The existence of multipath brings extra looks of targets. This paper considers the extended target detection problem with a narrow band Multiple-Input Multiple-Output(MIMO) radar in the presence of multipath from the view of waveform-filter design. The goal is to maximize the worst-case Signal-to-Interference-pulse-Noise Ratio(SINR) at the receiver against the uncertainties of the target and multipath reflection coefficients. Moreover, a Constant Modulus Constraint(CMC) is imposed on the transmit waveform to meet the actual demands of radar. Two types of uncertainty sets are taken into consideration. One is the spherical uncertainty set. In this case, the max-min waveform-filter design problem belongs to the non-convex concave minimax problems, and the inner minimization problem is converted to a maximization problem based on Lagrange duality with the strong duality property. Then the optimal waveform is optimized with Semi-Definite Relaxation(SDR) and randomization schemes. Therefore, we call the optimization algorithm Duality Maximization Semi-Definite Relaxation(DMSDR). Additionally, we further study the case of annular uncertainty set which belongs to non-convex non-concave minimax problems. In order to address it, the SDR is utilized to approximate the inner minimization problem with a convex problem, then the inner minimization problem is reformulated as a maximization problem based on Lagrange duality. We resort to a sequential optimization procedure alternating between two SDR problems to optimize the covariance matrix of transmit waveform and receive filter, so we call the algorithm Duality Maximization Double Semi-Definite Relaxation(DMDSDR). The convergences of DMDSDR are proved theoretically. Finally, numerical results highlight the effectiveness and competitiveness of the proposed algorithms as well as the optimized waveform-filter pair.
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