## 3 Closed-Form Expression of Estimation Error
\begin{array}{c|c}\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left( \begin{array}{c|c}\frac{\partial f_{kl}\left(\theta_{1},\theta_{2}\right)}{ \partial\theta_{1}}\end{array}\right|_{\begin{array}{c|c}\theta_{1}=\theta_ {1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right) \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\begin{array}{c|c }\frac{\partial f_{kl}\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}} \end{array}\right|_{\begin{array}{c|c}\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)\] \[=-\left[\begin{array}{c|c}\sum\limits_{k=1}^{M}\sum\limits_{l=1}^ {M}\left(\begin{array}{c|c}\frac{\partial f_{kl}\left(\theta_{1},\theta_{2} \right)}{\partial\theta_{1}}\end{array}\right|_{\begin{array}{c|c}\theta_{1 }=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)\end{array}\right]. \tag{25}\]  
We define \(\mathbf{C}^{(u=1)}\) and \(\mathbf{b}\) as follows:  
\[\mathbf{C}^{(u=1)}\equiv\left[\begin{array}{c|c}\sum\limits_{k=1}^{M}\sum \limits_{l=1}^{M}\left(\begin{array}{c|c}\frac{\partial f_{kl}\left(\theta_{1 },\theta_{2}\right)}{\partial\theta_{1}}\end{array}\right|_{\begin{array}{c |c}\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)&\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\begin{array}{c|c }\frac{\partial f_{kl}\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}} \end{array}\right|_{\begin{array}{c|c}\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)\\ \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\begin{array}{c|c}\frac{ \partial f_{kl}\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{1}}\end{array} \right|_{\begin{array}{c|c}\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)&\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\begin{array}{c|c }\frac{\partial g_{kl}\left(\theta_{1},\theta_{2}\right)}{\partial\theta_{2}} \end{array}\right|_{\begin{array}{c|c}\theta_{1}=\theta_{1}^{(0)}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}}\right)\left(\hat{R}_{lk}+\delta R_{lk} \right)\end{array}\right] \tag{26}\]  
\[\mathbf{b}\equiv-\left[\begin{array}{c|c}\sum\limits_{k=1}^{M}\sum\limits_{l= 1}^{M}f_{kl}\left(\theta_{1}^{(0)},\theta_{2}^{(0)}\right)\delta R_{lk}\\ \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}g_{kl}\left(\theta_{1}^{(0)},\theta_{ 2}^{(0)}\right)\delta R_{lk}\end{array}\right]. \tag{27}\]  
Using (26) and (27) in (25), (25) can be written as  
\[\mathbf{C}^{(u=1)}\left[\begin{array}{c|c}\delta\theta_{1}^{(u=1)}\\ \delta\theta_{2}^{(u=1)}\end{array}\right]=\mathbf{b}. \tag{28}\]  
The solution of (28) and the associated estimates are given by  
\[\left[\begin{array}{c}\delta\theta_{1}^{(u=1)}\\ \delta\theta_{2}^{(u=1)}\end{array}\right]=\left(\mathbf{C}^{(u=1)}\right)^{-1} \mathbf{b} \tag{29}\]\[\left\{\begin{array}{l}\hat{\theta}_{1}^{(u=1)}=\theta_{1}^{(0)}+\delta\theta_{1 }^{(u=1)}\\ \hat{\theta}_{2}^{(u=1)}=\theta_{2}^{(0)}+\delta\theta_{2}^{(u=1)}\end{array}\right. \tag{30}\]  
where the superscript \((u=1)\) indicates that the first order Taylor expansion is used. This approximation is called \(U\) approximation.  
At high SNR, it is true that \(\hat{R}_{lk}\) is much larger than \(\delta R_{lk}\):  
\[\hat{R}_{lk}+\delta R_{lk}\simeq\hat{R}_{lk}. \tag{31}\]  
This approximation is called \(V\) approximation.  
Using (31) in (25) yields  
\[\left[\begin{array}{l}\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M} \left(\frac{\partial f_{lk}(\theta_{1},\theta_{2})}{\partial\theta_{1}}\Big{|} _{\theta_{1}=\theta_{1}^{(0)}}\right)\hat{R}_{lk}\quad\sum\limits_{k=1}^{M} \sum\limits_{l=1}^{M}\left(\frac{\partial f_{lk}(\theta_{1},\theta_{2})}{ \partial\theta_{2}}\Big{|}_{\theta_{1}=\theta_{1}^{(0)}}\right)\hat{R}_{lk} \\ \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\frac{\partial g_{lk}(\theta_ {1},\theta_{2})}{\partial\theta_{1}}\Big{|}_{\theta_{1}=\theta_{1}^{(0)}} \right)\hat{R}_{lk}\quad\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\frac {\partial g_{lk}(\theta_{1},\theta_{2})}{\partial\theta_{2}}\Big{|}_{\theta_ {1}=\theta_{1}^{(0)}}\right)\hat{R}_{lk}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}\right] \tag{32}\] \[=-\left[\begin{array}{l}\sum\limits_{k=1}^{M}\sum\limits_{l=1} ^{M}f_{kl}\left(\theta_{1}^{(0)},\theta_{2}^{(0)}\right)\delta R_{lk}\\ \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\delta_{kl}\left(\theta_{1}^{(0)}, \theta_{2}^{(0)}\right)\delta R_{lk}\end{array}\right]\]  
where the superscript \((u=1,v)\) indicates that both \(U\) approximation and \(V\) approximation are used to get the estimates.  
Let \(\textbf{C}^{(u=1,v)}\) be defined by  
\[\textbf{C}^{(u=1,v)}\equiv\left[\begin{array}{l}\sum\limits_{k=1}^{M}\sum \limits_{l=1}^{M}\left(\left.\frac{\partial f_{lk}(\theta_{1},\theta_{2})}{ \partial\theta_{1}}\Big{|}_{\theta_{1}=\theta_{1}^{(0)}}\right)\hat{R}_{lk} \quad\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\left.\frac{\partial f_{ lk}(\theta_{1},\theta_{2})}{\partial\theta_{2}}\Big{|}_{\theta_{1}=\theta_{1}^{(0)}} \right)\hat{R}_{lk}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}\right)\hat{R}_{lk}\\ \sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left(\left.\frac{\partial g_{kl}( \theta_{1},\theta_{2})}{\partial\theta_{1}}\Big{|}_{\theta_{1}=\theta_{1}^{(0) }}\right)\hat{R}_{lk}\quad\sum\limits_{k=1}^{M}\sum\limits_{l=1}^{M}\left( \left.\frac{\partial g_{kl}(\theta_{1},\theta_{2})}{\partial\theta_{2}}\Big{|}_ {\theta_{1}=\theta_{1}^{(0)}}\right)\hat{R}_{lk}\\ \theta_{2}=\theta_{2}^{(0)}\end{array}