## Appendix C First and Second Order Derivatives of \(B\left(\theta_{1},\theta_{2}\right)\)
\begin{subarray}{c}\theta_{1}=\theta_{ 1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{subarray} }\right)\] (A51) \[+\delta\theta_{2}^{(u=1)}\left(\left.\frac{\partial f_{kl}\left( \theta_{1},\theta_{2}\right)}{\partial\theta_{2}}\right|_{ \begin{subarray}{c }\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{subarray} }\right)\]  
\[g_{kl}\left(\hat{\theta}_{1},\hat{\theta}_{2}\right) = g_{kl}\left(\theta_{1}^{(0)},\theta_{2}^{(0)}\right)+\delta \theta_{1}^{(u=1)}\left(\left.\frac{\partial g_{kl}\left(\theta_{1},\theta_{2 }\right)}{\partial\theta_{1}}\right|_{ \begin{subarray}{c}\theta_{1}=\theta_{ 1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{subarray} }\right)\] (A52) \[+\delta\theta_{2}^{(u=1)}\left(\left.\frac{\partial g_{kl}\left( \theta_{1},\theta_{2}\right)}{\partial\theta_{1}}\right|_{ \begin{subarray}{c }\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{subarray} }\right)\]  
where \(\delta\theta_{1}^{(u=1)}\) and \(\delta\theta_{2}^{(u=1)}\) denote estimation errors for the first incident signal and the second incident signal, respectively. **Appendix G. Derivation of \(E\left[\delta R_{lk}\delta R_{l^{\prime}k^{\prime}}^{*}\right]\)**  
\[\hat{R}_{lk}=\frac{1}{L}\sum_{i=1}^{L}\left(x_{l}(t_{i})x_{k}^{*}(t_{i})+x_{l} (t_{i})n_{k}^{*}(t_{i})+n_{l}(t_{i})x_{k}^{*}(t_{i})+n_{l}(t_{i})n_{k}^{*}(t_{ i})\right).\] (A53)  
\[\hat{R}_{lk}=\frac{1}{L}\sum_{i=1}^{L}\left(x_{l}(t_{i})x_{k}^{*}(t_{i})\right).\] (A54)  
\[\delta R_{lk}=\frac{1}{L}\sum_{i=1}^{L}\left(x_{l}(t_{i})n_{k}^{*}(t_{i})+n_{ l}(t_{i})x_{k}^{*}(t_{i})+n_{l}(t_{i})n_{k}^{*}(t_{i})\right).\] (A55)  
From (A55), \(E\left[\delta R_{lk}\delta R_{l^{\prime}k^{\prime}}^{*}\right]\) is given by  
\[=\frac{1}{L^{2}}\left(\sum_{i=1}^{L}\sum_{i^{\prime}=1}^{L}\left( \begin{array} []{c}x_{l}(t_{i})x_{l^{\prime}}^{*}(t_{\prime})E\left(n_{k}^{*}(t_{i})n_{l^ {\prime}}(t_{\prime})\right)+x_{l}(t_{i})x_{k^{\prime}}(t_{\prime})E\left(n_ {k}^{*}(t_{i})n_{l^{\prime}}^{*}(t_{\prime})\right)\\ +x_{l}(t_{i})E\left(n_{k}^{*}(t_{i})n_{l^{\prime}}^{*}(t_{\prime})n_{l^{ \prime}}(t_{\prime})\right)+x_{k}^{*}(t_{i})x_{k^{\prime}}^{*}(t_{\prime})E \left(n_{l}(t_{i})n_{l^{\prime}}(t_{\prime})\right)\\ +x_{k}^{*}(t_{i})x_{k^{\prime}}(t_{\prime})E\left(n_{l}(t_{i})n_{l^{\prime}}^{ *}(t_{\prime})\right)+x_{k}^{*}(t_{i})E\left(n_{l}(t_{i})n_{l^{\prime}}^{*}(t _{\prime})\right)\\ +x_{k}^{*}(t_{i})x_{k^{\prime}}(t_{\prime})E\left(n_{l}(t_{i})n_{k}^{*}(t_{ \prime})\right)+x_{k^{\prime}}(t_{\prime})E\left(n_{l}(t_{i})n_{k}^{*}(t_{i})n_ {l^{\prime}}^{*}(t_{\prime})\right)\\ +E\left(n_{l}(t_{i})n_{k}^{*}(t_{i})n_{l^{\prime}}^{*}(t_{\prime})n_{k^{ \prime}}(t_{\prime})\right)\end{array} \right)\] (A56)  
where the second moments, third moments, and the fourth moments in (A56) are derived in Appendices H-J, respectively. Finally, when \(i=i^{\prime}\), \(E\left[\left(\delta R\right)_{lk}\left(\delta R\right)_{l^{\prime}k^{\prime}}^ {*}\right]\) is given by  
\[E\left[\left(\delta R\right)_{lk}\left(\delta R\right)_{l^{\prime}k^{ \prime}}^{*}\right]\] (63) \[=\frac{1}{L^{2}}\left(\sum_{i=1}^{L}\sum_{\bar{t}^{\prime}=1}^{L} \left(\begin{array}{c}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\  
\[E\left[\left(\delta R\right)_{lk}\left(\delta R\right)_{l^{\prime}k^{ \prime}}^{*}\right]\] (A75) \[=\frac{1}{L^{2}}\left(\sum_{i=1}^{L}\sum_{i^{\prime}=1}^{L}\left( \begin{array}{c}x_{l}\left(t_{i}\right)x_{l^{\prime}}^{*}\left(t_{l^{\prime}} \right)\left(2\mu^{2}\right)+x_{l}\left(t_{i}\right)x_{l^{\prime}}^{*}\left(t_ {l^{\prime}}\right)\left(-2j\mu^{2}\right)\\ +x_{l}\left(t_{i}\right)\left(2\mu^{3}+2j\mu^{3}\right)+x_{k}^{*}\left(t_{i} \right)x_{l^{\prime}}^{*}\left(t_{l^{\prime}}\right)\left(2j\mu^{2}\right)\\ +x_{k}^{*}\left(t_{i}\right)x_{k^{\prime}}\left(t_{l^{\prime}}\right)\left(2 \mu^{2}\right)+x_{k}^{*}\left(t_{i}\right)\left(2\mu^{3}+2j\mu^{3}\right)\\ +x_{l^{\prime}}^{*}\left(t_{l^{\prime}}\right)\left(2\mu^{3}+2j\mu^{3}\right) \\ +\left(4\left(\frac{\sigma^{2}}{2}+\mu^{2}\right)^{2}\right)\end{array}