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In this study, a secondary network is underlaid in a conventional downlink orthogonal frequency division multiple access (OFDMA) system to formulate a new implementation of cognitive radio, where spectrum sharing is enabled to improve spectrum utilisation. This implementation is applicable to the scenario when the number of active users is larger than the number of subcarriers, which usually cannot be achieved in traditional OFDMA systems. To properly allocate radio resources, some solutions are proposed based on duality theory. The proposed solutions are partially distributed algorithms that can dynamically allocate resources to secondary users (SUs) with the co-operation of primary users (PUs) so that the capacity of secondary network can be maximised and the co-channel interference can be minimised. In addition, the effect of interference temperature limit on the capacity of secondary network is studied, which may serve as a leverage to balance the performance between the PUs and the secondary network.