## 4 Analysis
### Theoretical Error Analysis  
The analytical performance of the proposed RIS is investigated in this part. Equations (55) and (56) can be rewritten in matrix form:  
\[\mathbf{X}_{\mu} =\mathbf{A}^{T}\mathbf{\mathcal{W}}\mathbf{A}, \tag{74}\] \[\mathbf{w}_{\mu} =\mathbf{A}^{T}\mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{b}), \tag{75}\]  
where \(\mathbf{A}=[\mathbf{a}_{1}^{*},\dots,\mathbf{a}_{M}^{*}]^{T}\), \(\mathbf{a}_{i}^{*}=\partial\bar{r}_{i}(\mathbf{\mu})/\partial\mathbf{\mu}|_{\mathbf{\mu}= \mathbf{\mu}^{*}}\), \(\mathbf{b}=[b_{1}^{*},\dots,b_{M}^{*}]^{T}\), and \(b_{i}^{*}=\bar{r}_{i}(\tilde{\mathbf{\mu}}^{*})-\mathbf{a}_{i}^{T}\tilde{\mathbf{\mu} }^{*}\) represents the solution that reaches one of the threshold conditions. When the noise is relatively small so the convergence approaches, we have \(\widetilde{\mathbf{X}}_{k}\simeq\mathbf{X}_{\mathbf{\mu}}\) and \(\widetilde{\mathbf{w}}_{k}\simeq\mathbf{w}_{\mathbf{\mu}}\). The iteration converges after \(K^{*}\) times, obtaining the final values of (62)-(63):  
\[\mathbf{X}_{\mathbf{\mu}}^{(K^{*})} \approx c(K^{*})\cdot\rho_{K^{*}}\mathbf{A}^{T}\mathbf{\mathcal{W}} \mathbf{A}=\mathbf{A}^{T}\mathbf{\mathcal{W}}\mathbf{A}, \tag{76}\] \[\mathbf{w}_{\mathbf{\mu}}^{(K^{*})} \approx c(K^{*})\cdot\rho_{K^{*}}\mathbf{A}^{T}\mathbf{\mathcal{W}}( \mathbf{r}-\mathbf{b})=\mathbf{A}^{T}\mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{b}), \tag{77}\]  
where \(c(K^{*})=(1-\kappa^{K^{*}})/(1-\kappa)\) for \(\kappa\
eq 1\) and \(K^{*}\) if \(\kappa=1\). In this case, the converged solution is  
\[\tilde{\mathbf{\mu}}=\tilde{\mathbf{\mu}}^{(K^{*})}\approx(\mathbf{A}^{T}\mathbf{ \mathcal{W}}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{ b}), \tag{78}\]  
after substituting (76) and (77) into (65). In using (49), \(b_{i}\) is approximated by  
\[b_{i}=r_{i}(\tilde{\mathbf{\mu}}^{*})-\mathbf{a}_{i}^{T}\tilde{\mathbf{ \mu}}_{1}\approx r_{i}(\mathbf{\mu}^{o})-\mathbf{a}_{i}^{T}\mathbf{\mu}^{o}, \tag{79}\]  
where \(\mathbf{\mu}\) is asymptotically replaced by \(\mathbf{\mu}^{o}\). Substituting (79) into (78) leads to  
\[\tilde{\mathbf{\mu}}\approx(\mathbf{A}^{T}\mathbf{\mathcal{W}}\mathbf{A})^{-1}\mathbf{ A}^{T}\mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{r}(\mathbf{\mu}^{o}))+\mathbf{\mu}^{o}. \tag{80}\]  
Moving the true value to the left side, the estimation error is  
\[\delta\mathbf{\mu}=\tilde{\mathbf{\mu}}-\mathbf{\mu}^{o}\approx(\mathbf{A}^{T}\mathbf{ \mathcal{W}}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{ r}^{o}), \tag{81}\]  
leading to the bias and covariance matrix of the RIS as  
\[\mathrm{bias}(\tilde{\mathbf{\mu}}) =E[\delta\mathbf{\mu}], \tag{82}\] \[\mathrm{cov}(\tilde{\mathbf{\mu}}) =E[\delta\mathbf{\mu}\delta\mathbf{\mu}^{T}]\approx(\mathbf{A}^{T}\mathbf{ \mathcal{W}}\mathbf{A})^{-1}. \tag{83}\]When the noise is not significant so that \(\bar{\mathbf{\mu}}\approx\mathbf{\mu}^{o}\) holds, the bias (82) approximately equals zero. From (50), we have \(\mathbf{A}\approx\frac{\partial\mathbf{\sigma}^{2}}{\partial\mathbf{\mu}^{oT}}\). Comparing (83) with (73) can conclude that the covariance of the proposed RIS attains the CRLB if the noise and error are not significant.