\big{(}\boldsymbol{\mathbf{s}}_{DD}^{e}\big{)}^{\dagger}\boldsymbol{ \Theta}_{i}^{\dagger}\boldsymbol{\Theta}_{i}\boldsymbol{\mathbf{s}}_{DD}^{e} \Big{\}}, \tag{104}\]  
where  
\[\begin{array}{ll}\boldsymbol{\Theta}_{i}^{\dagger}\boldsymbol{\Theta}_{i}&= \Big{(}\big{(}\mathbf{F}_{N}\bigotimes\mathbf{I}_{\text{Rx}}\big{)}\Big{(} \mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(}\mathbf{F}_{N }^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}\Big{)}^{\dagger}\Big{(} \big{(}\mathbf{F}_{N}\bigotimes\mathbf{I}_{\text{Rx}}\big{)}\Big{(}\mathbf{T}^ {a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(}\mathbf{F}_{N}^{\dagger} \bigotimes\mathbf{I}_{\text{Tx}}\big{)}\Big{)},\\ &=\Big{(}\Big{(}\mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(} \mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}\Big{)}^{ \dagger}\big{(}\mathbf{F}_{N}\bigotimes\mathbf{I}_{\text{Rx}}\big{)}^{\dagger} \big{(}\mathbf{F}_{N}\bigotimes\mathbf{I}_{\text{Rx}}\big{)}\Big{(}\Big{(} \mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(}\mathbf{F}_{N}^ {\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}\Big{)}\\ &=N\Big{(}\Big{(}\mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(} \mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}^{\dagger} \mathbf{I}_{\text{MN}}\Big{(}\Big{(}\mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{ \beta_{i}}\Big{)}\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx }}\big{)}\Big{)}\\ &=N\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}^{ \dagger}\Big{(}\mathbf{T}^{a_{i}}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}\big{(} \mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}\\ &=N\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}^{ \dagger}\big{(}\boldsymbol{\Sigma}^{\beta_{i}}\Big{)}^{\dagger}\mathbf{E}^{ \beta_{i}}\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}} \big{)}\\ &=N\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}^{ \dagger}\mathbf{I}_{\text{MN}}\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{ \text{Tx}}\big{)}\\ &=\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)}^{ \dagger}\big{(}\mathbf{F}_{N}^{\dagger}\bigotimes\mathbf{I}_{\text{Tx}}\big{)} =N^{2}\mathbf{I}_{\text{MN}},\end{array} \tag{105}\]  
Then  
\[\text{Tr}\Big{\{}\boldsymbol{\varphi}_{e}^{\dagger}\boldsymbol{\varphi}_{e} \Big{\}}=\text{Tr}\Big{\{}\!\sum_{i=1}^{QP}\!(\boldsymbol{\mathbf{s}}_{DD}^{e} )^{\dagger}\boldsymbol{\Theta}_{i}^{\dagger}\boldsymbol{\Theta}_{i}\boldsymbol{ \mathbf{s}}_{DD}^{e}\Big{\}}=\text{Tr}\Big{\{}\!N^{2}\!\sum_{i=1}^{QP}\!( \boldsymbol{\mathbf{s}}_{DD}^{e})^{\dagger}\mathbf{I}_{\text{MN}}\boldsymbol{ \mathbf{s}}_{DD}^{e}\Big{\}}=\text{Q}PMN^{3}\sigma_{e}^{2}. \tag{106}\]