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This letter considers a downlink multi-input single-out (MISO) system where each user performs simultaneous wireless information and power transfer (SWIPT) based on a power splitting receiver architecture. Assuming imperfect channel state information (CSI) at the base station, we develop two robust joint beamforming and power splitting (BFPS) designs that minimize the transmission power under both the signal-to-interference-plus-noise ratio (SINR) and energy harvesting (EH) constraints per user. In the first design, we consider the worst-case (WC) SINR and EH constraints, and show that the WC-BFPS problem can be relaxed as a semidefinite program (SDP) through a linear matrix inequality representation for (infinitely many) robust quadratic matrix inequality constraints. In the second design, we consider the chance constraints (CCs) for SINR and EH, and resort to both semidefinite relaxation and Bernstein-type inequality restriction to transform the CC-BFPS problem into another convex SDP. Based on these convex reformulations, the (near-)optimal robust BFPS designs can be efficiently solved. Numerical results are provided to demonstrate the merit of the proposed robust designs.Index Terms-Bernstein-type inequality, energy harvesting, power splitting, robust beamforming, semidefinite program.

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