  Prior studies on Intelligent Reflecting Surface (IRS) have mostly assumed
perfect channel state information (CSI) available for designing the IRS passive
beamforming as well as the continuously adjustable phase shift at each of its
reflecting elements, which, however, have simplified two challenging issues for
implementing IRS in practice, namely, its channel estimation and passive
beamforming designs both under the constraint of discrete phase shifts. To
address them, we consider in this paper an IRS-aided single-user communication
system with discrete phase shifts and design the IRS training reflection matrix
for channel estimation as well as the passive beamforming for data
transmission, both subject to the constraint of discrete phase shifts. We show
that the training reflection matrix design for discrete phase shifts greatly
differs from that for continuous phase shifts, and thus the corresponding
passive beamforming should be optimized by taking into account the correlated
channel estimation error due to discrete phase shifts. Specifically, we
consider a practical block-based transmission, where each block has a finite
(insufficient) number of training symbols for channel estimation. A novel
hierarchical training reflection design is proposed to progressively estimate
IRS elements' channels over multiple blocks by exploiting IRS-elements grouping
and partition. Based on the resolved IRS channels in each block, we further
design the progressive passive beamforming at the IRS with discrete phase
shifts to improve the achievable rate for data transmission over the blocks.
