This paper proposes an interleaved transmission architecture for a multiple-input multiple-output (MIMO) integrated sensing and communication (ISAC) system that enables continuous data transmission while facilitating receiver processing reminiscent of pulse-based radar. A key feature of this architecture is that we can jointly design each transmitted ISAC block and its sensing filter to mitigate the high sidelobes that arise from the inter-block interference caused by targets at different ranges. We provide an exemplar of such a design in which constructive interference is integrated into the communication aspects, and the integrated mainlobe-to-sidelobe ratio (IMSR) of the beampattern is used to ensure desirable directivity of the sensing. In addition, the power of each time sample is constrained to manage the peak-to-average power ratio (PAPR). The joint transmitter-sensor design problem is addressed using an alternating optimization (AO) framework, with the subproblem for transmitted waveform design being solved via the successive convex approximation (SCA) method. To further enhance computational efficiency, the alternating direction penalty method (ADPM) is employed to solve the subproblems within the SCA iterations. The convergence of ADPM is established, with convergence of the case of more than two auxiliary variables being established for the first time. Numerical simulations validate the effectiveness of our proposed approach in achieving desirable performance in both radar sensing and communication, with the fast algorithm achieving comparable performance with greater computational efficiency.
