## Appendix B  
We simplify \(z(t)\) first; Note that for the signal with unit power, we drop \(x(t-\tau(t))\) and \(x(t-\tau_{i_{m},i_{n}}-\frac{\phi_{i_{m},i_{n}}(t)+\gamma_{i_{m},i_{n}}(t)}{2 \pi f_{c}})\); then, the (35) can be simplified as  
\[z(t)=A_{0}(t)\sqrt{P_{t}}\Theta_{0}(t)+\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_ {m},i_{n}}(t)\sqrt{P_{t}}\Theta_{i_{m},i_{n}}^{\text{HWI}}(t). \tag{13}\]  
Recall the definitions of \(\Theta_{0}(t)\) and \(\Theta_{i_{m},i_{n}}^{\text{HWI}}(t)\); (13) can be rewritten as  
\[z(t)=\sqrt{P_{t}}\bigg{\{}A_{0}(t)e^{-j2\pi f_{c}\tau_{0}(t)}+\sum_{i=1}^{K} \sum_{m=1,n=1}^{M,N}A_{i_{m},i_{n}}(t)e^{-j2\pi f_{c}\tau_{i_{m},i_{n}}(t)- \gamma_{i_{m},i_{n}}(t)-\gamma_{i_{m},i_{n}}}\bigg{\}}. \tag{14}\]  
Considering the designed phase shift set \(\phi_{i_{m},i_{n}}^{\text{opt}}(t)\) in (33), (14) can be rewritten as  
\[z(t)=\sqrt{P_{t}}\bigg{\{}\underbrace{A_{0}(t)e^{-j2\pi f_{c}\tau_{0}(t)}}_{z _{1}(t)}+\underbrace{\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_{m},i_{n}}(t)e^{- j2\pi f_{c}\tau_{0}(t)-\gamma_{i_{m},i_{n}}}}_{z_{2}(t)}\bigg{\}}. \tag{15}\]  
Next, we know that \(\mathbb{E}\{|z(t)|^{2}\}=P_{t}\mathbb{E}\{(z_{1}(t)+z_{2}(t))^{*}(z_{1}(t)+z_{ 2}(t))\}\). Then, we have \(\mathbb{E}\{(z_{1}(t)+z_{2}(t))^{*}(z_{1}(t)+z_{2}(t))\}=\mathbb{E}\{z_{1}^{*} (t)z_{1}(t)\}+\mathbb{E}\{z_{2}^{*}(t)z_{2}(t)\}+\mathbb{E}\{z_{1}^{*}(t)z_{2}( t)+z_{2}^{*}(t)z_{1}(t)\}\). At time \(t\), we omit the expression \((t)\) for brevity; then, we have  
\[\mathbb{E}\{z_{1}^{*}z_{1}\}=A_{0}^{2}, \tag{16}\]and  
\[\mathbb{E}\{z_{2}^{*}z_{2}\} =\mathbb{E}\bigg{\{}\bigg{|}\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_{ m}i_{n}}e^{-j2\pi f_{c}\gamma_{0}-\gamma_{i_{m}i_{n}}}\bigg{|}^{2}\bigg{\}}\] \[=\mathbb{E}\bigg{\{}\bigg{|}\sum_{i=1}^{K}\sum_{m=1,n=1}^{M,N}A_{i_ {m}i_{n}}e^{-j\gamma_{i_{m}i_{n}}}\bigg{|}^{2}\bigg{\}}. \tag{10}\]  
For the \(i\)-th RIS, let \(k=1,2,\ldots,MN\); then, we can rewrite (10) as  
\[\mathbb{E}\bigg{\{}\bigg{|}\sum_{k=1}^{MN}A_{k}e^{-j\gamma_{j}} \bigg{|}^{2}\bigg{\}} =\sum_{k=1}^{MN}A_{k}^{2}+A_{1}\bigg{|}\sum_{k\
eq 1}^{MN}A_{k} \mathbb{E}\bigg{\{}e^{-\gamma_{1}}e^{-\gamma_{k}}\bigg{\}}\bigg{|}+\] \[\cdots+A_{MN}\bigg{|}\sum_{k\
eq MN}^{MN}A_{k}\mathbb{E}\bigg{\{}e ^{-\gamma_{MN}}e^{-\gamma_{7j}}\bigg{\}}\bigg{|}. \tag{11}\]  
Because \(f(x)=e^{-jx}\) is a continuous function in \(\mathbb{R}\), and \(\gamma_{c}\) as well as \(\gamma_{y}\) are independent and identically distributed (i.i.d) random variables, then \(|\mathbb{E}\{e^{-\gamma_{7}z}e^{-\gamma_{7j}}\}|=|\mathbb{E}\{e^{-\gamma_{7j}} \}|\cdot|\mathbb{E}\{e^{-\gamma_{7j}}\}|\). Recall \(\gamma\sim\mathcal{U}(-a_{\text{HWI}},a_{\text{HWI}})\); due to the law of large numbers and the symmetry of the odd function \(\sin\left(\gamma\right)\), we have \(|\mathbb{E}\{e^{-\gamma_{7}}\}|^{2}=\text{sinc}^{2}(a_{\text{HWI}})\), where \(\text{sinc}(a_{\text{HWI}})=\sin\left(a_{\text{HWI}}\right)/a_{\text{HWI}}\). RIS HWI is i.i.d. ; therefore, we have  
\[(\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:  
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