In intelligent reflecting surface (IRS) aided wireless communication systems, channel state information (CSI) is crucial to achieve its promising passive beamforming gains. However, CSI errors are inevitable in practice and generally correlated over the IRS reflecting elements due to the limited training with discrete phase shifts, which degrade the data transmission rate and reliability. In this paper, we focus on investigating the effect of CSI errors to the outage performance in an IRS-aided multiuser downlink communication system. Specifically, we aim to jointly optimize the active transmit precoding vectors at the access point (AP) and passive discrete phase shifts at the IRS to minimize the APs transmit power, subject to the constraints on the maximum CSI-error induced outage probability for the users. First, we consider the single-user case and derive the users outage probability in terms of the mean signal power (MSP) and variance of the received signal at the user. Since there is a trade-off in tuning these two parameters to minimize the outage probability, we propose to maximize their weighted sum with the optimal weight found by one-dimensional search. Then, for the general multiuser case, since the users outage probabilities are difficult to obtain in closed-form due to the inter-user interference, we propose a novel constrained stochastic successive convex approximation (CSSCA) algorithm, which replaces the non-convex outage probability constraints with properly designed convex surrogate approximations. Simulation results verify the effectiveness of the proposed robust beamfoming algorithms and show their significant performance improvement over various benchmark schemes.
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