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Cognitive radio (CR) has been recognised as one of the promising technologies to provide efficient utilisation of the limited wireless spectrum. The authors investigate the power allocation problem for orthogonal frequency division multiplexing (OFDM)-based CR systems that coexist with primary networks. Specifically, in two scenarios with and without integral bit rate constraint, the authors study the secondary users (SUs) power allocation under both individual transmit power constraints of their own and interference power constraints of all the primary users. When the bit rate of SUs can be continuous on each OFDM subchannel, an efficient power allocation algorithm that contains two level Lagrangian dual variable iterations is proposed to maximise the CR system sum-rate. Besides, when integral bit requirement exists, a heuristic power allocation and bit loading methodology which based on the iterative power rescaling operation is developed. Simulation results show that the algorithm proposed for the continuous rate case can achieve the same rate performance as the standard interior-point algorithm, but converges much faster. The proposed allocation scheme for the discrete rate case has the same bit rate performance as the greedy-based Max-Min scheme.