## 3 Problem Formulation  
As mentioned before, we define the vector of user streams as \(\mathbf{s}\triangleq\left[s_{\text{pub}},s_{1},\cdots,s_{K}\right]\in\mathbb{C} ^{K+1}\) and the precoder matrix is \(\mathbf{w}\triangleq\left[w_{\text{pub}},w_{1},\cdots,w_{K}\right]\in\mathbb{ C}^{L\times(K+1)}\). The final linearly precoded signal that satellite broadcasts to \(K\) users is  
\[\mathbf{x}=\mathbf{w}\mathbf{s}=w_{\text{pub}}s_{\text{pub}}+\sum_{k=1}^{K}w_{ k}s_{k}, \tag{3}\]  
where \(w_{\text{pub}},w_{k}\in\mathbb{C}^{L}\) are used to execute simultaneous downlink transmissions. The received signal has the following form:  
\[\mathbf{y}=\mathbf{h}^{H}\mathbf{x}+\boldsymbol{n}, \tag{4}\]  
where \(\mathbf{h}=[h_{1},h_{2},\cdots,h_{K}]\) is the downlink channel matrix, \(\boldsymbol{n}\sim\mathcal{CN}(0,\mathbf{I}_{K})\) is the additive white Gaussian noise (AWGN) vector, which is normalized for brevity. For better understanding, we take user \(k\) as an example, its received signal will be  
\[y_{k}=h_{k}^{H}w_{\text{pub}}s_{\text{pub}}+h_{k}^{H}w_{k}s_{k}+h_{k}^{H}\sum_ {i\
eq k}w_{i}s_{i}+n_{k}, \tag{5}\]  
where item \(h_{k}^{H}\sum_{i\
eq k}w_{i}s_{i}\) is the multi-user interference and \(n_{k}\) is the noise. Without loss of generality, we assume the power of users' streams are normalized as well, which is \(\mathbb{E}\left[\mathbf{s}\mathbf{s}^{H}\right]=\mathbf{I}\). This will help us to obtain simpler and clearer mathematical expressions. Therefor, the SINR of the public part of the signal for user \(k\) is  
\[\gamma_{k}^{\text{pub}}=\frac{\left|h_{k}^{H}w_{\text{pub}}\right|^{2}}{\sum_ {i=1}^{K}\left|h_{i}^{H}w_{i}\right|^{2}+1}. \tag{6}\]  
Usually, we can now acquire the rate with [PERSON]'s formula. But in this scenario, we want to guarantee that all \(K\) users could successfully receive the signal. That means the rate of the public part of the signal should not exceed the user with lowest public part rate, which yields the following equation:  
\[R_{k}^{\text{pub}}=\min_{k}\Bigl{\{}\log_{2}\Bigl{(}1+\gamma_{k}^{\text{pub }}\Bigr{)}\Bigr{\}}. \tag{7}\]  
Similarly, the SINR of private part of the signal for user \(k\) will be  
\[\gamma_{k}^{\text{pri}}=\frac{\left|h_{k}^{H}w_{\text{pub}}\right|^{2}}{\sum_ {i\
eq k}^{K}\left|h_{i}^{H}w_{i}\right|^{2}+1}. \tag{8}\]  
For the private part of the signal, user \(k\) will discard others' private streams. So there is no rate requirement like the public part. The rate of private part of the signal for user \(k\) is  
\[R_{k}^{\text{pri}}=\log_{2}\Bigl{(}1+\gamma_{k}^{\text{pri}}\Bigr{)}. \tag{9}\]  
Now we can get the final rate of user \(k\), which is  
\[R_{k}=R_{k}^{\text{pub}}+R_{k}^{\text{pri}}. \tag{10}\]The total power consumption of the broadcast signal is \(|w_{\text{pub}}s_{\text{pub}}|^{2}+\sum_{k}|w_{k}s_{k}|^{2}\). Since we have normalized the power of user streams, we can obtain the power consumption as \(\text{tr}\big{(}\mathbf{ww}^{\text{H}}\big{)}\), in which \(\text{tr}(\cdot)\) is the trace of a matrix. By the definition of energy efficiency, the problem we aim to solve is given as  
\[\max_{\mathbf{w}}\ EE=\frac{\sum_{k=1}^{K}R_{k}}{\text{tr}(\mathbf{ww}^{\text{ H}})+P_{c}}. \tag{11}\]  
\[\text{s.t.}\ \text{tr}\Big{(}\mathbf{ww}^{\text{H}}\Big{)}\leq P_{t}, \tag{12}\]  
\[\mathbf{y}=\mathbf{h}^{H}\mathbf{x}+\boldsymbol{n}, \tag{4}\]0  
where \(P_{c}\) is the circuit power consumed in the satellite, \(P_{t}\) is the maximum transmission power of the satellite and \(R^{\text{th}}\) is the required minimum rate. Equation (11) is the sum energy efficiency in the network. Constraint (12) is the power limit, indicating that the transmit power cannot exceed the maximum power. Constraint (13) is the QoS constraint to guarantee successful transmission. Since this problem is non-convex and NP-hard, conventional approaches use SCA-based methods to approximate the optimal solution. However, the complexities of these approaches are very high, and the optimal solution sometimes cannot be obtained.