# Intelligent Reflecting Surface Aided Wireless Systems with Imperfect Hardware
## 1 Introduction
### Related Studies  
In other perspectives, ref. [23] analyzed the asymptotic performance of the large-scale MISO downlink system relying on spatial correlation statistics. The authors in [24; 25] considered imperfect channel state information (CSI) using discrete-resolution RIS to boost weighted sum-rate for downlink MISO users. The works in [26; 27] examined imperfect CSIs in robust beamforming design that results in less power consumption. To achieve low-overhead channel estimation, several works [28; 29; 30] have studied RIS reflection patterns and designed pilot sequences. The error probability of RIS-enabled systems is studied with quantized channel phase compensation in [31]. The authors provided analysis of exact and asymptotic bit error rate (BER) expressions in different practical channel scenarios and the existence of the direct link. Moreover in terms of mobility, refs. [32; 33] proposed novel RIS-aided wireless communication channel estimation schemes in mobility scenarios. Specifically, in [32], the authors proposed a channel estimation scheme for RIS-aided systems in mobility scenarios. The authors exploited Kalman filters to take advantage of the channel time correlation. Furthermore, the cascaded RIS channel is represented as a state-space model based on the mobility scenario. The optimal RIS reflection coefficients are then obtained via minimum mean square error (MMSE). Differently, in [33], the authors proposed two wide-band channel estimation frameworks integrating multi-path and single-path Doppler shift adjustments. Several recent research works have also studied the application of machine learning in RIS channel estimation. In [34], the authors studied the protection of users' location data during RIS-enhanced millimeter wave (mmWave) channel estimation by incorporating privacy-preserving federated learning. Ref. [35] proposed two residual neural network algorithms--single-scale enhanced deep residual (EDSR) and multi-scale enhanced deep residual (MDSR)--to obtain accurate CSI in a semi-passive RIS assisted network. The RIS device is equipped with a small number of active radio frequency units for capturing partial CSI. Similarly, in [36], the authors also proposed three residual neural networks--blind convolutional denoising network, convolutional denoising generative adversarial networks, and multiple residual dense networks--to obtain accurate CSI. In [37], the authors introduced an artificial intelligence-assisted channel estimation scheme in RIS-enabled networks. Furthermore, in [38], the authors explored a deep learning approach to obtain an achievable rate by studying RIS phase shifts and receiver location interaction without utilizing CSI. Once the model is trained adequately, the deep learning model can be used to predict the optimal number of RIS elements for any receiver location. In terms of RIS-assisted multiple-input multiple-output (MIMO) networks, the authors in [39] proposed a joint precoding and RIS phase adjustment design for RIS-assisted MIMO systems. Then, in [7], the authors examined information exchange and passive beamforming for RIS-aided multi-user MIMO networks. The authors designed a stochastic algorithm to solve the bilinear receiver estimation problem. Similarly, in [40], the authors designed RIS phase shifts in a RIS-assisted massive MIMO network with direct links based on the statistical CSI. Then, the authors derived the uplink ergodic rate, and afterward, the maximized sum rate was obtained via a genetic algorithm. Differently, in [41], the authors studied the outage probability of a RIS-enabled MIMO network by exploiting Kornecker channel models to characterize MIMO antennas spatial correlations and RIS-reflecting elements. The authors utilized random matrix theory and Mellin transforms to obtain the outage probability. In [42], the authors proposed a new signal-passing-based algorithm to factorize the RIS-aided multi-user MIMO cascaded channels.