  The concept of reconfiguring wireless propagation environments using
intelligent reflecting surfaces (IRS)s has recently emerged, where an IRS
comprises of a large number of passive reflecting elements that can smartly
reflect the impinging electromagnetic waves for performance enhancement.
Previous works have shown promising gains assuming the availability of perfect
channel state information (CSI) at the base station (BS) and the IRS, which is
impractical due to the passive nature of the reflecting elements. This paper
makes one of the preliminary contributions of studying an IRS-assisted
multi-user multiple-input single-output (MISO) communication system under
imperfect CSI. Different from the few recent works that develop least-squares
(LS) estimates of the IRS-assisted channel vectors, we exploit the prior
knowledge of the large-scale fading statistics at the BS to derive the Bayesian
minimum mean squared error (MMSE) channel estimates under a protocol in which
the IRS applies a set of optimal phase shifts vectors over multiple channel
estimation sub-phases. The resulting mean squared error (MSE) is both
analytically and numerically shown to be lower than that achieved by the LS
estimates. Joint designs for the precoding and power allocation at the BS and
reflect beamforming at the IRS are proposed to maximize the minimum user
signal-to-interference-plus-noise ratio (SINR) subject to a transmit power
constraint. Performance evaluation results illustrate the efficiency of the
proposed system and study its susceptibility to channel estimation errors.
