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A cognitive radio is a smart software-defined radio that can adapt its transmission parameters, such as transmit power level and modulation type, based on the wireless channel conditions. In this paper, we introduce a joint power allocation and constellation design algorithm for cognitive radios assuming spectrum sensing imperfections. The proposed algorithm minimizes the symbol error rate of the secondary user by designing optimized 2-D constellation points and assigning transmit power levels. The constellation points are assumed to be equally probable with zero-mean and unit average symbol energy. The transmit power levels are assigned such that they do not exceed a predefined maximum transmit power threshold, and that the interference resulting from the secondary user to the primary user does not exceed a predefined value. The outcomes of the proposed algorithm, which are constellation points and …