  Most works in IRS-assisted systems have ignored the impact of the inevitable
residual hardware impairments (HWIs) at both the transceiver hardware and the
IRS while any relevant works have addressed only simple scenarios, e.g., with
single-antenna network nodes and/or without taking the randomness of phase
noise at the IRS into account. In this work, we aim at filling up this gap by
considering a general IRS-assisted multi-user (MU) multiple-input single-output
(MISO) system with imperfect CSI and correlated Rayleigh fading. In parallel,
we present a general computationally efficient methodology for IRS reflect
beamforming (RB) optimization. Specifically, we introduce an advantageous
channel estimation (CE) method for such systems accounting for the HWIs.
Moreover, we derive the uplink achievable spectral efficiency (SE) with
maximal-ratio combining (MRC) receiver, displaying three significant advantages
being: 1) its closed-form expression, 2) its dependence only on large-scale
statistics, and 3) its low training overhead. Notably, by exploiting the first
two benefits, we achieve to perform optimization with respect to the reflect
beamforming matrix (RBM) that can take place only at every several coherence
intervals, and thus, reduces significantly the computational cost compared to
other methods which require frequent phase optimization. Among the insightful
observations, we highlight that uncorrelated Rayleigh fading does not allow
optimization of the SE, which makes the application of an IRS ineffective.
Also, in the case that the phase drifts, describing the distortion of the
phases in the RBM, are uniformly distributed, the presence of an IRS provides
no advantage. The analytical results outperform previous works and are verified
by Monte-Carlo (MC) simulations.
