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This paper addresses the problem of designing channel-aware access control (CAAC) algorithms for cognitive radio networks. To protect the primary transmission, the access probabilities of secondary users are adjusted based on the channel-state information and the measured interference temperature. It is shown that the proposed CAAC algorithm always converges to a fixed point. Furthermore, the CAAC algorithm can be interpreted as a noncooperative game. Then, the game model is extended to include general access utilities, unsuccessful transmission discount, and interference constraints. Sufficient conditions are established to guarantee the uniqueness of the Nash equilibrium (NE) of the proposed general game model, and a distributed iteration algorithm is proposed to find the unique NE. The convergence properties of this algorithm in continuous-time iteration, as well as in the discrete-time version, are proved under some sufficient conditions. Simulation results demonstrate the convergence and effectiveness of the distributed channel-access algorithms.