## 3 Closed-Form Expression of Estimation Error
In addition, due to very complicated expression, expressing each entry of \(\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A}\left( \theta_{1},\cdots,\theta_{d}\right)\right)^{-1}\) in terms of the entries of \(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A}\left(\theta _{1},\cdots,\theta_{d}\right)\) may impair the readability of this paper. Therefore, the number of signals in this paper is set to two. Using (12) in (3), the entries of \(\mathbf{P}_{\mathbf{A}\left(\theta_{1},\theta_{2}\right)}\) can be written as  
\[P_{kl}\left(\theta_{1},\theta_{2}\right) =\frac{1}{\det\left(\theta_{1},\theta_{2}\right)}\left[\left(a_{ k}\left(\theta_{1}\right)M-a_{k}\left(\theta_{2}\right)B^{*}\left(\theta_{1}, \theta_{2}\right)\right)a_{l}^{*}\left(\theta_{1}\right)\right] \tag{16}\] \[+\frac{1}{\det\left(\theta_{1},\theta_{2}\right)}\left[\left(a_{ k}\left(\theta_{2}\right)M-a_{k}\left(\theta_{1}\right)B\left(\theta_{1}, \theta_{2}\right)\right)a_{l}^{*}\left(\theta_{2}\right)\right]\]  
where \(\det\left(\theta_{1},\theta_{2}\right)\) is given in Appendix B. The numerator of \(P_{kl}\left(\theta_{1},\theta_{2}\right)\) is defined as \(Q_{kl}\left(\theta_{1},\theta_{2}\right)\):  
\[Q_{kl}\left(\theta_{1},\theta_{2}\right) =\left(a_{k}\left(\theta_{1}\right)M-a_{k}\left(\theta_{2}\right) B^{*}\left(\theta_{1},\theta_{2}\right)\right)a_{l}^{*}\left(\theta_{1}\right)\] \[+\left(a_{k}\left(\theta_{2}\right)M-a_{k}\left(\theta_{1}\right) B\left(\theta_{1},\theta_{2}\right)\right)a_{l}^{*}\left(\theta_{2}\right). \tag{17}\]  
Let \(\theta_{1}^{(0)}\) and \(\theta_{2}^{(0)}\) denote the true incident angles of two incident signals and let \(\hat{\theta}_{1}\) and \(\hat{\theta}_{2}\) denote the estimates of two incident signals. From (9) and (10), \(f_{kl}\left(\theta_{1},\theta{2}\right)\) and \(g_{kl}\left(\theta_{1},\theta{2}\right)\) in Appendix I, the estimates, \(\left[\hat{\theta}_{1},\hat{\theta}_{2}\right],\) satisfy \[\frac{\partial}{\partial\theta_{1}}\text{tr}\left(\mathbf{P}_{\mathbf{A} \left(\theta_{1},\theta_{2}\right)}\mathbf{\hat{R}}^{\prime}\right)\bigg{|}_{ \begin{subarray}{c}\theta_{1}=\theta_{1}\\ \theta_{2}=\theta_{2}\end{subarray} } = \sum_{k=1}^{M}\sum_{l=1}^{M}\left.\frac{\partial P_{kl}\left( \theta_{1},\theta_{2}\right)}{\partial\theta_{1}}R_{lk}^{\prime}\right|_{ \begin{subarray}{c}\theta_{1}=\theta_{1}\\ \theta_{2}=\theta_{2}\end{subarray} } \tag{18}\] \[= \sum_{k=1}^{M}\sum_{l=1}^{M}f_{kl}\left(\hat{\theta}_{1},\hat{ \theta}_{2}\right)R_{lk}^{\prime}=0\] \[\frac{\partial}{\partial\theta_{2}}\text{tr}\left(\mathbf{P}_{ \mathbf{A}\left(\theta_{1},\theta_{2}\right)}\mathbf{\hat{R}}^{\prime}\right) \bigg{|}_{ \begin{subarray}{c}\theta_{1}=\theta_{1}\\ \theta_{2}=\theta_{2}\end{subarray} } = \sum_{k=1}^{M}\sum_{l=1}^{M}\left.\frac{\partial P_{kl}\left( \theta_{1},\theta_{2}\right)}{\partial\theta_{2}}R_{lk}^{\prime}\right|_{ \begin{subarray}{c}\theta_{1}=\theta_{1}\\ \theta_{2}=\theta_{2}\end{subarray} }\] (19) \[= \sum_{k=1}^{M}\sum_{l=1}^{M}g_{kl}\left(\hat{\theta}_{1},\hat{ \theta}_{2}\right)R_{lk}^{\prime}=0.\]  
The derivatives of ML cost function with respect to \(\theta_{1}\) and \(\theta_{2}\) should be zero at the true incident angles for the noiseless sample covariance matrix:  
\[\frac{\partial\left(\mathbf{P}_{\mathbf{A}\left(\theta_{1}, \theta_{2}\right)}\mathbf{\hat{R}}\right)}{\partial\theta_{1}}\Bigg{|}_{ \begin{subarray}{c}\theta_{1}=\theta_{1}^{\left(0\right)}\\ \theta_{2}=\theta_{2}^{\left(0\right)}\end{subarray} } = \sum_{k=1}^{M}\sum_{l=1}^{M}\left.\frac{\partial P_{kl}\left( \theta_{1},\theta_{2}\right)}{\partial\theta_{1}}\,\hat{R}_{lk}\right|_{ \begin{subarray}{c}\theta_{1}=\theta_{1}^{\left(0\right)}\\ \theta_{2}=\theta_{2}^{\left(0\right)}\end{subarray} } \tag{20}\] \[= \sum_{k=1}^{M}\sum_{l=1}^{M}f_{kl}\left(\theta_{1}^{\left(0 \right)},\theta_{2}^{\left(0\right)}\right)\,\hat{R}_{lk}=0\]  
\[\frac{\partial\left(\mathbf{P}_{\mathbf{A}\left(\theta_{1}, \theta_{2}\right)}\mathbf{\hat{R}}\right)}{\partial\theta_{2}}\Bigg{|}_{