# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 3 MUSIC Algorithm
Since the number of signal eigenvectors is equal to the number of the incident signals, there are \(d\) signal eigenvectors associated with \(d\) incident signals. Difference between the True Covariance Matrix and the Sample Covariance Matrix: \(\delta\mathbf{R}\)  
The true covariance matrix, obtained from the ensemble average, is expressed as  
\[\mathbf{R}=\mathbf{A}\left(\theta_{1}^{\left(0\right)}\ldots\theta_{d}^{\left(0 \right)}\right)E\left(\mathbf{s}\left(t\right)\mathbf{s}^{H}\left(t\right) \right)\mathbf{A}\left(\theta_{1}^{\left(0\right)}\ldots\theta_{d}^{\left(0 \right)}\right)^{H}+E\left(\mathbf{n}\left(t\right)\mathbf{n}^{H}\left(t \right)\right), \tag{5}\]  
where \(E\left(\cdot\right)\) denotes statistical expectation, and  
\[E\left(\mathbf{s}\left(t\right)\mathbf{s}^{H}\left(t\right)\right)=\left[ \begin{array}{ccc}E\left(s_{1}\left(t\right)s_{1}^{*}\left(t\right)\right)& \cdots&E\left(s_{1}\left(t\right)s_{d}^{*}\left(t\right)\right)\\ \vdots&\ddots&\vdots\\ E\left(s_{d}\left(t\right)s_{1}^{*}\left(t\right)\right)&\cdots&E\left(s_{d} \left(t\right)s_{d}^{*}\left(t\right)\right)\end{array} \right] \tag{6}\]  
is the source covariance matrix. The superscript asterisk denotes a complex conjugate of a complex quantity. The noise covariance matrix in (5) is simplified to  
\[E\left(\mathbf{n}\left(t\right)\mathbf{n}^{H}\left(t\right)\right)=\sigma^{2} \mathbf{I}_{M} \tag{7}\]  
when the noises on antenna elements are spatially uncorrelated. Note that \(\mathbf{I}_{M}\) denotes \(M\)-by-\(M\) identity matrix. (5) can be expressed in matrix form:  
\[\mathbf{R} = \left[ \begin{array}{ccc}\sum\limits_{p=1}^{d}\sum\limits_{i=1} ^{d}\exp\left(j\left(\psi_{1}\left(\theta_{i}\right)-\psi_{1}\left(\theta_{p} \right)\right)\right)E\left(s_{i}s_{p}^{*}\right)&\cdots&\sum\limits_{p=1}^{d} \sum\limits_{i=1}^{d}\exp\left(j\left(\psi_{1}\left(\theta_{i}\right)-\psi_{M }\left(\theta_{p}\right)\right)\right)E\left(s_{i}s_{p}^{*}\right)\\ \vdots&\ddots&\vdots\\ \sum\limits_{p=1}^{d}\sum\limits_{i=1}^{d}\exp\left(j\left(\psi_{M}\left( \theta_{i}\right)-\psi_{1}\left(\theta_{p}\right)\right)\right)E\left(s_{i}s _{p}^{*}\right)&\cdots&\sum\limits_{p=1}^{d}\sum\limits_{i=1}^{d}\exp\left(j \left(\psi_{M}\left(\theta_{i}\right)-\psi_{M}\left(\theta_{p}\right)\right) \right)E\left(s_{i}s_{p}^{*}\right)\end{array} \right]+\sigma^{2}\mathbf{I}_{M}.\]  
The sample covariance matrix, which is an ML estimate of the true covariance matrix, is obtained from time-average:  
\[\mathbf{\hat{R}}^{\prime} = \frac{1}{L}\sum\limits_{i=1}^{L}\mathbf{x}^{\prime}\left(t_{i} \right)\mathbf{x}^{\prime}\left(t_{i}\right)^{H}=\frac{1}{L}\sum\limits_{i=1} ^{L}\left(\mathbf{x}\left(t_{i}\right)+\mathbf{n}\left(t_{i}\right)\right)( \mathbf{x}\left(t_{i}\right)+\mathbf{n}\left(t_{i}\right))^{H} \tag{9}\] \[= \frac{1}{L}\left[ \begin{array}{ccc}\sum\limits_{i=1}^{L}\left(x _{1}\left(t_{i}\right)x_{1}^{*}\left(t_{i}\right)+x_{1}\left(t_{i}\right)n_{1} ^{*}\left(t_{i}\right)+n_{1}\left(t_{i}\right)x_{1}^{*}\left(t_{i}\right)+n_{1 }\left(t_{i}\right)n_{1}^{*}\left(t_{i}\right)\right)&\cdots\\ \vdots&\ddots\\ \sum\limits_{i=1}^{L}\left(x_{M}\left(t_{i}\right)x_{1}^{*}\left(t_{i}\right)+ x_{M}\left(t_{i}\right)n_{1}^{*}\left(t_{i}\right)+n_{M}\left(t_{i}\right)x_{1}^{*} \left(t_{i}\right)+n_{M}\left(t_{i}\right)n_{1}^{*}\left(t_{i}\right)\right)& \cdots\\ \vdots&\\ \sum\limits_{i=1}^{L}\left(x_{M}\left(t_{i}\right)x_{M}^{*}\left(t_{i}\right)+ x_{M}\left(t_{i}\right)n_{M}^{*}\left(t_{i}\right)+n_{M}\left(t_{i}\right)x_{M}^{*} \left(t_{i}\right)+n_{M}\left(t_{i}\right)n_{M}^{*}\left(t_{i}\right)\right) \end{array}