## 5 Spectral Efficiency Analysis  
In this section, we analyze the downlink SE for the proposed system using the phase shift set \(\phi^{\rm opt}_{i_{in},i_{in}}(t)\) in (33), considering that RIS and transceiver HWI exists. The result will reveal that the RIS and transceiver HWI can decrease SE in different ways, and as the number of RIS \(K\) and transmit power \(P_{t}\) becomes large, the main impairment factor is transceiver HWI instead of RIS HWI. Recall (6) and (14), and denote the received signal as  
\[z(t)\triangleq S_{0}^{\rm HWI}(t)\Theta_{0}(t)+\sum_{i=1}^{K}\sum_{m=1,n=1}^{ M,N}S_{i_{in},i_{in}}^{\rm HWI}(t)\Theta_{i_{in},i_{in}}^{\rm HWI}(t). \tag{35}\]  
Then, the spectral efficiency \(\mathrm{SE}(t)\) at time \(t\) can be obtained as  
\[\mathrm{SE}(t)=\mathbb{E}\bigg{\{}\log_{2}\bigg{\{}1+\frac{|z(t)|^{2}}{(\kappa _{t}+\kappa_{r})|z(t)|^{2}+N_{0}}\bigg{\}}\bigg{\}}. \tag{36}\]  
By substituting the optimal phase shift set \(\phi^{\rm opt}_{i_{in},i_{in}}(t)\) in (33) into (36), we characterize the impact of the HWI on the downlink SE, given in the following proposition. **Proposition 2**. : _Consider a system with \(K\) identical RISs. Each of them uses the optimal phase shift set \(\phi^{\rm opt}_{i_{in},i_{in}}(t)\) in (33), the RIS HWI \(\gamma_{i_{in},i_{in}}\sim\mathcal{U}[-a_{\rm HWI},a_{\rm HWI}]\), where \(a_{\rm HWI}\in[0,\pi/2]\), and the transceiver HWI \(\eta_{t}\) and \(\eta_{r}\). Then, at time \(t\), the downlink SE of the proposed system with HWI can be obtained as_  
\[\mathrm{SE}(t)\xrightarrow{a.s.}\log_{2}\bigg{\{}1+\frac{\mathcal{Q}(t)}{( \kappa_{t}+\kappa_{r})\mathcal{Q}(t)+N_{0}}\bigg{\}}, \tag{37}\]  
_where \(\mathcal{Q}(t)\) is the received power that can be obtained as_  
\[\mathcal{Q}(t)= \mathbb{E}\{|z(t)|^{2}\}\xrightarrow{a.s.}P_{t}\bigg{\{}A_{0}^{ 2}(t)+\mathrm{sinc}^{2}(a_{\rm HWI})A^{\star}(t) \tag{38}\] \[+\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_{in},i_{in}}^{2}(t)+2A_{0 }(t)\,\mathrm{sinc}(a_{\rm HWI})\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_{in},i_ {in}}(t)\bigg{\}},\]  
_where_  
\[A^{\star}(t)=A_{1}(t)\sum_{k\
eq 1}^{KMN}A_{k}(t)+A_{2}(t)\sum_{k\
eq 2}^{ KMN}A_{k}(t)+\cdots+A_{KMN}(t)\sum_{k\
eq KMN}^{KMN}A_{k}(t), \tag{39}\]  
_and \(k=1,2,\ldots,KMN\)._  
Proof of Proposition 2.: See Appendix B. _Special Case 1_: For an ideal multi-RIS communication system without any HWI, \(\kappa_{t}=\kappa_{r}=0\) and \(a_{\rm HWI}=0\). Thus, the downlink spectral efficiency \(\mathrm{SE}(t)\) reduces to  
\[\mathrm{SE}(t)=\log_{2}\bigg{\{}1+\frac{\mathcal{Q}_{\rm ideal}(t)}{N_{0}} \bigg{\}}, \tag{40}\]  
where  
\[\mathcal{Q}_{\rm ideal}(t)=P_{t}\bigg{\{}A_{0}^{2}(t)+M^{2}N^{2}A^{\star}(t)+ 2 MNA_{0}\sum_{k=1}^{KMN}A_{k}\bigg{\}}. \tag{41}\]  
**Remark 5**. : _Mathematically, the RIS HWI \(\gamma_{i_{m},i_{n}}\) leads to two extra scalable factors, \(\operatorname{sinc}(a_{\text{HWI}})\) and \(\operatorname{sinc}^{2}(a_{\text{HWI}})\), to the received power \(\mathcal{Q}(t)\). Note that \(\operatorname{sinc}(a_{\text{HWI}})\) is a monotonically decreasing function when \(a_{\text{HWI}}\in[0,\pi/2]\)._  
_Special Case 2_: If the total number of RISs is large enough, i.e., \(K\gg 1\), then the downlink SE can be rewritten as  
\[\operatorname{SE}(t)\big{|}_{K\gg 1}\overset{a.s.}{\longrightarrow}\log_{2} \bigg{\{}1+\frac{1}{\kappa_{t}+\kappa_{r}}\bigg{\}}. \tag{42}\]  
**Remark 6**. : _This result reveals that the only factor that can influence SE is the HWI in transceivers when the proposed system has enough RISs. Furthermore, if the transmit power is high, the SE saturates, and (42) is also the upper bound of SE for a high \(P_{t}\) regime. In other words, the SE cannot be further enhanced by increasing \(K\) or by increasing the transmit power \(P_{t}\).