Online power allocation for maximizing mutual information in cognitive radio system

The design of online algorithms for maximizing the achievable rate in a cognitive radio (CR) over a fixed number of slots under a sum-power constraint is considered. CR is allowed to transmit in slots unoccupied by the primary transmitter (PT), however, no information about future PT slot occupancy is known at the CR. The fading coefficients are also assumed to be arbitrarily varying over time, with only causal information available at the CR. For a maximization problem, the utility of an online algorithm is tested by finding its competitive ratio or competitiveness that is defined to be the maximum of the ratio of the gain of the optimal offline algorithm and the gain of the online algorithm over all input sequences. We show that the lower bound on the optimal competitive ratio for maximizing the achievable CR rate is arbitrarily close to the number of slots. Conversely, we propose a simple strategy that invests available power uniformly assuming all the future slots are going to be unoccupied, and show that its competitive ratio is equal to the number of slots, to conclude that it is an optimal online algorithm.