Total and minimum energy efficiency tradeoff in robust multigroup multicast satellite communications

First of all, authors present the model of multigroup multicast satellite communication systems and the optimization problem formulation. A downlink multigroup multicast satellite communication system is depicted in Fig. 1, where N_u single antenna users are simultaneously served by N_t beams. The information data is sent by a GW to the users within the coverage areas of the satellite beams. It is assumed that the feeder link between the GW and satellite is ideal and no intercluster interference is considered. In addition, each beam contains a multicast group, and there are a total of M = N_t multicast groups. The users are grouped according to the method by You et al ^[1]. Every user can only belong to one of the multicast groups. The signal received by the i^th user in group m is represented as (1), where h_i ∊ ℂ^Nt×1 and w_i ∊ ℂ^Nt×1 are the forward link beam domain channel vector from all N_t beams to the i^th user and the beamforming vector for group m, respectively. s_m is denoted as the transmit signal for all users in group m and n_i denotes the additive white Gaussian noise.

Moreover, authors also pay attention to modeling uncertainty e_i of the channel phase θ_i in h_i, which also implies the accurate satellite channel vector h_i is troublesome to acquire. To this end, the robust beamformer design based on expectation of accurate channel vector is introduced next. According to the signal model in (1), we can obtain the signal-to-interference-plus-noise ratio (SINR) of the i^th user in group m, i.e. SINR_i,m. The data rate of the i^th user in group m is the average value related to the satellite channel phase uncertainty q_i, i.e. R_i,m ≜ ��{log₂(1 + SINR_i,m)}, in order to ensure robustness. The multicast rate of the users in group m can be denoted by R_m ≜ min R_i,m. The sum rate of all multicast groups is expressed as R_tot ≜ Σ R_m (m = 1, …, M). The power consumption of the m^th group is modeled as P_m ≜ ξ_m‖w_i‖₂² + P_0,m, where ξ_m>1 is the inefficiency of the power amplifier and P_0,m is the basic power consumption for group m. The total power consumption of all groups is P_tot ≜ Σ P_m (m = 1, …, M). With the multigroup multicast rate and power consumption, the EE of group m is EE_m ≜ B∙R_m/P_m and the system TEE is EEtot ≜ B∙R_tot/P_tot, where B is the bandwidth. Specifically, a new metric, namely, the weighted product (WP) of TEE and MEE is introduced as F_WP(β). Therefore, the robust beamforming design can be formulated as (2).

Fig. 1. Model of a downlink multigroup multicast satellite communication system.

Then, authors transform the optimization problem and present the TEE-MEE trade-off algorithm. To make the problem easier to tackle, the original problem ��₁ undergoes several transformations. Introducing auxiliary variables {α_m}, ��₁ can be equivalently converted into ��₂ where B is also omitted with loss of generality. Adopting a variable transformation by introducing auxiliary variables 2^u and 2^v, ��₂ is converted to ��₃. Moreover, for R_i,m, an explicit tight approximate function of the average rate is also used. An efficient approximation method called as semidefinite relaxation (SDR) is invoked and the optimization variables are transformed into {W_m}, thus relaxing ��₃ as ��₄. To make the notations more concise, {W₁, …, W_m} is denoted by W, and then problem ��₄ is rewritten as ��₅ which is a differential convex (DC) programming. The concave-convex procedure (CCCP) is utilized to handle this DC problem and the constraints in ��₅ can be re-expressed, thus transforming ��₅ into a series of optimization subproblems which is expressed as ��₆. Now, problem ��₆ is convex and can be handled utilizing classical convex optimization methods. If the ranks of W_m^*, ∀ m, to problem ��₆ are all equal to one, the corresponding W_m^*, ∀ m, will be a feasible solution to problem ��₂. Then, the robust multicast beamformers can be obtained. When the ranks of W_m^*, ∀ m, to problem ��₆ are not all one, the Gaussian randomization method is adopted to handle the rank issue. In summary, the whole approach for the considered problem is described in Algorithm 1.

Finally, authors discuss the performance of the proposed robust algorithm in multigroup multicast satellite communication systems through numerical simulations and give a brief conclusion. In simulation, the number of users in each group is set to be 5, and the number of beams is set to be 7. The simulation results are based on 10⁶ channel realizations. Figure 2 depicts the convergence curves of Algorithm 1 for some typical system settings. It can be observed that the objective function has a fast convergence speed and usually converges within about 5 iterations for the given simulation parameters. Simulation results also indicate that using more antennas can improve the EE performance. Figure 3 demonstrates the TEE-MEE tradeoff curves attained by Algorithm 1 under different priority weights. It can be observed that all the TEE-MEE points of the proposed approach lie on or above the line where TEE = MEE. In addition, the conventional baseline approach in which the obsolete CSI is directly utilized is compared. It can be observed that the proposed robust beamforming design approach outperforms the conventional one. In conclusion, those numerical results illustrated that the proposed robust beamforming design approach could effectively balance the tradeoff between TEE and MEE. Meanwhile, the proposed robust algorithm outperforms the conventional baselines in terms of the EE performance.

Fig. 2. Convergence trajectories of Algorithm 1 versus the number of iterations for different system setup parameters.

Fig. 3. TEE-MEE tradeoff curves of the proposed and conventional approaches with P = 46 dBm, σ = 45.