We deal with the waveform design problem for the radar-communications coexistence (RCC) that ensures low correlations of the waveform and meanwhile implements efficient fast-time information embedding (IE). Two designs of unimodular waveforms are respectively proposed, wherein we evaluate the integrated sidelobe level (ISL) to find optimal waveform correlations for the radar transmission and exploit the attributes of waveform phases with large degrees of freedom to embed communication symbols. The first design is an offline strategy, which simply uses the phase initialization for waveform generation to embed communication symbols. It requires only a predefined database indexing between the waveform initialization and its emitted version, which enables the lowest ISL. While the second design employs the phase difference between any two waveform elements for IE, whose accuracy to approaching desirable phases is elaborately transformed into a quadratic form and jointly designed with the minimization of ISL. We formulate this design into a nonconvex optimization problem with a compound objective involving the outer product of the waveform vector. To solve this problem, we apply the majorization-minimization technique, based on which an efficient algorithm that iterates with a closed-form solution is devised. Simulation results verify the effectiveness of our designs developed for the RCC.

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