  Intelligent reflecting surface (IRS) is a cost-effective solution for
achieving high spectrum and energy efficiency in future wireless networks by
leveraging massive low-cost passive elements that are able to reflect the
signals with adjustable phase shifts. Prior works on IRS mainly consider
continuous phase shifts at reflecting elements, which are practically difficult
to implement due to the hardware limitation. In contrast, we study in this
paper an IRS-aided wireless network, where an IRS with only a finite number of
phase shifts at each element is deployed to assist in the communication from a
multi-antenna access point (AP) to multiple single-antenna users. We aim to
minimize the transmit power at the AP by jointly optimizing the continuous
transmit precoding at the AP and the discrete reflect phase shifts at the IRS,
subject to a given set of minimum signal-to-interference-plus-noise ratio
(SINR) constraints at the user receivers. The considered problem is shown to be
a mixed-integer non-linear program (MINLP) and thus is difficult to solve in
general. To tackle this problem, we first study the single-user case with one
user assisted by the IRS and propose both optimal and suboptimal algorithms for
solving it. Besides, we analytically show that as compared to the ideal case
with continuous phase shifts, the IRS with discrete phase shifts achieves the
same squared power gain in terms of asymptotically large number of reflecting
elements, while a constant proportional power loss is incurred that depends
only on the number of phase-shift levels. The proposed designs for the
single-user case are also extended to the general setup with multiple users
among which some are aided by the IRS. Simulation results verify our
performance analysis as well as the effectiveness of our proposed designs as
compared to various benchmark schemes.
