## 4 Energy Efficiency
### Optimization of Energy Efficiency
The closed update form for \(r\) and \(w\) can be derived as  
\[r_{k}=\frac{\mathbf{F}_{k}^{H}\bar{\mathbf{G}}^{H}\mathbf{A}_{i}^{H}\mathbf{h }_{k}}{\mathbf{h}_{k}^{H}\sum\limits_{i=1}^{K}\mathbf{A}_{i}\bar{\mathbf{G}} \mathbf{F}_{i}\mathbf{F}_{i}^{H}\bar{\mathbf{G}}^{H}\mathbf{A}_{i}^{H} \mathbf{h}_{k}+\mathbf{h}_{k}^{H}\sum\limits_{i=1}^{K}\mathbf{A}_{i}\mathbf{ D}\mathbf{A}_{i}^{H}\mathbf{h}_{k}+\sigma^{2}}, \tag{25}\]  
\[w_{k}=e_{k}{}^{-1}. \tag{26}\]  
Fixing \(\mathbf{r}\) and \(\mathbf{w}\), the problem in Equation (23) can be transformed into  
\[ \begin{split}\min\limits_{\mathbf{F}}T_{mmse}(\mathbf{F})\\ \text{s.t.}&\mathbb{E}\Big{\{}\|f(\mathbf{F}_{i} \mathbf{s})\|^{2}\Big{\}}\leq P,\forall l\in\{1,2,\cdots,L\},\end{split}  \tag{27}\]  
where \(T_{mmse}(\mathbf{F})=B_{w}\sum\limits_{k=1}^{K}\left(w_{k}e_{k}-\log w_{k} \right)+\frac{\chi}{\xi_{\max}}\sum\limits_{l=1}^{L}\mathbb{E}\Big{\{}\|f( \mathbf{F}_{i}\mathbf{s})\|^{2}\Big{\}}\). This problem equation (Equation (27)) is non-convex for the precoding matrix due to the complex nonlinear distortion terms involved in the rate utility and power constraints. We utilize an enhanced projected gradient descent method to solve this problem efficiently [22]. The gradient of the objective function \(\Phi(\mathbf{F})\) is evaluated using a gradient descent method with [PERSON]'s accelerated momentum at the estimated position, which can be expressed as [22]  
\[ \begin{split}\Delta\mathbf{F}^{(i)}&=\alpha\Delta \mathbf{F}^{(i-1)}-\eta^{(i-1)}\
abla T_{mmse}\Big{(}\mathbf{F}^{(i-1)}+\mu \Delta\mathbf{F}^{(i-1)}\Big{)},\\ \bar{\mathbf{F}}^{(i)}&=\mathbf{F}^{(i-1)}+\Delta \mathbf{F}^{(i)},\end{split}  \tag{28}\]  
where \(\alpha\) reflects the coupling of the gradient update at the current iteration with the previous iteration, and \(\mu\) is involved in predicting the previous position step. The parameter \(\eta^{(i-1)}\) corresponds to the learning rate in the gradient update. \(\eta^{(i-1)}=\varepsilon/\varphi^{(i)}\), where \(\varphi^{(i)}\) denotes the weighted mean square of the gradient and is expressed as  
\[\varphi^{(i)}=\varsigma\varphi^{(i-1)}+(1-\varsigma)\sum\Big{|}\
abla T_{mmse }\Big{(}\mathbf{F}^{(i-1)}+\mu\Delta\mathbf{F}^{(i-1)}\Big{)}\Big{|}^{2}. \tag{29}\]  
The gradient of the objective function \(\Phi(\mathbf{F})\) is calculated by [22; 23]  
\[ \begin{split}\left[\
abla_{\mathbf{F}}T_{mmse}(\mathbf{F}) \right]_{ln,k}\approx\frac{\Big{(}T_{mmse}\Big{(}\mathbf{F}+\Delta_{re}^{( ln,k)}\Big{)}-T_{mmse}(\mathbf{F})\Big{)}}{\delta}\\ +\frac{j\Big{(}T_{mmse}\Big{(}\mathbf{F}+\Delta_{im}^{(ln,k)} \Big{)}-T_{mmse}(\mathbf{F})\Big{)}}{\delta},\end{split}  \tag{30}\]where \(l=1,2,\cdots,L\), \(n=1,2,\cdots,N\), \(k=1,2,\cdots,K\) and \(\delta\) is an extremely small normal number. \(\Delta_{re}^{(lnk)}\) is an \(LN\times K\) matrix with \(\delta\) at the \(l\times n\)th row and \(k\)th column and zero entries at all other positions. Analogously, \(\Delta_{im}^{(lnk)}\) is an \(LN\times K\) matrix with only non-zero entry being \(j\delta\) at the \(l\times n\)th row and \(k\)th column. After calculating \(\tilde{\mathbf{F}}^{(i)}\), \(\tilde{\mathbf{F}}^{(i)}\) needs to be projected to the feasible region to ensure that the power constraints through the bisection method. \[\mathbf{F}^{(i)}\tilde{\mathbf{F}}^{(i)}=\Pi_{C}\Big{\{}\tilde{\mathbf{F}}^{(i )}\Big{\}}. \tag{31}\]  
This WMMSE-based precoding algorithm is summarized in Algorithm 2. ```
Input:\(\mathbf{h}_{k,l},\forall k,l,P\). Output:\(\mathbf{F}\). Initialization: set parameters \(\varphi\), \(\alpha\), \(\mu\), \(\zeta\), \(\epsilon\), the tolerance of accuracy \(\xi\) and the maximum number of iterations \(I\). Initialize \(\mathbf{F}^{(0)}\) that satisfies the transmit power constraint and set \(i=1\).