## 2 System Model
### MUSIC Algorithm
* \(\delta_{k,mn}\) denotes the attenuation factor or received gain of the array element with \(x\)-axis sequence number \(m\) and \(y\)-axis sequence number \(n\) for the \(k\)th signal. * \(s_{k}(t)=\exp(j2\pi f_{k}t)\) is the \(k\)th signal when it arrives at the antenna array. The signal frequency is \(f_{k}\), and the signals are non-coherent. * \(\tau_{k,mn}\) denotes the delay of the \(k\)th signal in the \(x\)-axis sequence numbered \(m\) and the \(y\)-axis sequence numbered \(n\) with respect to the origin reference array element. * \(n_{mn}(t)\) denotes the noise sample of the \(mn\)th antenna at time \(t\). It has amplitude 1, mean 0, and follows a circularly symmetric complex normal distribution with variance \(\sigma^{2}\), and is additive Gaussian white noise. Based on the above signal model, the signal delay \(\tau_{k,mn}\) can be expressed as follows:  
\[\tau_{k,mn}=d/\lambda\cdot(m\cos\varphi_{k}\sin\theta_{k}+n\sin\varphi_{k}\sin \theta_{k}) \tag{17}\]  
Let \(\boldsymbol{a}(\theta_{k},\varphi_{k})\) represent the direction vector corresponding to the \(k\)th source. The outputs of all array elements can be expressed in vector form, where the vector of received signals at the outputs of the antenna array is given by  
\[\mathbf{y}(t)=\mathbf{A}s_{k}(t)+\mathbf{N}(t) \tag{18}\]  
**A** defined as  
\[\mathbf{A}=\sum_{k=1}^{K}\sum_{m=0}^{M-1}\sum_{n=0}^{N-1}\boldsymbol{a}(\theta _{k},\varphi_{k}) \tag{19}\]  
The spatial correlation matrix of the array output is widely utilized for direction of arrival estimation, as it encapsulates the necessary information about the received signal. According to Equation (18), the spatial correlation matrix \(\mathbf{R}\) of the received signal, including noise, can be expressed as  
\[\mathbf{R} =\mathbb{E}\big{[}\mathbf{y}(t)\mathbf{y}^{H}(t)\big{]} \tag{20}\]  
where \(\mathbb{E}[\cdot]\) is the statistical expectation operator, the superscript H denotes the complex conjugate transpose operation, and \(R_{ij}\) is the element of the \(i\)th row of the \(j\)th column. \[ \begin{cases}\mathbf{y}(t)\mathbf{y}^{H}(t)=(\mathbf{A}s_{k}(t)+ \mathbf{N}(t))(\mathbf{A}s_{k}(t)+\mathbf{N}(t))^{H}\\ =\mathbf{A}s_{k}(t)s_{k}^{H}(t)\mathbf{A}^{H}+\mathbf{A}s_{k}(t)\mathbf{N}^{H }(t)+\mathbf{N}(t)s_{k}^{H}(t)\mathbf{A}^{H}+\mathbf{N}(t)\mathbf{N}^{H}(t)\\ \text{Let}\ \mathbb{E}\big{[}s_{k}(t)s_{k}^{H}(t)\big{]}=\mathbf{R}_{s}\\ \text{Then,}\ \mathbf{A}\mathbb{E}\big{[}s_{k}(t)s_{k}^{H}(t)\big{]}\mathbf{A}^{H}= \mathbf{A}\mathbf{R}_{s}\mathbf{A}^{H}\\ \text{Since}\ s_{k}(t)\ \mathbf{N}(t)\ \text{and}\ \mathbf{N}(t)\ \text{are uncorrelated, therefore:}\ \mathbb{E}\big{[}s_{k}(t)\mathbf{N}^{H}(t)\big{]}=0\\ \text{Similarly}\ \mathbb{E}\big{[}\mathbf{N}(t)s_{k}^{H}(t)\mathbf{A}^{H}\big{]}=0\\ \mathbb{E}\big{[}\mathbf{N}(t)\mathbf{N}^{H}(t)\big{]}=\sigma^{2}\mathbf{I} \end{cases}  \tag{21}\]After applying the expectation operator, Equation (20) can be rewritten as  
\[\mathbf{R}=\mathbf{A}\mathbf{R}_{s}\mathbf{A}^{H}+\sigma^{2}\mathbf{I} \tag{22}\]  
The task now focuses on estimating the azimuth and elevation angles (i.e., \(\phi\) and \(\theta\)) between the center of the receiving system and the signal source, using the correlation matrix \(\mathbf{R}\) of the receiving system as the available data. The correlation matrix can be unbiasedly estimated using the sample correlation matrix. \[\overline{\mathbf{R}}=\frac{1}{O-1}\sum_{k=1}^{N}\mathbf{y}(k)\mathbf{y}^{H}(k) \tag{23}\]  
where \(O\) denotes the number of independent observations, also referred to as the number of snapshots. When the number of matrix elements \(M\times N\) exceeds the number of signals \(K\), the rank of the signal component \(\mathbf{A}\mathbf{R}_{s}\mathbf{A}^{H}\) in the covariance matrix equals \(K\).