# OTFS-Based Handover Triggering in UAV Networks
## 6 Mathematical Analysis of the Channel Estimation Error Effect
### MMSE-Based OTFS DD Channel Estimation  
For evaluating \(MSE_{\mathcal{H},MMSE}\), we consider the MMSE channel estimation technique given in [26]. In this technique, pilot and data symbols are superimposed in the DD domain. Then, the MMSE channel estimator is used to estimate the DD channel based on the channel variance, the noise variance, and the variance of the pilot and the transmitted data. In this scheme, the transmitted \(MN\times 1\) DD vector containing both pilot and data symbols is \(s_{DD}=s_{DD}^{\epsilon}+s_{DD}^{\rho}\), where \(e\) indicates the pilot symbols, and \(o\) refers to the data symbols. From (23), the received DD vector becomes the following:  
\[\mathbf{y}_{DD}=\frac{1}{N}\Big{(}\big{(}\mathbf{F}_{N}\bigotimes\mathbf{B}_{ K\chi}\big{)}\mathbf{H}_{DD}\Big{(}\mathbf{F}_{N}^{\dagger}\bigotimes B_{ \mathrm{Tx}}\Big{)}\Big{(}s_{DD}^{q}+s_{DD}^{\rho}\Big{)}+\Big{(}\mathbf{F}_{N }\bigotimes\mathbf{B}_{R\chi}\Big{)}\mathbf{g}_{DT}\Big{)}, \tag{43}\]By considering rectangular pulses at Tx and Rx, i.e., \(\mathbf{B}_{Tx}=\mathbf{B}_{Rx}=\mathbf{I}_{M}\) and \(\mathbf{H}_{DD}\) given in (12), \(\mathbf{y}_{DD}\) can be re-written as follows:  
\[\mathbf{y}_{DD}=\frac{1}{N}\Big{(}\Big{(}\sum_{q=1}^{Q}\sum_{i_{q}=1}^{P_{q}}h_ {i_{q}}\mathbf{\Theta}^{i_{q}}\Big{)}(s_{DD}^{\epsilon}+s_{DD}^{\epsilon})+ \widetilde{g}_{DT}\Big{)}, \tag{44}\]  
where \(\mathbf{\Theta}^{i_{q}}=(\mathbf{F}_{N}\bigotimes\mathbf{I}_{Rx})\Big{(} \mathbf{T}^{\epsilon_{iq}}\mathbf{\Sigma}^{\theta_{iq}}\Big{)}\big{(}\mathbf{F }_{N}^{\epsilon}\bigotimes\mathbf{I}_{Tx}\big{)}\in\mathbb{C}^{MN\times MN}\), and \(\widetilde{g}_{DT}=(\mathbf{F}_{N}\bigotimes\mathbf{I}_{Rx})g_{DT}\). In compact form, and by assuming all coherence periods have the same number of paths, i.e., \(P_{q}=P\ \forall q\), then (44) can be re-written as follows:  
\[\mathbf{y}_{DD}=\frac{1}{N}\Big{(}\boldsymbol{\varphi}_{e}\boldsymbol{h}+ \boldsymbol{\varphi}_{o}\boldsymbol{h}+\widetilde{g}_{DT}\Big{)}=\frac{1}{N} \Big{(}\boldsymbol{\varphi}_{e}\boldsymbol{h}+\widetilde{g}_{go,DT}\Big{)} \tag{45}\]  
where \(\boldsymbol{h}\in\mathbb{C}^{QP\times 1}\) is the vector of the concatenated channel path gains. \(\widetilde{g}_{go,DT}\) is the random vector combining both the AWGN effect and the interference from data symbols on the pilot symbols. Also, \(\boldsymbol{\varphi}_{e}\), \(\boldsymbol{\varphi}_{o}\in\mathbb{C}^{MN\times QP}\) are the concatenated matrices corresponding to the pilot and data symbols, respectively. \[\boldsymbol{\varphi}_{e}=\left[\mathbf{\Theta}^{1}s_{DD}^{\epsilon},\ \mathbf{\Theta}^{2}s_{DD}^{\epsilon},\ \ldots,\ \mathbf{\Theta}^{QP}s_{DD}^{\epsilon}\ \right], \tag{46}\]  
\[\boldsymbol{\varphi}_{o}=\left[\mathbf{\Theta}^{1}s_{DD}^{\epsilon},\ \mathbf{\Theta}^{2}s_{DD}^{\epsilon},\ \ldots,\ \mathbf{\Theta}^{QP}s_{DD}^{\epsilon}\ \right], \tag{47}\]  
As proven in [26], the MMSE of the channel \(\boldsymbol{h}_{MMSE}\) is given as follows:  
\[\boldsymbol{\hat{h}}_{MMSE}=\Big{(}\boldsymbol{\varphi}_{e}^{\intercal} \boldsymbol{\Omega}_{go,DT}^{-1}\boldsymbol{\varphi}_{e}+\boldsymbol{\Omega} _{h}^{-1}\Big{)}^{-1}\boldsymbol{\varphi}_{e}^{\intercal}\boldsymbol{\Omega} _{co,DT}^{-1}\mathbf{y}_{DD}, \tag{48}\]  
where \(\boldsymbol{\Omega}_{co,DT}^{-1}\) is the co-variance matrix of the AWGN, interference comes from the superimposed data, and \(\boldsymbol{\Omega}_{h}^{-1}\) is the DD channel covariance matrix, which are expressed as follows:  
\[\boldsymbol{\Omega}_{go,DT}=\Big{(}\Big{(}\sum_{q=1}^{Q}\sum_{i_{q}=1}^{P} \sigma_{hi_{q}}^{2}\Big{)}\sigma_{o}^{2}+\sigma_{g}^{2}\Big{)}\mathbf{I}_{MN}, \tag{49}\]  
\[\boldsymbol{\Omega}_{h}^{-1}=\text{diag}\Big{\{}\sigma_{h_{1}}^{2},\sigma_{2} ^{2},\ldots,\sigma_{h_{i}}^{2},\ldots,\ \sigma_{h_{QP}}^{2}\Big{\}}, \tag{50}\]  
where \(\sigma_{hi_{q}}^{2}\) is the channel gain of path \(i_{q}\) in the coherence period \(q\), where \(i=i_{q}+(q-1)\times P\), \(1\leq q\leq Q\), and \(\sigma_{o}^{2}\) is the variance of the random data.