In this paper, we study the power allocation problem in a relay assisted multi-band underlay cognitive radio network. Such a network allows unlicensed users (secondary users) to access the spectrum bands under a transmission power constraint. Due to the concave increasing property of logarithm function, it is not always wise for secondary users to expend all the transmission power in one band if their aim is to maximize achievable data rate. In particular, we study a scenario where two secondary users and a half-duplexing relay exist with two available bands. The two users choose different bands for direct data transmission and use the other band for relay transmission. By properly allocating the power on two bands, each user may be able to increase its total achievable data rate while satisfying the power constraint. We formulate the power allocation problem as a non-cooperative game and investigate its Nash equilibria. We prove the power allocation game is a supermodular game and that Nash equilibria exist. We further find the best response function of users and propose a best response update algorithm to solve the corresponding dynamic game. Numerical results show the overall performance in terms of achievable rates is improved through our proposed transmission scheme and power allocation algorithm. Our proposed algorithm also shows satisfactory performance in terms of convergence speed.

Spectrum sharing effectively improves the spectrum usage by allowing secondary users (SUs) to dynamically and opportunistically share the licensed bands with primary users (PUs). The concept of cooperative spectrum sharing allows SUs to use portions of the PUs' radio resource for their own data transmission, under the condition that SUs help the PUs' transmission. The key issue with designing such a scheme is how to deal with the resource splitting of the network. In this paper we propose a relay-based cooperative spectrum sharing scheme in which the network consists of one PU and multiple SUs. The PU asks the SUs to relay its data in order to improve its energy efficiency, in return it rewards the SUs with a portion of its authorized spectrum. However each SU is only allowed to transmit its data via the rewarded channel at a power level proportional to the contribution it makes to the PU. Since energy cost is considered, the SUs must carefully determine their power level. This scheme forms a non-cooperative Stackelberg resource allocation game where the strategy of PU is the bandwidth it rewards and the strategy of each SU is power level of relay transmission. We first investigate the second stage of the sub-game which is addressed as power allocation game. We prove there exists an equilibrium in the power allocation game and provide a sufficient condition for the uniqueness of the equilibrium. We further prove a unique Stackelberg equilibrium exists in the resource allocation game. Distributed algorithms are proposed to help the users with incomplete information achieve the equilibrium point. Simulation results validate our analysis and show that our proposed scheme introduces significant utility improvement for both PU and SUs.