## 3 Energy-Efficient Optimal Power Allocation
### Energy-Efficient Optimal Power Allocation for Non-Real-Time Applications
#### 3.1.1 Average Transmit Power Constraint  
Ergodic capacity (EC) is an appropriate performance metric for non-real-time applications, which can be obtained by averaging over all states of an ergodic fading channel. Therefore, EE for non-real-time applications can be denoted as the ratio of the EC to the average power consumption [37]. Employing the ATP constraint, EE maximization problem is formulated as  
\[ \begin{array}{l}\mathop{\max}\limits_{P_{t}}H\left(P_{t}\right)=\frac{ \mathop{\mathrm{E}}\left[\log_{2}\left(1+\frac{P_{t}}{N_{\mathrm{s}}}\right) \right]}{\mathop{\mathrm{E}}\left(\zeta P_{t}+P_{t}\right)},\\ s.t.\left\{\begin{array}{l}\mathop{\mathrm{E}}\left(P_{t}\right)\leq P_{ av}\quad(a1)\\ \mathop{\mathrm{E}}\left(P_{t}\right)\leq I_{th}\quad\quad(b)\end{array}\right. \end{array}  \tag{8}\]  
where \(N_{\mathrm{s}}\) represents the noise power, \(\xi\) and \(P_{\mathrm{c}}\) are the amplifier coefficient and the constant circuit power consumption of the satellite user, respectively and \(\mathop{\mathrm{E}}\left(\cdot\right)\) denotes the statistical expectation. Moreover, \(P_{av}\) and \(I_{th}\) denote the ATP constraint limit and the AIP constraint limit, respectively. It can be proved that (8) is a nonlinear concave fractional programming problem. Therefore, the following conclusion can be obtained. **Theorem 1**: _Any local maximum in (8) is a global maximum and there is at most one maximum since (8) is strictly quasiconcave._Proof. : Because the numerator of \(H\left(P_{t}\right)\) is strictly concave, (8) is strictly quasiconcave. In addition, since the numerator and denominator of \(H\left(P_{t}\right)\) are differentiable and the numerator is strictly concave, (8) is strictly pseudoconcave [27]. Based on these results, when \(\frac{dH\left(P_{s}\right)}{dP_{s}}=0\), \(H\left(P_{t}\right)\leq H\left(P_{s}\right)\) would hold at any \(P_{t}\). Thus, \(H\left(P_{s}\right)\) can be proved to be the global maximum. Because (8) is a nonlinear fractional program, according to [PERSON]'s method [26], it can be equivalently formulated as the problem below with a parameter \(\eta\)  
\[T\left(\eta\right)=\max_{P_{t}\in S_{1}}\mathrm{E}\left[\log_{2}\left(1+\frac{ P_{r}}{N_{s}}\right)\right]-\eta\mathrm{E}\left(\xi P_{t}+P_{c}\right), \tag{9}\]  
where \(\eta\) is a non-negative parameter and \(S_{1}\) denotes the set \(S_{1}=\{P_{t}|P_{t}\in\left(a1\right)\cap\left(b\right)\}\). We can obtain the global maximum of (8) by solving (9). Furthermore, it is easy to prove that (9) is a convex problem. Thus, we can solve (9) by employing the Lagrange duality method since the duality gap is zero [27]. The Lagrangian function of (9) can be expressed as  
\[L\left(P_{t},\tau,\mu\right)=\mathrm{E}\left[\log_{2}\left(1+\frac{P_{r}}{N_{ s}}\right)\right]-\eta\mathrm{E}\left(\xi P_{t}+P_{c}\right)-\tau\left[ \mathrm{E}\left(P_{t}\right)-P_{uv}\right]-\mu\left[\mathrm{E}\left(P_{i} \right)-I_{th}\right], \tag{10}\]  
where \(\tau\) and \(\mu\) are the non-negative Lagrangian multipliers related to \(\left(a1\right)\) and \(\left(b\right)\) in (8), respectively. Hence, the Lagrange dual function of (9) is given as  
\[g\left(\tau,\mu\right)=\max_{P_{t}\geq 0}L\left(P_{t},\tau,\mu\right). \tag{11}\]  
Then, the dual problem of (9) can be presented as  
\[\min_{\tau,\mu}g\left(\tau,\mu\right). \tag{12}\]  
Similar to [38], (12) can be decoupled into multiple parallel subproblems based on the Lagrange dual-decomposition method [27]. These subproblems have the same structure for each fading state. Therefore, given a particular fading state, the corresponding subproblem can be formulated as  
\[\max_{P_{t}\geq 0}D\left(P_{t}\right)=\log_{2}\left(1+\frac{P_{r}}{N_{s}} \right)-\eta\xi P_{t}-\tau P_{t}-\mu P_{i}. \tag{13}\]  
We can obtain the global maximum of (9) by iteratively solving (13) for all fading states with the fixed \(\tau\) and \(\mu\), and updating \(\tau\) and \(\mu\) by subgradient method [27]. Then, we can derive the optimal transmit power \(P_{t}^{*}\) of (9) as shown in Theorem 2. **Theorem 2**.