Optimization of Joint Power and Bandwidth Allocation for Multiple Users in a Multi-spot-Beam Satellite Communication

Abstract

Multi-spot-beam techniques have been widely applied in modern satellite communication systems, due to the advantages of reusing the frequency of different spot beams and constructing flexible service networks. As the on-board resources of bandwidth and power in a multi-spot-system are scarce, it is important to enhance the resource utilization efficiency. To this end, this paper initially presents the formulation of the problem of joint power and bandwidth allocation for multi-users, and demonstrates that the problem is one of convex minimization. An algorithm based on the Karush-Kuhn-Tucker (KKT) conditions is then proposed to obtain an optimal solution of the problem. Compared with existing separate power or bandwidth algorithms, the proposed joint allocation algorithm improves the total system capacity and the fairness between users. A suboptimal algorithm is also proposed, to further reduce computational complexity, with a performance level much closer to that of the optimal allocation algorithm.

Appendix

From the analysis in Sect. 3, it is shown that the constraints can indeed be ignored.

Taken together with the fact that the constraints (4, 5, 6 and 7) are linear, to prove the optimization problem is convex, we only need to prove that $\sum\nolimits_{i = 1}^{M} {\left( {T_{i} - C_{i} } \right)^{2} }$ is convex [10].

It is known that the sum of convex functions is also convex. Therefore, to prove that $\sum\nolimits_{i = 1}^{M} {\left( {T_{i} - C_{i} } \right)^{2} }$ is convex, we just need to prove the following function is convex:

$$ f\left( {P_{i} ,W_{i} } \right) = \left( {T_{i} - C_{i} } \right)^{2} $$

(22)

where $C_{i} = W_{i} \log_{2} \left( {1 + \frac{{\alpha_{i}^{2} P_{i} }}{{W_{i} N_{0} }}} \right)$.

It is known that is the Hessian of one function is semi-definite, thus the function is convex [10]. The Hessian of f(P_i, W_i) is given as follows:

$$ H_{f} = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i}^{2} }}} & {\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i} \partial W_{i} }}} \\ {\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i} \partial W_{i} }}} & {\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial W_{i}^{2} }}} \\ \end{array} } \right] $$

(23)

To prove that H_f is positive semi-definite, we obtain the following equations:

$$ \begin{aligned} \frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i}^{2} }} & = 2\left( {\frac{{\partial C_{i} }}{{\partial P_{i} }}} \right)^{2} - 2\left( {T_{i} - C_{i} } \right)\frac{{\partial^{2} C_{i} }}{{\partial P_{i}^{2} }} \\ & = \,2\left( {\frac{{\partial C_{i} }}{{\partial P_{i} }}} \right)^{2} + 2\left( {T_{i} - C_{i} } \right)\frac{{W_{i} }}{{\ln 2\left( {N_{0} W_{i} /\alpha_{i}^{2} + P_{i} } \right)^{2} }} \\ \end{aligned} $$

(24)

$$ \begin{aligned} & \left| {H_{f} } \right| = \frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i}^{2} }}\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial W_{i}^{2} }} - \frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i} \partial W_{i} }}\frac{{\partial^{2} f\left( {P_{i} ,W_{i} } \right)}}{{\partial P_{i} \partial W_{i} }} \hfill \\ & \, = 4\left( {T_{i} - C_{i} } \right)\frac{{C_{i}^{2} }}{{W_{i} \ln 2\left( {N_{0} W_{i} /\alpha_{i}^{2} + P_{i} } \right)^{2} }} \hfill \\ \end{aligned} $$

(25)

When T_i ≥ C_i, it is obvious that (24) and (25) are non-negative. Therefore, H_f is positive semi-definite, and $\sum\nolimits_{i = 1}^{M} {\left( {T_{i} - C_{i} } \right)^{2} }$ is convex, thus the optimization problem is convex. As a result, the solution obtained from the joint bandwidth and power algorithm based on KKT conditions is the global optimal solution of the optimization problem.