## 8 Multi-Segmental Channel Estimation
### MSE Lower Bound Analysis  
In this subsection, we derive an overall MSE lower bound as a benchmark for the MSCE discussed above. As can be seen from above, the MSCE exploits the linear interpolation to approximate the channels between segments, that is an error approximation in the channel estimates rather than the noise itself. Firstly, the channel estimates after the interpolation can be rewritten in another form as:  
\[\mathbf{\tilde{h}}_{tl}=\mathbf{\Phi}\mathbf{\hat{c}} \tag{61}\]  
where:  
\[\mathbf{\tilde{h}}_{tl}=[\mathbf{\tilde{h}}_{tl}^{T}(0),\ldots,\mathbf{ \tilde{h}}_{tl}^{T}(l),\ldots,\mathbf{\tilde{h}}_{tl}^{T}(L-1)]^{T}\in\mathbb{ C}^{N_{s}L\times 1},\]  
\[\mathbf{\tilde{h}}_{tl}(l)=[\hat{h}_{tl}(0),\ldots,\hat{h}_{tl}(n),\ldots,\hat {h}_{tl}(N_{s}-1)]^{T}\in\mathbb{C}^{N_{s}\times 1},\]  
\[\mathbf{\hat{c}}=[\mathbf{\hat{h}}_{tl}^{T}(0),\ldots,\mathbf{\hat{h}}_{tl}^{ T}(l),\ldots,\mathbf{\hat{h}}_{tl}^{T}(L-1)]\in\mathbb{C}^{TL\times 1}\]  
\[\mathbf{\hat{h}}_{tl}(l)=[\hat{h}_{tl}(0,l),\ldots,\hat{h}_{tl}(t,l),\ldots, \hat{h}_{tl}(T-1,l)]^{T},\]  
and:  
\[\mathbf{\Phi}=\mathbf{I}_{L}\otimes\mathbf{\Psi},\]  
\[\mathbf{\Psi}=[\mathbf{\psi}(0),\ldots,\mathbf{\psi}(n),\ldots,\mathbf{\psi }(N_{s}-1)]^{T}\in\mathbb{R}^{N_{s}\times T} \tag{63}\]  
\[\mathbf{\psi}(n)=[\psi(n,0),\ldots,\psi(n,q),\ldots,\psi(n,T-1)]^{T}\]  
where \(\psi(n,q)\) denotes the interpolation coefficients. Defining the noise term and the residual interference term in as \(\mathbf{z}_{t}\), Equation (59) can be rewritten in a compact matrix form:  
\[\mathbf{y} =\underbrace{\left[ \begin{array}{ccc}\mathbf{\Lambda}_{0}&&\\ &\ddots&\\ &&\mathbf{\Lambda}_{T-1}\\ \end{array} \right]}_{\mathbf{\theta}\in TN_{s}\times TN_{s}}\underbrace{ \left[ \begin{array}{ccc}\mathbf{F}^{H}&&\\ &\ddots&\\ &&\mathbf{F}^{H}\\ \end{array} \right]}_{\mathbf{F}^{\prime}\in TN_{s}\times TN_{s}}\underbrace{ \left[\begin{array}[]  
and:  
\[ \begin{split}&\mathbf{\Lambda}_{t}\in\mathbb{C}^{N_{s}\times M},\\ &\mathbf{\gamma}_{k}=[\underbrace{0,\ldots,}_{k+1}e^{-j\frac{2\pi}{N_{ s}}k\cdot 0},\underbrace{0,}_{lN_{s}},e^{-j\frac{2\pi}{N_{s}}kl},\underbrace{0,}_{(L-l)N_ {s}},e^{-j\frac{2\pi}{N_{s}}k(L-1)},\\ &\underbrace{0,}_{N_{s}-k+1}]^{T}\in\mathbb{C}^{LN_{s}\times 1}, \end{split}  \tag{66}\]  
Hence, Equation (64) can be represented as:  
\[\mathbf{y}=\underbrace{\theta\mathbf{F}^{\prime}\mathbf{S}\mathbf{\Gamma}\mathbf{ \Phi}}_{\mathbf{B}}\mathbf{c}+\mathbf{z}, \tag{67}\]  
where the matrix \(\theta\) denotes the partial FFT and the matrix \(\mathbf{F}^{\prime}\) denotes the standard IFFT. The channel estimates \(\mathbf{\hat{c}}=\mathbf{\hat{h}}_{tl}\) can be obtained by LS estimation accordingly. The overall MSE is based on the approximation errors of linear interpolation and the estimation error of \(\mathbf{\hat{c}}\), namely MSE\({}_{appx}\) and MSE\({}_{est}\), respectively. According to the derivation of [35], the overall MSE can be expressed as:  
\[\text{MSE}_{overall}=\text{MSE}_{appx}+\text{MSE}_{est}. \tag{68}\]  
where:  
\[ \begin{split}\text{MSE}_{appx}=\frac{1}{N_{s}L}\text{tr}\{( \mathbf{I}_{N_{s}}-\mathbf{\Psi}\mathbf{\Psi}^{H})\mathbf{R}\}\\ \text{MSE}_{est}=\frac{1}{N_{s}L}\text{tr}\{\mathbb{E}((\mathbf{ c}-\mathbf{\hat{c}})(\mathbf{c}-\mathbf{\hat{c}})^{H})\}\end{split}  \tag{69}\]  
Note that the unity power of the multipath channel is assumed. Thus, \(\mathbb{E}(\mathbf{h}_{tl}^{H}\mathbf{h}_{tl})=1\). Additionally, the matrix \(\mathbf{R}\) represents the autocorrelation matrix of the channels in the time domain. The autocorrelation matrix of \(\mathbb{E}((\mathbf{c}-\mathbf{\hat{c}})(\mathbf{c}-\mathbf{\hat{c}})^{H})\) can be given by:  
\[\mathbb{E}((\mathbf{c}-\mathbf{\hat{c}})(\mathbf{c}-\mathbf{\hat{c}})^{H})= \mathbb{E}(\mathbf{zz}^{H})\mathbb{E}(\mathbf{BB}^{H}) \tag{70}\]  
where the cross-correlation function of noise is \(\mathbb{E}\{z_{t}(d_{1})z_{t}(d_{2})\}=\frac{\sigma_{v}^{2}}{T}e^{-j\frac{\pi( d_{1}-d_{2})(2t-1)}{T}}\text{sinc}(\frac{\pi(d_{1}-d_{2})}{2T})\), and the notation sinc denotes the sinc function.