## 3 Closed-Form Expression of Estimation Error  
From (1) and (3), we get  
\[a_{m}\left(\theta_{c}\right)\times a_{m}^{\ast}\left(\theta_{c}\right)=1. \tag{11}\]  
From (11), we obtain  
\[ \begin{array}{l}\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H} \mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)\\ =\left[\begin{array}{cccc}M&B\left(\theta_{1},\theta_{2}\right)&B\left( \theta_{1},\theta_{3}\right)&\cdots&B\left(\theta_{1},\theta_{d}\right)\\ B^{\ast}\left(\theta_{1},\theta_{2}\right)&M&B\left(\theta_{2},\theta_{3} \right)&\cdots&B\left(\theta_{2},\theta_{d}\right)\\ B^{\ast}\left(\theta_{1},\theta_{3}\right)&B^{\ast}\left(\theta_{2},\theta_{3} \right)&M&\cdots&B\left(\theta_{3},\theta_{d}\right)\\ \vdots&\vdots&&\ddots&\vdots\\ B^{\ast}\left(\theta_{1},\theta_{d}\right)&B^{\ast}\left(\theta_{2},\theta_{d} \right)&B^{\ast}\left(\theta_{3},\theta_{d}\right)&\cdots&M\end{array}\right], \end{array}  \tag{12}\]  
where, from (1), \(B(\theta_{k},\theta_{l})\) is defined as  
\[B(\theta_{k},\theta_{l}) \equiv \sum\limits_{m=1}^{M}\exp\left[j\frac{2\pi}{\lambda}\left(m-1 \right)\Delta\left(\sin\theta_{l}-\sin\theta_{k}\right)\right] \tag{13}\] \[= \frac{1-\exp\left[j\frac{2\pi}{\lambda}\Delta\left(\sin\theta_{l} -\sin\theta_{k}\right)M\right]}{1-\exp\left[j\frac{2\pi}{\lambda}\Delta\left( \sin\theta_{l}-\sin\theta_{k}\right)\right]}.\]  
Using (12), \(\left(\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A} \left(\theta_{1},\cdots,\theta_{d}\right)\right)^{-1}\right)_{k,l}\) can be written as \[\left(\left(\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right) ^{H}\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)\right)^{-1}\right)_{k,l}\right. \tag{14}\] \[=\frac{1}{\det\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d} \right)^{H}\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)\right)}\cdot \left(\operatorname{adj}\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d} \right)^{H}\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)\right)\right)_{k,l}.\]  
\(\operatorname{adj}\left(\cdot\right)\) denotes adjoint of a matrix. \(k,l\)-th element of the adjoint of \(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A}\left(\theta _{1},\cdots,\theta_{d}\right)\) can be expressed as  
\[\left(\operatorname{adj}\left(\mathbf{A}\left(\theta_{1},\cdots, \theta_{d}\right)^{H}\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right) \right)\right)_{k,l} \tag{15}\] \[=\left(-1\right)^{k+l}\cdot\det\left(\left[ \begin{array}{cccccc} M&\cdots&B^{*}\left(\theta_{1},\theta_{k-1}\right)&B^{*}\left(\theta_{1}, \theta_{k+1}\right)&\cdots&B^{*}\left(\theta_{1},\theta_{d}\right)\\ \vdots&\ddots&\vdots&\vdots&&\vdots\\ B\left(\theta_{1},\theta_{l-1}\right)&\cdots&B^{*}\left(\theta_{l-1},\theta_{ k-1}\right)&B^{*}\left(\theta_{l-1},\theta_{k+1}\right)&\cdots&B^{*}\left( \theta_{l-1},\theta_{d}\right)\\ B\left(\theta_{1},\theta_{l+1}\right)&\cdots&B^{*}\left(\theta_{l+1},\theta_{ k-1}\right)&B^{*}\left(\theta_{l+1},\theta_{k+1}\right)&\cdots&B^{*}\left( \theta_{l+1},\theta_{d}\right)\\ \vdots&&\vdots&\vdots&\ddots&\vdots\\ B\left(\theta_{1},\theta_{d}\right)&\cdots&B\left(\theta_{k-1},\theta_{d} \right)&B\left(\theta_{k+1},\theta_{d}\right)&\cdots&M\end{array} \right] \right),\]  
where the determinant can be obtained in many ways, one of which is a cofactor expansion. Generally, the number of incident signals is \(d\). From (14) and (15), an explicit expression of the entry at the \(k\)-th row and \(l\)-th column of \(\left(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A}\left( \theta_{1},\cdots,\theta_{d}\right)\right)^{-1}\) can be obtained. However, it is very complicated to express the determinant in (15) in terms of the entries of \(\mathbf{A}\left(\theta_{1},\cdots,\theta_{d}\right)^{H}\mathbf{A}\left(\theta _{1},\cdots,\theta_{d}\right)\) for all \(k=1,\ldots,d\) and \(l=1,\ldots,d\).