In this paper, we investigate the power allocation for maximizing weighted sum rate (WSR) in downlink multiple carriers non-orthogonal multiple access (MC-NOMA) systems with imperfect successive interference cancellation (SIC). We formulate the power allocation problem as a non-convex optimization problem with the total power constraint of all sub-channels while considering often-neglected issues of SIC error and power order constraints at users. First, we discuss that the optimization problem assuming receivers can perform perfect SIC, and we provide a concavity condition of the WSR maximization problem for the MC-NOMA system. When the concavity condition is not satisfied, a fractional quadratic transformation is used to overcome the difficulty of problem non-convexity. Based on the transformation, we propose an iterative power allocation algorithm. Then, we consider the SIC error and the power order constraints in the optimization problem and present a power allocation method with imperfect SIC. Moreover, for both the perfect and imperfect SIC, we derive some propositions of the optimal power allocation solution to the WSR maximization problem and propose a low-complexity power allocation algorithm based on these propositions. Finally, we provide a joint user scheduling and power allocation algorithm for maximizing the WSR. The simulation results illustrate that the proposed resource allocation methods have a better performance than the existing schemes.

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