  In this paper, we consider an active reconfigurable intelligent surface (RIS)
to assist the multiuser downlink transmission in the presence of practical
hardware impairments (HWIs), including the HWIs at the transceivers and the
phase noise at the active RIS. The active RIS is deployed to amplify the
incident signals to alleviate the multiplicative fading effect, which is a
limitation in the conventional passive RIS-aided wireless systems. We aim to
maximize the sum rate through jointly designing the transmit beamforming at the
base station (BS), the amplification factors and the phase shifts at the active
RIS. To tackle this challenging optimization problem effectively, we decouple
it into two tractable subproblems. Subsequently, each subproblem is transformed
into a second order cone programming problem. The block coordinate descent
framework is applied to tackle them, where the transmit beamforming and the
reflection coefficients are alternately designed. In addition, another
efficient algorithm is presented to reduce the computational complexity.
Specifically, by exploiting the majorization-minimization approach, each
subproblem is reformulated into a tractable surrogate problem, whose
closed-form solutions are obtained by Lagrange dual decomposition approach and
element-wise alternating sequential optimization method. Simulation results
validate the effectiveness of our developed algorithms, and reveal that the
HWIs significantly limit the system performance of active RIS-empowered
wireless communications. Furthermore, the active RIS noticeably boosts the sum
rate under the same total power budget, compared with the passive RIS.
