This paper focuses on designing a set of transmit sequences for coexisting cognitive Multiple-Input Multiple-Output (MIMO) radar and MIMO communications (MRMC) systems. The aim is to shape the transmit beampattern to minimize the spatial Integrated Side-lobe Level (ISL) while having a good response on a desired direction in cognitive paradigm. The resulting non-convex, multi-variable optimization problem for the sequence design is addressed using the proposed Coordinate Descent (CD) framework under constant modulus and discrete phase constraints. The CD method has the capability of directly designing M-ary Phase Shift Keying (MPSK), which is important in practical applications. Numerical results indicate the superior performance of the proposed method over solutions based on Semi-definite Relaxation (SDR).

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