Joint design of transmit beamforming and receiving filter for airborne MIMO(multiple-input multiple-output) radar in the background of signal-dependent clutter was studied. Airborne MIMO radar transmit beamspace STAP(space-time adaptive processing) signal model was established. In order to ensure the target detection performance in clutter environment, the SINR(signal-to-interference-plus-noise ratio) was maximized and a new iterative optimization algorithm based on the MM(maximization-minimization) framework was designed to solve the joint design problem. By properly finding the lower bound of the objective function, the convergence speed of the proposed algorithm was effectively improved and the running time of the proposed algorithm was reduced. In addition, compared with the traditional phased array radar and MIMO radar, the optimized transmit beamforming and receive filter can significantly improve the output SINR. Simulation results demonstrate the effectiveness of the proposed method.