###### Abstract  
In this paper, we consider a multiple reconfigurable intelligent surface (RIS)-assisted system using positioning information (PI) to explore the potential of Doppler effect mitigation and spectral efficiency (SE) enhancement in high-speed communications (HSC) in the presence of hardware impairments (HWI). In particular, we first present a general multi-RIS-assisted system model for HSC with HWI. Then, based on PI, different phase shift optimization strategies are designed and compared for maximizing SE, eliminating Doppler spread, and maintaining a very low delay spread. Moreover, we compare the performance of different numbers of RISs with HWI in terms of SE and delay spread. Finally, we extend our channel model from line-of-sight to the Rician channel to demonstrate the effectiveness and robustness of our proposed scheme. Numerical results reveal that the HWI of RISs increases the delay spread, but has no impact on Doppler shift and spread. Additionally, the multiple RIS system not only suffers a more severe delay spread, but is limited in SE due to the HWI. When the number of RISs increases from 2 to 16, the range of average spectral efficiency and delay spread are from 4 to 4.6 Bit/s/Hz and from 0.7 us to 2.5 us, respectively. In contrast to conventional RIS-assisted systems that require channel estimation, the proposed PI-based RIS system offers simplicity without compromising effectiveness and robustness in both SE enhancement and Doppler mitigation. m +
Footnote †: journal: Applied Sciences  
an instantaneous controllable phase and/or amplitude shift on the incoming wave [8]. By doing so, the reflected signal can be reoriented to improve transmission quality [10]. On the other hand, the Doppler effect is a major challenge for HSC systems with dense data sources [2]. Recently, there have already been studies that focused on Doppler mitigation using RIS. [PERSON] [12] discussed the Doppler effect and mitigation in the context of RIS. With the help of RIS, the author investigated the possibility of eliminating or mitigating the multipath fading effect caused by mobile receiver movement. It has been shown that by implementing a real-time tunable RIS, the dramatic changes in signal strength caused by Doppler effects can be mitigated effectively. However, the Doppler shift cannot be compensated for if a direct light-of-sight (LoS) link exists. [PERSON] et al. [13] introduced a general time-varying model for wireless communication with a single RIS and derived the received signal expression in the complex baseband communication system. Moreover, using the proposed model, they proved that if an uncontrollable direct channel link exists, it is impossible to simultaneously maximize the signal-to-noise ratio (SNR) and compensate for the Doppler shifts introduced by the RIS. This result matches well with that of [12]. Based on the model in [13], [PERSON] et al. [14] utilized predictable information such as mobility patterns to design RIS phase shift sets in advance to reduce the complexity of beamforming. The above works [12; 13; 14] commonly assume perfect RIS hardware and transceivers. Unfortunately, in reality, inherent hardware impairments (HWI) such as phase noises, imperfect channel state information (CSI), and quantization errors, which limit system performances, have to be considered due to the imperfections of the electronic devices [15; 16; 17; 18]. In [17], the authors considered the HWI that appears at both RIS and the signal transceivers. Then, closed-form expressions for the average achievable rate (ACR) and the RIS utility were derived; the HWI results in reductions of ACR and RIS utility in more reflecting elements. In [18], the authors analyzed the spectral and energy efficiency of RIS-assisted multiple-input single-output downlink systems with HWI. It is shown that the spectral efficiency (SE) is limited due to HWI even when the numbers of transceiver antennas and RIS elements grow infinitely large, which is in contrast with the conventional case with ideal hardware.