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An important problem in radar waveform optimization is the synthesis of discrete time constant modulus signals from Fourier magnitude data. Iterative algorithms for solving this problem have been proposed in the literature, but the algorithms are only applicable in limited cases, and the convergent behavior of these algorithms has not been established. We connect waveform design to the well-studied problem of phase retrieval. This is useful for explaining the success of the proposed iterative methods. We generalize and extend the existing algorithms to handle the case in which the dimensions of the time domain waveform and the frequency domain data are unequal, and we provide a convergence analysis. We also relate the phase retrieval problem to the problem of synthesizing discrete time constant modulus signals from power spectral density (PSD) data, which is different and more appropriate for the waveform design problem. We compare the iterative methods to direct search gradient methods for both problems, and establish that the proposed algorithms can provide comparable performance with reduced computational complexity.