# Asymptotic Performance Analysis of the MUSIC Algorithm for Direction-of-Arrival Estimation
## 3 MUSIC Algorithm  
Let \(\theta_{1}^{(0)},\ldots,\theta_{d}^{(0)}\) represent the true incident angles of \(d\) incident signals. The noisy array output vector is written as\[\mathbf{x}^{\prime}\left(t_{i}\right) = \left[ \begin{array}{c}x_{1}^{\prime}\left(t_{i}\right)\\ x_{2}^{\prime}\left(t_{i}\right)\\ \vdots\\ x_{M}^{\prime}\left(t_{i}\right)\end{array} \right]=\left[ \begin{array}{c}x_{ 1}\left(t_{i}\right)+n_{1}\left(t_{i}\right)\\ x_{2}\left(t_{i}\right)+n_{2}\left(t_{i}\right)\\ \vdots\\ x_{M}\left(t_{i}\right)+n_{M}\left(t_{i}\right)\end{array} \right] \tag{1}\] \[= \left[ \begin{array}{ccc}\exp\left(j\psi_{1}\left(\theta_{1}^{ \left(0\right)\right)}\right)&\cdots&\exp\left(j\psi_{1}\left(\theta_{d}^{ \left(0\right)}\right)\right)\\ \exp\left(j\psi_{2}\left(\theta_{1}^{\left(0\right)}\right)\right)&\cdots&\exp \left(j\psi_{2}\left(\theta_{d}^{\left(0\right)}\right)\right)\\ \vdots&\ddots&\vdots\\ \exp\left(j\psi_{M}\left(\theta_{1}^{\left(0\right)}\right)\right)&\cdots&\exp \left(j\psi_{M}\left(\theta_{d}^{\left(0\right)}\right)\right)\end{array}  \right]\left[ \begin{array}{c}s_{1}\left(t_{i}\right)\\ \vdots\\ s_{d}\left(t_{i}\right)\end{array} \right]+\left[ \begin{array}{c}n_{1}\left( t_{i}\right)\\ n_{2}\left(t_{i}\right)\\ \vdots\\ n_{M}\left(t_{i}\right)\end{array} \right]\] \[= \left[ \begin{array}{ccc}\mathbf{a}\left(\theta_{1}^{\left(0 \right)}\right)&\cdots&\mathbf{a}\left(\theta_{d}^{\left(0\right)}\right)\\ \end{array} \right]\left[ \begin{array}{c}s_{1}\left(t_{i}\right)\\ \vdots\\ s_{d}\left(t_{i}\right)\end{array} \right]+\left[ \begin{array}{c}n_{1}\left( t_{i}\right)\\ n_{2}\left(t_{i}\right)\\ \vdots\\ n_{M}\left(t_{i}\right)\end{array} \right]\] \[= \mathbf{A}\left(\theta_{1}^{\left(0\right)}\cdots\theta_{d}^{ \left(0\right)}\right)\left[ \begin{array}{c}s_{1}\left(t_{i}\right)\\ \vdots\\ s_{d}\left(t_{i}\right)\end{array} \right]+\mathbf{n}(t_{i}),\]  
where \(M\) denotes the number of antenna elements, and \(\mathbf{n}\left(t_{i}\right)\) denotes zero-mean Gaussian noise vector. Note that \(\mathbf{x}^{\prime}(t_{n})\) and \(\mathbf{x}(t_{n})\) denote the noisy array output vector and the noiseless array output vector, respectively. The Gaussian noise vector is a random vector in that entries of the vector are zero-mean Gaussian random variables. The variances of all the random variables are equal, and given by \(\sigma^{2}\). The real part and the imaginary part of each random variable are independent. The variance of the real part of each random variable is denoted by \(\frac{\sigma^{2}}{2}\), and the variance of the imaginary part of each random variable is also given by \(\frac{\sigma^{2}}{2}\). Assume that the uniform linear array (ULA) is adopted to estimate azimuth angles of the incident angles. The phase at the \(m\)-th antenna, with phase reference at the origin (\(x=0\), \(y=0\)), can be expressed as  
\[\psi_{m}\left(\theta_{n}^{\left(0\right)}\right)=\Delta\sin\theta_{n}^{\left(0 \right)}\frac{2\pi}{\Lambda}\left(m-1\right),\ \ \ \ \ m=1,\ldots,M,\ \ \ \ \ n=1,\ldots,d \tag{2}\]  
where \(\Lambda\) denotes the wavelength of the incident signal, and \(\Delta\) is the distance between the adjacent antenna elements. Let the superscript \(H\) denote conjugate transpose of a matrix. In the signal subspace-based MUSIC algorithm, the DOAs are estimated from the angles which maximize the MUSIC spectrum:  
\[\hat{F}(\theta)=\mathbf{a}^{\mathrm{H}}(\theta)\mathbf{\hat{E}}_{\mathrm{s}}^{ \prime}\mathbf{\hat{E}}_{\mathrm{s}}^{\prime\mathrm{H}}\mathbf{a}(\theta), \tag{3}\]  
where the steering vector can be written as  
\[\mathbf{a}\left(\theta\right)=\left[\exp\left(j\psi_{1}\left(\theta\right) \right)\,\exp\left(j\psi_{2}\left(\theta\right)\right)\,\ldots\,\exp\left(j \psi_{M}\left(\theta\right)\right)\right]^{T} \tag{4}\]  
where \(\mathbf{\hat{E}}_{\mathrm{s}}^{\prime}\) is a matrix whose columns consist of the signal eigenvectors of the sample covariance matrix.