Optimum power allocation in a hierarchical spectrum sharing scheme

We consider a hierarchical spectrum sharing scheme wherein a secondary wireless system is allowed to share spectrum while maintaining priority for the primary system. Under this setting, we consider the problem of allocating transmit powers for the primary and secondary transmissions so as to maximize the sum rate of the two systems, while maintaining a minimum rate for the primary system. We determine the optimum power allocation for two cases corresponding to power values being (i) continuous real in a given range, and (ii) from a discrete set. We show that the optimal power allocation for both cases can be found to be one among a small set of points, and can be determined by an O(1) algorithm.