## 2 Signal Model  
Consider a UCA with radius \(r\) and \(M(M>K)\) isotropic sensors uniformly distributed over the circumference in the \(xy\)-plane, the geometry configuration of which is depicted in Figure 1. Assume that there are \(K\) co-frequency far-field narrowband signals impinging on the UCA from distinct angles \((\theta_{1},\varphi_{1}),(\theta_{2},\varphi_{2}),\cdots,(\theta_{K},\varphi_{K})\), where \(\theta_{k}\in[0,\pi/2)\) and \(\varphi_{k}\in[-\pi,\pi)\) are elevation and azimuth angle of the \(k\)-th signal, respectively. The elevation angle is measured downward from the \(z\)-axis and the azimuth angle is measured counterclockwise from the \(x\)-axis. The \(M\times 1\) array output vector observed at the \(t\)th snapshot is modeled as:  
\[\mathbf{x}(t)=\mathbf{A}(\mathbf{\theta},\mathbf{\varphi})\mathbf{s}(t)+\mathbf{n}(t),\quad t=1,2,\cdots,N \tag{1}\]  
where \(\mathbf{\theta}=[\theta_{1},\theta_{2},\cdots,\theta_{K}]^{T}\), \(\mathbf{\varphi}=[\varphi_{1},\varphi_{2},\cdots,\varphi_{K}]^{T}\), \(\mathbf{A}(\mathbf{\theta},\mathbf{\varphi})=[\mathbf{a}(\theta_{1},\varphi_{1}),\mathbf{a}( \theta_{2},\varphi_{2}),\cdots,\)\(\mathbf{a}(\theta_{K},\varphi_{K})]\) is the array manifold matrix, \(\mathbf{a}(\theta_{k},\varphi_{k})\) is the steering vector of the \(k\)th signal and \(\{\mathbf{a}(\theta_{k},\varphi_{k})\}_{k=1}^{K}\) is assumed to be linearly independent, i.e, \(\text{rank}(\mathbf{A}(\mathbf{\theta},\mathbf{\varphi}))=K\), \(\mathbf{s}(t)=[s_{1}(t),s_{2}(t),\cdots,s_{K}(t)]^{T}\) is the source signal vector, \(\mathbf{n}(t)\) is the additive white Gaussian noise vector with zero-mean and assumed to be both spatially and temporally uncorrelated with the source signals, i.e., \(\text{E}\{\mathbf{n}(t)\mathbf{n}^{H}(t)\}=\sigma_{\varphi}^{2}\mathbf{I}\), and \(N\) denotes the number of snapshots. According to geometry configuration of the UCA, \(\mathbf{a}(\theta_{k},\varphi_{k})\) is given by:  
\[\mathbf{a}(\theta_{k},\varphi_{k})=[a_{1}(\theta_{k},\varphi_{k}),a_{2}(\theta_{k },\varphi_{k}),\cdots,a_{M}(\theta_{k},\varphi_{k})]^{T} \tag{2}\]  
where \(a_{m}(\theta_{k},\varphi_{k})=e^{i(2\pi r/\lambda)\sin\theta_{k}\cos(\varphi_ {k}-\beta_{m})}\), \(\beta_{m}=2\pi(m-1)/M\) and \(\lambda\) denotes the wavelength. We assume that the waveform of \(s_{k}(t)\) is known, while its complex amplitude is unknown. Thus, \(s_{k}(t)\) can be expressed as:  
\[s_{k}(t)=\gamma_{k}y_{k}(t),\quad k=1,2,\cdots,K \tag{3}\]  
where \(y_{k}(t)\) denotes the synchronous known waveform and \(\gamma_{k}\) denotes the unknown complex amplitude due to propagation through different paths. Meanwhile, the number of signals \(K\) is known, since we even know their waveforms. Then, \(\mathbf{x}(t)\) becomes:  
\[\mathbf{x}(t)=\mathbf{A}\mathbf{s}(t)+\mathbf{n}(t)=\mathbf{A}\mathbf{\Gamma}\mathbf{y}(t)+\mathbf{n}(t)=\mathbf{ B}\mathbf{y}(t)+\mathbf{n}(t) \tag{4}\]  
where \(\mathbf{s}(t)=\mathbf{\Gamma}\mathbf{y}(t)\), \(\mathbf{\Gamma}=\text{diag}\{\gamma\}\) denotes the unknown amplitude matrix, \(\mathbf{\gamma}=[\gamma_{1},\gamma_{2},\cdots,\gamma_{K}]^{T}\), \(\mathbf{y}(t)=[y_{1}(t),y_{2}(t),\cdots,y_{K}(t)]^{T}\) denotes the known waveform vector, \(\mathbf{B}=\mathbf{A}\mathbf{\Gamma}\) denotes the spatial signature matrix and the argument \((\mathbf{\theta},\mathbf{\varphi})\) is dropped for convenience here and in the following. Define cross-correlation matrix and auto-correlation matrix of the known waveforms and array output vectors as:  
\[ \begin{cases}\mathbf{R}_{xy}\triangleq\text{E}\Big{\{}\mathbf{x}(t)\mathbf{y}^{H}(t) \Big{\}}=\lim_{N\to\infty}\frac{1}{N}\sum_{t=1}^{N}\mathbf{x}(t)\mathbf{y}^{H}(t)\\ \mathbf{R}_{yy}\triangleq\text{E}\Big{\{}\mathbf{y}(t)\mathbf{y}^{H}(t) \Big{\}}=\lim_{N\to\infty}\frac{1}{N}\sum_{t=1}^{N}\mathbf{y}(t)\mathbf{y}^{H}(t)\end{cases}