# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 5 Eigenvector Perturbation  
Let \(\mathbf{E}_{S}\) and \(\mathbf{\hat{E}}_{S}^{\prime}\) represent matrices whose columns consist of the signal eigenvectors of \(\mathbf{R}\) and \(\mathbf{\hat{R}}^{\prime}\), respectively. The projection matrices onto the column space of \(\mathbf{E}_{S}\) and the column space of \(\mathbf{\hat{E}}_{S}^{\prime}\) are given by  
\[\mathbf{P} \equiv \mathbf{E}_{S}\mathbf{E}_{S}^{H} \tag{15}\] \[\mathbf{\hat{P}}^{\prime} \equiv \mathbf{\hat{E}}_{S}^{\prime}\mathbf{\hat{E}}_{S}^{\prime}. \tag{16}\]  
Using (15) and (16), the MUSIC spectrum in (3) is rewritten as  
\[F(\theta)=\mathbf{a}^{H}(\theta)\mathbf{\hat{P}}\mathbf{a}(\theta). \tag{17}\]  
The symmetric positive semi-definite matrix \(\mathbf{R}\) has the decomposition:  
\[\mathbf{R}=\mathbf{E}\mathbf{V}\mathbf{E}^{H}, \tag{18}\]  
with the unitary matrix \(\mathbf{E}=\left[ \begin{array}{cc}\mathbf{e}_{1}&\cdots&\mathbf{e}_{M}\\ \end{array} \right]\) and the diagonal matrix \(\mathbf{V}=\text{diag}\left(\lambda_{1},\cdots,\lambda_{M}\right)\). Note that \(\mathbf{e}_{1}\), \(\cdots\), \(\mathbf{e}_{M}\) and \(\lambda_{1}\), \(\cdots\), \(\lambda_{M}\) are the eigenvectors and eigenvalues of \(\mathbf{R}\), respectively. The eigenvalues are in descending order. For \(d\) incident signals, \(\mathbf{E}_{s}\) is given by  
\[\mathbf{E}_{s}=\left[ \begin{array}{cc}\mathbf{e}_{1}&\cdots&\mathbf{e}_{d}\\ \end{array} \right]. \tag{19}\]  
Let \(\delta\mathbf{E}_{s}\) represent the difference between \(\mathbf{\hat{E}}_{s}^{\prime}\) and \(\mathbf{E}_{s}\):  
\[\delta\mathbf{E}_{s}=\mathbf{\hat{E}}_{s}^{\prime}-\mathbf{E}_{s}. \tag{20}\]Using (20) in (16) yields  
\[\mathbf{\hat{P}}^{\prime} \approx \left(\mathbf{E}_{S}+\delta\mathbf{E}_{S}\right)\left(\mathbf{E}_{S }+\delta\mathbf{E}_{S}\right)^{H} \tag{21}\] \[= \mathbf{P}+\left[ \begin{array}{cccc}\mathbf{e}_{1}&\cdots& \mathbf{e}_{d}\end{array} \right]\left[ \begin{array}{cccc}\delta\mathbf{e}_ {1}^{\left(g-1\right)}\cdots\delta\mathbf{e}_{d}^{\left(g-1\right)}\end{array}  \right]^{H}+\left[ \begin{array}{cccc}\delta\mathbf{e}_{1}^{\left(g-1\right)}& \cdots&\delta\mathbf{e}_{d}^{\left(g-1\right)}\end{array} \right]\left[ \begin{array}{cccc}\mathbf{e}_{1}\cdots\mathbf{e}_{d}\end{array} \right]^{H}\] \[+\left[ \begin{array}{cccc}\delta\mathbf{e}_{1}^{\left(g-1\right) }&\cdots&\delta\mathbf{e}_{d}^{\left(g-1\right)}\end{array} \right]\left[ \begin{array}{cccc}\delta\mathbf{e}_{1}^{\left(g-1\right)}\cdots\delta \mathbf{e}_{d}^{\left(g-1\right)}\end{array} \right]^{H}\] \[\equiv \mathbf{P}+\delta\mathbf{P}^{\left(a\right)}+\delta\mathbf{P}^{ \left(b\right)}+\delta\mathbf{P}^{\left(c\right)}=\mathbf{P}+\delta\mathbf{P}.\]  
The first order perturbation of the eigenvectors due to the perturbation in the entries of the covariance matrix can be expressed as [27]  
\[\delta\mathbf{e}_{k}^{\left(g-1\right)} = \sum_{i=1,j\
eq k}^{M}t_{i,k}^{\left(g=1\right)}\mathbf{e}_{i} \tag{22}\] \[t_{a,b}^{\left(g-1\right)} \equiv \frac{\mathbf{e}_{a}^{H}\delta\mathbf{e}_{b}}{\lambda_{b}-\lambda _{a}}=\frac{\sum\limits_{p=1}^{M}\left(\sum\limits_{q=1}^{M}e_{a,p}^{*}\left( \delta\mathbf{R}\right)_{pq}e_{b,d}\right)}{\lambda_{b}-\lambda_{a}}, \tag{23}\]  
where the superscript \(\left(g=1\right)\) denotes the first order perturbation, \(e_{a,p}\) and \(e_{b,q}\) denote the \(p\)-th entry of \(\mathbf{e}_{a}\) and \(q\)-th entry of \(\mathbf{e}_{b}\), respectively.