The power spectral density (PSD) shape of the transmit waveform plays an important role in some fields of radar, such as electronic counter-countermeasures (ECCM), target detection, and target classification. In addition, radar hardware generally requires the waveform to have constant modulus (CM) characteristics. Therefore, it is a significant problem to synthesize the discrete-time CM waveform from a given PSD. To address this problem, some algorithms have been proposed in the existing literature. In this paper, based on the majorization–minimization (MM) framework, a novel algorithm is proposed to solve this problem. The proposed algorithm can be proved to converge to the stationary point, and the error reduction property can be obtained without the unitary requirements on the discrete Fourier transform (DFT) matrix. To accelerate the convergence rate of the proposed algorithm, three acceleration schemes are developed for the proposed algorithm. Considering a specific algorithm stopping condition, one of the proposed acceleration schemes shows better computation efficiency than the existing algorithms and is more robust to the initial points. Besides, when the DFT matrix is not unitary, the numerical results show that the proposed acceleration scheme has better matching performance compared with the existing algorithms.