## 1 Introduction
In another approach, radar sub-sampling matrix optimization has been performed for coexistence between a multi-input multi-output (MIMO) communication system and MIMO matrix-completion (MIMO-MC) radar [31; 32], which gives a non-optimal performance to balance a trade-off between radar and communication. Because the co-design approaches with integrated waveforms allow simultaneous radar and communication operations, these generally utilize orthogonal frequency division multiplexing (OFDM) technology [33; 34; 35; 36], which already has wide applications in radar and communication. Additionally, pulsed OFDM waveforms have been used in [37; 38], where each pulse consists of several OFDM symbols. [PERSON] et al. considered the transmit waveform design problem via a hybrid linear-nonlinear precoding signaling scheme where the joint waveform was designed as a superposition of nonlinearly precoded waveform and linearly precoded communication symbols [39]. Moreover, in [40; 41], the radar waveform is modified by embedding communication information. However, performance of one of the systems is compromised in each of the aforementioned cases. On the other hand, the DFRC schemes have been proposed in [42; 43; 44], which facilitate the weight vector design to generate the desired beampatterns. In [43], the mainlobes in the beampatterns have been kept constant for the radar, while different phases [42], different sidelobe levels, or both [44] are used to encode communication bits. However, the waveforms do not follow constant modulus constraints in such schemes. An important consideration in the design of suitable waveforms is the requirement to maintain a constant modulus [45; 46; 47] to obtain an improved performance. Because the constant modulus waveforms ensure energy-efficient transmission by avoiding signal distortion in amplifiers [48], these are well-suited for the current era radar and communication-based equipment. However, inclusion of the constant modulus constraint (CMC) in the design leads to non-convex optimization problem formulations which are NP-hard to solve. Different methods have been proposed in the literature to deal with the non-convex CMC problem [49; 50], e.g., semidefinite relaxation is used in [49] to design code radar waveforms and sequential optimization procedures are proposed in [50] to design constant modulus waveforms for MIMO radar. Additionally, a manifold-based algorithm has been proposed in [51] to solve the constant envelope precoding problem. Moreover, a strategy based on antenna selection is proposed in [52] for MIMO DFRC systems where several sparse antenna selection and permutation methods are used for embedding communication bits in the radar emissions. However, as full array sensors have not been used, emissions have wide mainlobes and high sidelobes problems. Additionally, different optimization methods have been proposed in [53] to decrease the downlink multi-user interference in communication operation and generate the desired radar beampatterns. However, because the waveform formulation in the radar direction is unconstrained, the radar detection performance deteriorates drastically. In [46], different methods have been presented for integrated waveforms through a waveform synthesis (WS) constraint. These methods synthesize the desired radar and communication waveforms in different directions; however, they give a high peak-to-average power ratio (PAPR) or high sidelobe levels. In this paper, we investigate a constant modulus waveforms design to approximate a desired beampattern for dual-function radar-communication based on a MIMO system. Note that the desired beampattern consists of a high-gain radar main beam with a slightly high gain communication beam while maintaining the desired low sidelobe levels. First, we formulate the waveform design problem as an optimization problem. Because the constant modulus constraint makes the problem non-convex and NP-hard, traditional methods cannot be applied to solve the problem.