## 4 Problem Formulate  
In this section, we design the precoder and combiner according to the MMSE principle with detailed mathematical derivation and propose the form of the optimization problem. By solving this optimization problem, we are able to derive the optimal precoder and combiner that minimize the MSE of the received and transmitted signals. We take the MSE as the performance measure and optimization objective for the joint precoder and combiner design, which is defined as  
\[ \begin{split} J_{\text{MSE}}&=\mathbb{E}\Big{\{}|| \mathbf{F}^{T}\mathbf{y}-\mathbf{s}||^{2}\Big{\}}\\ &=\mathbb{E}\Big{\{}\text{tr}\Big{(}\mathbf{F}^{T}\mathbf{y}-\mathbf{s} \Big{)}\Big{(}\mathbf{F}^{T}\mathbf{y}-\mathbf{s}\Big{)}^{T}\Big{\}}\\ &=\text{tr}\Big{\{}\mathbf{F}^{T}\mathbb{E}\Big{(}\mathbf{yy}^{T} \Big{)}\mathbf{F}-2\mathbf{F}^{T}\mathbb{E}\Big{(}\mathbf{yy}^{T}\Big{)}\Big{\}}+ \Theta,\end{split}  \tag{7}\]  
where the expectation is taken over variables \(\mathbf{y}\) and \(\mathbf{s}\). Consider OOK modulation in the BS, and equal probability for the code bit \(\mathbf{s}\), i.e., \(\text{Pr}(s=-1)=\text{Pr}(s=1)=\frac{1}{2}\)[36]. Hence,  
\[\Theta=\text{tr}\Big{\{}\mathbb{E}\Big{(}\mathbf{s}\mathbf{s}^{T}\Big{)}\Big{\}}=N_{ \text{s}}. \tag{8}\]  
Define the correlation matrix \(\mathbf{R}_{\mathbf{yy}}=\mathbb{E}\big{(}\mathbf{yy}^{T}\big{)}\in\mathbb{R}^{N_{ \text{r}}\times N_{\text{r}}},\mathbf{R}_{\mathbf{ys}}=\mathbb{E}\big{(}\mathbf{y} \mathbf{s}^{T}\big{)}\in\mathbb{R}^{N_{\text{r}}\times N_{\text{s}}}\). Then, we have  
\[J_{\text{MSE}}=\text{tr}\Big{\{}\mathbf{F}^{T}\mathbf{R}_{yy}\mathbf{F}-2 \mathbf{F}^{T}\mathbf{R}_{\mathbf{zs}}\Big{\}}+N_{\text{s}}. \tag{9}\]  
First, we have that \(\mathbf{R}_{\mathbf{yy}}\) is given by  
\[\mathbf{R}_{\mathbf{yy}}=\mathbb{E} \begin{bmatrix}y_{1}^{2}&y_{1}y_{2}&\cdots&y_ {1}y_{N_{\text{r}}}\\ y_{2}y_{1}&y_{2}^{2}&\cdots&y_{2}y_{N_{\text{r}}}\\ \vdots&\vdots&\ddots&\vdots\\ y_{N_{\text{r}}}y_{1}&y_{N_{\text{r}}}y_{2}&\cdots&y_{N_{\text{r}}}^{2}\end{bmatrix} . \tag{10}\]  
For \(1\leqslant i\leqslant N_{\text{r}}\), we have  
\[\mathbb{E}\Big{(}y_{i}^{2}\Big{)}=\sum_{\mathbf{s}}\mathbb{E}\Big{(}y_{i}^{2}|\bm {s}\Big{)}\text{Pr}(\mathbf{s}). \tag{11}\]  
Define a set \(\mathcal{S}_{0}=\{\text{s}^{(j)}\in\{-1,1\},j=1,\cdots,2^{N_{\text{s}}}\}\) with alphabet size \(2^{N_{\text{s}}}\), where each element of \(\mathcal{S}_{0}\) has the _a priori_ probability \(1/2^{N_{\text{s}}}\). Then, (11) can further expand as  
\[\mathbb{E}\Big{(}y_{i}^{2}\Big{)}=\sum_{j=1}^{2^{N_{\text{s}}}}\mathbb{E} \Big{(}y_{i}^{2}|\mathbf{s}^{(j)}\Big{)}\text{Pr}\Big{(}\mathbf{s}^{(j)}\Big{)}. \tag{12}\]  
Based on (6), we have \(\mathbb{E}\Big{(}y_{i}|\mathbf{s}^{(j)}\Big{)}=\mathbf{g}_{i}\mathbf{W}\mathbf{s}_{j}+n_{b}\). Since \(\mathbb{E}\big{(}X^{2}\big{)}=\mathbb{E}^{2}(X)+\text{Var}(X)\) for a random variable \(X\), (12) is equal to  
\[ \begin{split}\mathbb{E}\Big{(}y_{i}^{2}\Big{)}&=\sum_{j=1}^{2 ^{N_{\text{s}}}}\bigg{\{}\Big{[}\mathbf{g}_{i}\mathbf{W}\mathbf{s}^{(j)}+n_{b}\Big{]}^ {2}+\mathbf{g}_{i}\mathbf{W}\mathbf{s}^{(j)}+n_{b}\bigg{\}}\text{Pr}\Big{(}\mathbf{s}^{(j )}\Big{)}\\ &=\frac{1}{2^{N_{\text{s}}}}\sum_{j=1}^{2^{N_{\text{s}}}}\bigg{\{} \Big{[}\mathbf{g}_{i}\mathbf{W}\mathbf{s}^{(j)}+n_{b}\Big{]}^{2}+\mathbf{g}_{i}\mathbf{W} \mathbf{s}^{(j)}+n_{b}\bigg{\}}\\ &=\mathbf{g}_{i}\mathbf{W}\mathbf{W}^{T}\mathbf{g}_{i}^{T}+n_{b}^{2}+n_{ b}.\end{split}  \tag{13}\]Note that we have used the fact that \(\sum_{j=1}^{2N_{s}}\mathbf{s}^{(j)}\Big{[}\mathbf{s}^{(j)}\Big{]}^{T}=2^{N_{s}} \mathbf{I}_{N_{s}}\) and \(\sum_{j=1}^{2N_{s}}\mathbf{s}^{(j)}=\mathbf{0}\) based on our definition. For \(1\leqslant i\leqslant N_{r},1\leqslant j\leqslant N_{r},i\
eq j\), we have  
\[\mathbb{E}\big{(}y_{i}y_{j}\big{)}=\mathbb{E}(y_{i})\mathbb{E}\big{(}y_{j} \big{)}=\sum_{\mathbf{s}}\mathbb{E}(y_{i}|\mathbf{s})\mathrm{Pr}(\mathbf{s}) \sum_{\mathbf{s}}\mathbb{E}\big{(}y_{j}|\mathbf{s}\big{)}\mathrm{Pr}(\mathbf{s })=0. \tag{14}\]  
According to (13) and (14), we have that  
\[\mathbf{R}_{yy} =