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In this paper, we propose an adaptive-power-control scheme for a cognitive radio system (CRS) in a Rayleigh fading channel. By allowing transmit power adaptation at the secondary user (SU) transmitter to maintain a constant output SNR to the SU receiver, this scheme maximizes the output SNR and limits the interference to a primary user (PU) within an interference constraint. To calculate this maximum constant output SNR at the SU receiver subject to the interference constraint of the PUs, we have developed an analytical model for the distribution of the interference to the PU while considering the detection performance at the SU. The analytical model takes into account multiple antennas at SUs, including a single antenna as a special case. It is found that the proposed adaptive-power control scheme outperforms the fixed-power control scheme. Typically, with SUs using two transmit and two receive antennas at the bit error rate (BER) of 10-3, the proposed adaptive-power control scheme achieves a 3-dB gain in the SNR.