## 4 The Proposed LSE Scheme
### MSE Analysis  
Next, the analytical expression for MSE of Equations (23) and (24) is obtained in the multipath fading channel. For notational convenience, we may rewrite Equation (10) as  
\[\tilde{Y}_{l}(k)=|H_{l}(k)|^{2}\mathcal{E}_{X}^{2}e^{j2\pi\varphi(k)\rho}\left[ 1+\frac{\{\tilde{C}_{l}(k)+\tilde{Z}_{l}(k)\}e^{-j2\pi\varphi(k)\rho}}{|H_{l}(k )|^{2}\mathcal{E}_{X}^{2}}\right] \tag{25}\]  
where \(|H_{l}(k)|^{2}=\sum_{r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}\), \(\tilde{C}_{l}(k)=\sum_{r=1}^{N_{R}}\tilde{C}_{l}^{r}(k)\), and \(\tilde{Z}_{l}(k)=\sum_{r=1}^{N_{R}}\tilde{Z}_{l}^{r}(k)\). Since CPs are transmitted at a boosted power level of 8.52 dB, under high SNR condition, one safely assumes that \[\angle\tilde{Y}_{l}(k)\approx 2\pi\varphi(k)\rho+\frac{\hat{C}_{l}^{Q}(k)+\hat{Z}_{l} ^{Q}(k)}{|H_{l}(k)|^{2}E_{X}^{2}} \tag{26}\]  
where \(\hat{C}_{l}(k)=\hat{C}_{l}(k)e^{-j2\pi\varphi(k)\rho}\) and \(\hat{Z}_{l}(k)=\hat{Z}_{l}(k)e^{-j2\pi\varphi(k)\rho}\). Notice that \(\hat{C}_{l}(k)\) and \(\hat{Z}_{l}(k)\) are statistically identical to \(\hat{C}_{l}(k)\) and \(\hat{Z}_{l}(k)\), respectively. With the use of optimal subsets \(\mathcal{S}_{\hat{b}}^{+}\) and \(\mathcal{S}_{\hat{d}}^{-}\), the estimation error for \(\varepsilon\) and \(\xi\) is computed by  
\[\hat{\varepsilon}-\varepsilon=\frac{1}{2\pi\rho N_{p}E_{X}^{2}}\sum_{k\in \mathcal{S}_{\hat{b}}^{+}\cup\mathcal{S}_{\hat{d}}^{-}}\frac{\hat{C}_{l}^{Q}( k)+\hat{Z}_{l}^{Q}(k)}{|H_{l}(k)|^{2}} \tag{27}\]  
and  
\[\hat{\xi}-\xi=\frac{1}{2\pi\rho M_{p}E_{X}^{2}}\sum_{k\in\mathcal{S}_{\hat{b} }^{+}\cup\mathcal{S}_{\hat{d}}^{-}}\frac{k\hat{C}_{l}^{Q}(k)+k\hat{Z}_{l}^{Q}( k)}{|H_{l}(k)|^{2}}. \tag{28}\]  
Recognizing that \(E\{\hat{C}_{l}^{Q}(k)\}=E\{\hat{Z}_{l}^{Q}(k)\}=0\), we obtain the following expressions:  
\[ \begin{split} E\left\{|\hat{\varepsilon}-\varepsilon|^{2}\right\} &=\left(\frac{1}{2\pi\rho N_{p}E_{X}^{2}}\right)^{2}\sum_{k\in \mathcal{S}_{\hat{b}}^{+}\cup\mathcal{S}_{\hat{d}}^{-}}E\left\{\frac{|\hat{C} _{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}\\ &+\left(\frac{1}{2\pi\rho N_{p}E_{X}^{2}}\right)^{2}\sum_{k\in \mathcal{S}_{\hat{b}}^{+}\cup\mathcal{S}_{\hat{d}}^{-}}E\left\{\frac{|\hat{Z} _{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}\end{split}  \tag{29}\]  
and  
\[ \begin{split} E\left\{|\hat{\xi}-\xi|^{2}\right\}& =\left(\frac{1}{2\pi\rho M_{p}E_{X}^{2}}\right)^{2}\sum_{k\in \mathcal{S}_{\hat{b}}^{+}\cup\mathcal{S}_{\hat{d}}^{-}}k^{2}E\left\{\frac{| \hat{C}_{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}\\ &+\left(\frac{1}{2\pi\rho M_{p}E_{X}^{2}}\right)^{2}\sum_{k\in \mathcal{S}_{\hat{b}}^{+}\cup\mathcal{S}_{\hat{d}}^{-}}k^{2}E\left\{\frac{|\hat {Z}_{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}.\end{split}  \tag{30}\]  
By placing the antennas sufficiently apart, the channel between all transmit-receive antenna couples is considered to be statistically independent. Referring from Equation (11), the variance of \(\hat{C}_{l}^{Q}(k)\) in Equation (29) can be readily shown to be  
\[E\left\{\frac{|\hat{C}_{l}^{Q}(k)|^{2}}{|H_{l}(k)|^{4}}\right\}=E_{X}^{3}\sigma _{C}^{2}\sum_{r=1}^{N_{R}}E\left\{\frac{|H_{l}^{r}(k)|^{2}}{\left(\sum_{r=1}^{ N_{R}}|H_{l}^{r}(k)|^{2}\right)^{2}}\right\}+N_{R}E_{X}^{2}\sigma_{C}^{2}E \left\{\frac{1}{|H_{l}(k)|^{4}}\right\}\left(\sigma_{Z}^{2}+\frac{\sigma_{C}^ {2}}{2}\right) \tag{31}\]  
where the first expectation of the RHS is further derived into  
\[ \begin{split}\sum_{r=1}^{N_{R}}E\left\{\frac{|H_{l}^{r}(k)|^{2}}{ \left(\sum_{r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}\right)^{2}}\right\}&=E \left\{\sum_{r=1}^{N_{R}}\frac{|H_{l}^{r}(k)|^{2}}{\left(\sum_{r=1}^{N_{R}}|H _{l}^{r}(k)|^{2}\right)^{2}}\right\}\\ &=E\left\{\frac{\sum_{r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}}{\left(\sum_{ r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}\right)^{2}}\right\}\\ &=E\left\{\frac{1}{\sum_{r=1}^{N_{R}}|H_{l}^{r}(k)|^{2}}\right\} \end{split} .