## 4 Energy Efficiency Optimization
Therefore, by substituting (35) and (37) into (38) and exploiting the definition of SE, the EE optimization problem can be rewritten as:  
\[\min_{\tau,\{p_{\}\}_{l=1}^{L},\bar{N}_{x},\bar{N}_{y}}\tau\sum_{ l=1}^{L}p_{l}+\bar{N}_{x}\bar{N}_{y}p_{\text{ant}}, \tag{39}\] \[\text{s.t.} \begin{cases}(1-\tau)\log\left(1+\frac{\tau N_{x}^{2 }N_{y}^{2}}{1-\tau}\alpha_{l}p_{l}-\frac{N_{x}N_{y}}{1-\tau}\gamma_{l}\right) =r_{T},\\ l=1,\cdots,L,\\ C_{2}\sim C_{4}.\end{cases} \]  
By rearranging the constraint \(C_{1}\), we can obtain:  
\[p_{l}=\frac{1-\tau}{\tau\alpha_{l}\bar{N}_{x}^{2}\bar{N}_{y}^{2}}\left\{\exp \left(\frac{r_{T}}{1-\tau}\right)-1+\frac{\bar{N}_{x}\bar{N}_{y}}{1-\tau} \gamma_{l}\right\}. \tag{40}\]  
With this equality, we can rewrite the problem (39) as:  
\[\min_{\tau,\bar{N}_{x},\bar{N}_{y}}\sum_{l=1}^{L}\frac{1-\tau}{ \alpha_{l}}\left\{\exp\left(\frac{r_{T}}{1-\tau}\right)-1\right\}\frac{1}{ \bar{N}_{x}^{2}\bar{N}_{y}^{2}}+\sum_{l=1}^{L}\frac{\gamma_{l}}{\bar{N}_{x} \bar{N}_{y}}+\bar{N}_{x}\bar{N}_{y}p_{\text{ant}}, \tag{41}\] \[\text{s.t.} \begin{cases}\sum_{l=1}^{L}\frac{1-\tau}{\tau\alpha_{l }\bar{N}_{x}^{2}\bar{N}_{y}^{2}}\left\{\exp\left(\frac{r_{T}}{1-\tau}\right)-1+ \frac{\bar{N}_{x}\bar{N}_{y}}{1-\tau}\gamma_{l}\right\}\leq p_{\text{max}},\\ C_{3},C_{4}.\end{cases} \]With a few algebraic manipulations, we can see that the inequality constraint in (41) gives rise to a lower bound on \(\bar{N}_{x}\bar{N}_{y}\), that is:  
\[\bar{N}_{x}\bar{N}_{y} \geq\frac{\sum\limits_{l=1}^{L}\frac{\gamma_{l}}{\pi\alpha_{l}}+ \sqrt{\left(\sum\limits_{l=1}^{L}\frac{\gamma_{l}}{\pi\alpha_{l}}\right)^{2}-4p _{\max}\sum\limits_{l=1}^{L}\frac{1-\tau}{\pi\alpha_{l}}\left\{\exp\left(\frac {\tau_{T}}{1-\tau}\right)-1\right\}}}{2p_{\max}} \tag{42}\] \[\stackrel{{\Delta}}{{=}}\mathcal{N}_{\text{LB}}.\]  
Moreover, setting the first-order derivative of the target function with respect to \(\bar{N}_{x}\bar{N}_{y}\) to zero, we can obtain:  
\[-\frac{2\sum\limits_{l=1}^{L}\frac{1-\tau}{\pi\alpha_{l}}\left\{\exp\left(\frac {\tau_{T}}{1-\tau}\right)-1\right\}}{\bar{N}_{x}^{3}\bar{N}_{y}^{3}}-\sum \limits_{l=1}^{L}\frac{\gamma_{l}}{\pi\alpha_{l}}\frac{1}{\bar{N}_{x}^{2}\bar {N}_{y}^{2}}+p_{\text{ant}}=0. \tag{43}\]  
Since (43) is a monotonically increasing function of \(\bar{N}_{x}\bar{N}_{y}\), it has a single solution (denoted as \(\bar{N}_{x}\bar{N}_{y}=\mathcal{N}^{\dagger}\)) when \(\bar{N}_{x}\bar{N}_{y}>0\). Therefore, the target function achieves its minimum at \(\bar{N}_{x}\bar{N}_{y}=\mathcal{N}^{\dagger}\). Note that \(\mathcal{N}^{\dagger}\) can be simply obtained using the method of bisection. Considering the lower bound in (42), if we discard the integer constraints on \(\bar{N}_{x}\) and \(\bar{N}_{y}\), the optimal \(\bar{N}_{x}\) and \(\bar{N}_{y}\) satisfy:  
\[\bar{N}_{x}\bar{N}_{y}=\max\left\{\mathcal{N}_{\text{LB}},\mathcal{N}^{ \dagger}\right\}. \tag{44}\]  
When the right-hand side of (44) is greater than \(N_{x}N_{y}\), the problem is infeasible. As long as the problem is feasible, \(\bar{N}_{x}\) and \(\bar{N}_{y}\) can be selected as arbitrary positive integers, which can make (44) satisfied. After determining the number of active antennas, the optimal transmit power of AP can be derived directly using (40). Finally, the optimal \(\tau\) can be obtained by one-dimensional search over the interval \((0,1)\). From the above solutions, we can obtain two important observations on the design of system parameters. * First, for the above EE and widely-investigated SE optimization problems [12], we note that the optimal transmit powers derived have distinct scaling behaviors with respect to the number of the AP's (active) antennas. According to (40), the optimal transmit power that maximizes the EE reduces approximately with \(\bar{N}_{x}^{2}\bar{N}_{y}^{2}\) if the processing energy \(E_{p}\) is negligible and reduces with \(\bar{N}_{x}\bar{N}_{y}\) if the processing energy is significant. In contrast, in SE optimization, the optimal transmit power is independent of the number of AP antennas [12].