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A joint radar-communication (JRC) system utilizes the integrated transmit waveform and a single platform to perform radar and communication functions simultaneously. Admittedly, the multibeam waveform design approach could transmit the assigned waveforms in different beams with the aid of spatial and temporal degrees of freedom. However, a high sidelobe level (SLL) in the beampattern reduces energy efficiency and expands exposure probability. In this study, we propose a novel spatial–temporal joint waveform design method based on the beamforming algorithm to form a low SLL beampattern. Waveform synthesis constraints are considered to synthesize desired radar and communication waveforms at designated directions. Furthermore, we impose the constant modulus constraint to lessen the impact of the high peak-to-average ratio (PAPR). The optimization process of the whole model can be summarized as two stages. First, the covariance matrix is created by convex optimization with respect to the minimum SLL. Second, the integrated transmit waveform is tuned through an alternating projection algorithm. Based on the simulation findings, we demonstrate that the proposed method outperforms the traditional methods in terms of low SLL and waveform synthesis. Meanwhile, we validate the effectiveness of the proposed method using semi-physical experiment results. [ABSTRACT FROM AUTHOR]