## 4 Energy Efficiency
### Optimization of Energy Efficiency  
The average total power consumption of a single satellite can be written as  
\[P_{l}{}^{total}=\frac{1}{\xi_{\max}}\mathbb{E}\Big{\{}\|f(\mathbf{F_{l}} \mathbf{s})\|^{2}\Big{\}}+P_{c}+P_{0}. \tag{18}\]  
Hence, the objective function in Equation (16) can be further written as  
\[\frac{\frac{B_{w}R_{\max}}{\frac{1}{\xi_{\max}}\sum\limits_{l=1}^{L}\mathbb{E} \Big{\{}\|f(\mathbf{F_{l}}\mathbf{s})\|^{2}\Big{\}}+LP_{c}+LP_{0}}}{\rm.} \tag{19}\]  
To further process the objective function (19), a nonlinear variable transformation is used to transform the objective function into a linear programming problem by introducing an auxiliary variable \(\chi\). Function (19) can be reformulated into the following univariate equation:  
\[B_{w}R_{sum}-\chi\Big{(}\frac{1}{\xi_{\max}}\sum\limits_{l=1}^{L}\mathbb{E} \Big{\{}\|f(\mathbf{F_{l}}\mathbf{s})\|^{2}\Big{\}}+LP_{c}+LP_{0}\Big{)}. \tag{20}\]  
The range of values of \(\chi\) can be obtained from [29] as  
\[0\leq\chi\leq\frac{B_{w}\sum\limits_{k=1}^{K}\log_{2}(1+\frac{\left|\mathbf{h }_{k}^{H}\mathbf{A_{k}}\mathbf{G_{F_{l}}}\right|^{2}}{\sigma^{2}})}{LP_{c}+LP_ {0}}. \tag{21}\]  
If \(\chi\) makes the optimal value of Function (20), then the solution of the above optimization function (Function (20)) is the optimal solution of Function (19). The optimal value of \(\chi\) can be found based on the algorithm for finding the optimal univariate parameter in [29]. Assuming below that the value of \(\chi\) is given, the objective function (Function (20)) can be further rewritten as  
\[\Phi(\mathbf{F})=B_{w}R_{sum}-\frac{\chi}{\xi_{\max}}\sum\limits_{l=1}^{L} \mathbb{E}\Big{\{}\|f(\mathbf{F_{l}}\mathbf{s})\|^{2}\Big{\}}. \tag{22}\]  
The objective function and constraints in (16) are non-convex and the relevant optimization variables are coupled. In the following, we approach this problem by using a WMMSE-based algorithm to exploit the relationship between the achievable rate and the MSE of the optimal receiver [22]. Based on the analysis above, the problem (16) can be equated to the following WMMSE problem [30]:  
\[ \begin{array}{l}\mathop{\min}\limits_{\mathbf{r},\mathbf{w},\mathbf{F}}B_{ w}\sum\limits_{k=1}^{K}\left(w_{k}\epsilon_{k}-\log w_{k}\right)+\frac{\chi}{\xi_{ \max}}\sum\limits_{l=1}^{L}\mathbb{E}\Big{\{}\|f(\mathbf{F_{l}}\mathbf{s})\|^ {2}\Big{\}}\\ {\rm s.t.\ }\mathbb{E}\Big{\{}\|f(\mathbf{F_{l}}\mathbf{s})\|^{2}\Big{\}}\leq P,\forall l\in\{1,2,\cdots,L\},\end{array}  \tag{23}\]where \(w_{k}\) denotes the weight coefficient of the \(k\)th GU and \(e_{k}\) denotes the mean square error (MSE) of the \(k\)th GU, which can be written as  
\[ \begin{split} e_{k}&=\mathbb{E}\Big{\{}\big{(}\hat{s }_{k}-s_{k}\big{)}(\hat{s}_{k}-s_{k})^{H}\Big{\}}\\ &=r_{k}\mathbf{h}_{k}^{H}\sum\limits_{i=1}^{K}\mathbf{A}_{i} \bar{\mathbf{G}}\mathbf{F}_{i}\mathbf{F}_{i}^{H}\bar{\mathbf{G}}^{H}\mathbf{A }_{i}^{H}\mathbf{h}_{k}r_{k}^{H}+r_{k}\mathbf{h}_{k}^{H}\sum\limits_{i=1}^{K} \mathbf{A}_{i}\mathbf{D}\mathbf{A}_{i}^{H}\mathbf{h}_{k}r_{k}^{H}+\sigma^{2}r_ {k}r_{k}^{H}\\ &\qquad-r_{k}\mathbf{h}_{k}^{H}\mathbf{A}_{k}\bar{\mathbf{G}} \mathbf{F}_{k}-\mathbf{F}_{k}^{H}\bar{\mathbf{G}}^{H}\mathbf{A}_{k}^{H} \mathbf{h}_{k}r_{k}^{H}+1,\end{split}  \tag{24}\]  
where \(\hat{s}_{k}=r_{k}y_{k}\) denotes the estimated signal for the \(k\)th GU, and \(r_{k}\) is the receiver gain. We use a block coordinate descent (BCD) method to solve this problem. The WMMSE objective function is minimized by fixing two of the three variables sequentially and updating the third.