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Spectrum sharing has received an increasing amount of attention in cognitive radio over the past few years as an effective method of alleviating the spectrum scarcity problem in wireless communications by allowing unlicensed users to use the same spectrum as the licensed users under the condition of protecting the latter from harmful interference using a received interference power constraint at the licensed receivers. In this paper, we study the outage capacity and the truncated channel inversion with fixed rate (TIFR) capacity of a sensing-enhanced spectrum sharing cognitive radio system under two different scenarios, namely with and without missed-detection interference power constraints for the protection of the primary users, for both Rayleigh and Nakagami-m fading channels. In our analysis, we consider various constraints on the capacity that include: (i) average transmit power constraints, (ii) peak interference power constraints, (iii) average interference power constraints and (iv) target detection probability constraints, and derive the power allocation strategy, as well as the TIFR and outage capacity for each scenario. Finally, we provide simulation results, which indicate that the sensing-enhanced spectrum sharing cognitive radio system can achieve higher outage and TIFR capacity compared to the conventional non-sensing spectrum sharing cognitive radio system.