## 2 The MUSIC Algorithm of 2-D UCA  
Consider a UCA with \(M\) elements impinged by \(D\) narrowband signals \(s_{i}(t)\), \(i=1\), \(2\),..., \(D\) (\(D<M\)), where \(t\) is the time variable, as shown in Figure 1. The \(D\) sources are assumed to be from far-field with azimuth \(\theta_{i}\) and elevation \(\varphi_{i}\), \(i=1\), \(2\),..., \(D\). Assume the radius of UCA is \(r\) and the noise is additive white Gaussian noise (AWGN). Consider the reference point is 0, the ideal steering matrix can be obtained from the array geometry as  
\[\mathcal{A}(\theta,\varphi)=[\mathbf{a}(\theta_{1},\varphi_{1}),\mathbf{a}(\theta_{2}, \varphi_{2}),\cdots,\mathbf{a}(\theta_{D},\varphi_{D})] \tag{1}\]  
\[\mathbf{a}(\theta_{i},\varphi_{i})=[a_{1}(\theta_{i},\varphi_{i}),a_{2}(\theta_{i},\varphi_{i}),\cdots,a_{M}(\theta_{i},\varphi_{i})]^{\mathrm{T}} \tag{2}\]  
\[a_{k}(\theta_{i},\varphi_{k})=\exp(-\mathrm{j}\omega\tau_{i})=\exp(-\mathrm{ j}dr\cos\big{(}2\pi(k-1)/\,M-\theta_{i}\big{)}\cos\varphi_{i}) \tag{3}\]  
where \(k=1\), \(2\),..., \(M\); \(i=1\), \(2\),..., \(D\); \(\lambda\) is wavelength; \(d=2\pi/\,\lambda\). The array output of the _k_th element at time \(t\) can be written as  
\[x_{k}(t)=\sum_{i=1}^{D}a_{k}(\theta_{i},\varphi_{i})s_{i}(t)+n_{k}(t) \tag{4}\]  
Equation (4) can be written in matrix form as  
\[\mathbf{X}(t)=\mathbf{AS}(t)+\mathbf{N}(t) \tag{5}\]  
where \(\mathbf{X}(t)\) is \(M\times 1\) array output vector; \(\mathbf{A}\) is \(M\times D\) array steering matrix; \(\mathbf{S}(t)\) is \(D\times 1\) signal vector; \(\mathbf{N}(t)\) is \(M\times 1\) noise vector. \(\mathbf{X}(t)\), \(\mathbf{S}(t)\) and \(\mathbf{N}(t)\) are abbreviated as \(\mathbf{X}\), \(\mathbf{S}\) and \(\mathbf{N}\), respectively, and the array covariance matrix of \(\mathbf{X}\) can be written as  
\[\mathbf{R}=\mathrm{E}[\mathbf{XX}^{\mathrm{H}}]=\mathbf{APA}^{\mathrm{H}}+\sigma^{2}\mathbf{I} \tag{6}\]  
Figure 1: Uniform circular array diagram. where \(\mathbf{P}=\mathrm{E}[\mathbf{SS}^{\mathrm{H}}]=\mathrm{diag}[P_{1},\,P_{2},\,...,\,P_{i},\,...,\,P_{D}]\); \(P_{i}\) is the power of the \(i\)th signal; \(\sigma^{2}\) is noise power; \(\mathbf{I}\) is \(M\times 1\) identity matrix. \(\mathrm{E}[\bullet]\,\mathrm{and}\,[\bullet]^{\mathrm{H}}\) denote the statistical expectation and the Hermitian transpose, respectively. In real systems, the covariance matrix \(\mathbf{R}\) can be estimated from a finite set of sample snapshots as  
\[\hat{\mathbf{R}}= \frac{1}{L}\sum_{n=1}^{L}\mathbf{X}(n)\mathbf{X}^{\mathrm{H}}(n) \tag{7}\]  
where \(L\) is the total number of snapshots. Then the eigendecomposition of \(\hat{\mathbf{R}}\) can be written as  
\[\hat{\mathbf{R}}=\mathbf{U}\,\mathbf{\sum}\,\mathbf{U}^{\mathrm{H}}=\sum_{i=1}^{M}\lambda_{i} \mathbf{v}_{i}\mathbf{v}_{i}^{\mathrm{H}} \tag{8}\]  
where \(\mathbf{\sum}=\mathrm{diag}\{\lambda_{2}\geq\lambda_{2},\cdots,\lambda_{M}\}\) ; \(\lambda_{i}\) and \(\mathbf{v}_{i}\) are the eigenvalue and corresponding eigenvector of \(\hat{\mathbf{R}}\), respectively. In ideal conditions, \(\lambda_{1}\geq\lambda_{2}\cdots\geq\lambda_{D}\geq\lambda_{D+1}=\cdots= \lambda_{M}=\sigma^{2}\). Assume that the number of incident signals \(D\) is known, \(\hat{\mathbf{R}}\) can be described as  
\[\mathcal{A}(\theta,\varphi)=[\mathbf{a}(\theta_{1},\varphi_{1}),\mathbf{a}(\theta_{2}, \varphi_{2}),\cdots,\mathbf{a}(\theta_{D},\varphi_{D})] \tag{1}\]0  
where \(\mathbf{U}_{\mathrm{S}}=[\mathbf{v}_{1},\mathbf{v}_{2},\cdots,\mathbf{v}_{D}]\) and \(\mathbf{U}_{\mathrm{N}}=[\mathbf{v}_{D+1},\mathbf{v}_{D+2},\cdots,\mathbf{v}_{M}]\) are the signal subspace and noise subspace, respectively; \(\mathbf{\sum}_{\mathrm{S}}=\mathrm{diag}\{\lambda_{1},\cdots,\lambda_{D}\}\) and \(\mathbf{\sum}_{\mathrm{N}}=\mathrm{diag}\{\lambda_{D+1},\cdots,\lambda_{M}\}\) are diagonal matrices related to the signal and noise power, respectively.