# Hybrid Precoding Design for Wireless Fading Channels
## 3 Proposed Hybrid Precoding
### Linear OMMSE Precoder Design  
Here, we explain how to design the average-based linear precoding matrix, \(\mathbf{G}_{1}\). Since \(\mathbf{H}_{1}\mathbf{x}_{2}=\mathbf{0}\), \(\mathbf{y}_{1}\) can be simply written as follows:  
\[\mathbf{y}_{1}=\beta_{1}\mathbf{H}_{1}\mathbf{W}_{1}\mathbf{s}_{1}+\mathbf{z} _{1}=\beta_{1}\mathbf{H}_{1}\mathbf{V}_{22}\mathbf{G}_{1}\mathbf{s}_{1}+ \mathbf{z}_{1}. \tag{11}\]  
If we define the equalizer for the received signal in group 1 as \(\mathbf{A}_{1}=\text{diag}(a_{1,1},\ldots,a_{1,K})\), the estimate \(\hat{\mathbf{s}}_{1}\) for \(\mathbf{s}_{1}\) can be obtained as follows:  
\[\hat{\mathbf{s}}_{1}=\frac{\mathbf{A}_{1}}{\beta_{1}}\mathbf{y}_{1}=\mathbf{ A}_{1}\mathbf{H}_{1}\mathbf{V}_{22}\mathbf{G}_{1}\mathbf{s}_{1}+\frac{\mathbf{A}_{1}}{ \beta_{1}}\mathbf{z}_{1}. \tag{12}\]  
The mean-squared error (MSE), \(J_{1}\), for group 1 can be defined as follows:  
\[J_{1}=E_{\mathbf{s}_{1},\;\mathbf{z}_{1}}\Bigg{[}\parallel\frac{1}{\beta_{1}} \mathbf{A}_{1}\mathbf{y}_{1}-\mathbf{s}_{1}\parallel^{2}\Bigg{]}=E_{\mathbf{ s}_{1},\;\mathbf{z}_{1}}\Bigg{[}\parallel\mathbf{A}_{1}\mathbf{H}_{1} \mathbf{V}_{22}\mathbf{G}_{1}\mathbf{s}_{1}+\frac{1}{\beta_{1}}\mathbf{A}_{1} \mathbf{z}_{1}-\mathbf{s}_{1}\parallel^{2}\Bigg{]}. \tag{13}\]  
If we calculate the expectation for \(\mathbf{s}_{1}\) and \(\mathbf{z}_{1}\) in this equation, \(J_{1}\) can be rewritten as follows:  
\[J_{1}=\text{tr}\Big{\{}(\mathbf{A}_{1}\mathbf{H}_{1}\mathbf{V}_{22}\mathbf{G} _{1}-\mathbf{I}_{K})(\mathbf{A}_{1}\mathbf{H}_{1}\mathbf{V}_{22}\mathbf{G}_{1 }-\mathbf{I}_{K})^{H}\Big{\}}+\frac{\sigma_{z}^{2}}{\beta_{1}^{2}}\text{tr}( \mathbf{A}_{1}\mathbf{A}_{1}^{H}). \tag{14}\]  
Meanwhile, by defining \(\mathbf{Q}_{1}=\mathbf{A}_{1}\mathbf{H}_{1}\mathbf{V}_{22}\) and using the relationship \(\beta_{1}^{2}=\frac{P_{1}}{\text{tr}(\mathbf{W}_{1}\mathbf{W}_{1}^{H})}=\frac {P_{1}}{\text{tr}(\mathbf{G}_{1}\mathbf{G}_{1}^{H})}\), \(J_{1}\) can be expressed as follows:  
\[J_{1}=\text{tr}\Big{\{}(\mathbf{Q}_{1}\mathbf{G}_{1}-\mathbf{I}_{K})(\mathbf{ Q}_{1}\mathbf{G}_{1}-\mathbf{I}_{K})^{H}\Big{\}}+\frac{\sigma_{z}^{2}}{P_{1}} \text{tr}(\mathbf{A}_{1}\mathbf{A}_{1}^{H})\;\text{tr}(\mathbf{G}_{1}\mathbf{ G}_{1}^{H}). \tag{15}\]  
Now, if we design the linear precoding matrix, \(\mathbf{G}_{1}\), to minimize \(J_{1}\), the optimization problem can be expressed as follows:  
\[\min_{\mathbf{G}_{1}}J_{1}=\min_{\mathbf{G}_{1}}\;\;\text{tr}\Big{\{}(\mathbf{ Q}_{1}\mathbf{G}_{1}-\mathbf{I}_{K})(\mathbf{Q}_{1}\mathbf{G}_{1}-\mathbf{I}_{K})^{H} \Big{\}}+\frac{\sigma_{z}^{2}}{P_{1}}\text{tr}(\mathbf{A}_{1}\mathbf{A}_{1}^{H })\;\text{tr}(\mathbf{G}_{1}\mathbf{G}_{1}^{H}). \tag{16}\]  
In order to find a solution to this optimization problem, the differentiation of \(J_{1}\) by \(\mathbf{G}_{1}^{*}\) is designed to be \(\mathbf{0}\), where \(\mathbf{G}_{1}^{*}\) represents the complex conjugate of \(\mathbf{G}_{1}\).  
\[\begin{array}{ll}\frac{\partial}{\partial\mathbf{G}_{1}^{*}}J_{1}&=\mathbf{ Q}_{1}^{H}(\mathbf{Q}_{1}\mathbf{G}_{1}-\mathbf{I}_{K})+\frac{\sigma_{z}^{2}}{P_{1}} \text{tr}(\mathbf{A}_{1}\mathbf{A}_{1}^{H})\;\mathbf{G}_{1}\\ &=\Big{\{}\mathbf{Q}_{1}^{H}\mathbf{Q}_{1}+\frac{\sigma_{z}^{2}}{P_{1}}\text{ tr}(\mathbf{A}_{1}\mathbf{A}_{1}^{H})\mathbf{I}_{K}\;\Big{\}}\mathbf{G}_{1}-\mathbf{Q}_{1}^{H }=\mathbf{0}.\end{array} \tag{17}\]  
By solving this equation, the solution to the optimization problem can be obtained as follows:  
\[\mathbf{G}_{1,\text{opt}}=\Big{\{}\mathbf{Q}_{1}^{H}\mathbf{Q}_{1}+\frac{ \sigma_{z}^{2}}{P_{1}}\text{tr}(\mathbf{A}_{1}\mathbf{A}_{1}^{H})\mathbf{I}_{K }\;\Big{\}}^{-1}\mathbf{Q}_{1}^{H}. \tag{18}\]