## 2 Array Signal Model  
The AOA estimation setup can be explained with reference to Figure 1. Suppose D is the number of incoming signals, \(\mathbf{S}(t)\), arriving from different directions and received by M sensors. The time sample of each component, \(\mathbf{X}(t)\), of the received signal contains AWGN, \(\mathbf{N}(t)\), and can be defined by:  
\[\mathbf{X}(t)\ =\mathbf{A}(\theta)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{1}\]  
where \(\mathbf{A}(\theta)=[\ \mathbf{a}(\theta_{1}\ )\ \mathbf{a}(\theta_{2}\ )\ \dots\ \mathbf{a}(\theta_{D}\ )]\) is the arriving signal steering matrix for \(D\) signals, \(k=1,2,\dots\), \(D\). \(\mathbf{S}(t)=[\mathbf{s_{1}}(t),\ \mathbf{s_{2}}(t),\ \dots,\ \mathbf{s_{D}}(t)]^{\mathrm{T}}\) is the (\(D\times N\)) incident signals, \(\mathbf{N}(t)=[\mathbf{n_{1}}(t),\ \mathbf{n_{2}}(t),\ \dots,\ \mathbf{n_{M}}(t)]\) is an array of AWGN for each channel, \(\mathbf{a}(\theta_{k})=\left[\ 1\ \ e^{j\beta\delta sin\theta_{k}}\ \ \dots\ \ \ e^{j\beta d(M-1)sin\theta_{k}}\ \ \right]\) is the steering vector of a linear array, \(d\) is the separation between adjacent elements and \(\beta=\frac{2\pi}{\lambda}\) is the spatial frequency. A circular array is a common geometry which can be used to increase the gain. Also, it provides beam steering in three dimensions that is not possible with a linear array. A uniform circular array of M elements in the x-y plane is assumed in this work. With the circular array, both elevation angle (\(\theta\)) and azimuth angle (\(\phi\)) can be estimated: the total received signal is:  
\[\mathbf{X}(t)=\mathbf{A}(\theta,\ \ \phi)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{2}\]  
where \(\mathbf{A}(\theta,\ \ \phi)\) is the \(M\times D\) matrix of steering vectors:  
\[\mathbf{A}(\theta,\ \ \phi)=\ \left[\ \mathbf{a}(\theta_{1},\ \ \phi_{1})\ \ \ \mathbf{a}(\theta_{2},\ \ \phi_{2})\ \ \dots\ \ \ \mathbf{a}(\theta_{D},\ \ \phi_{D})\ \right] \tag{3}\]Consider there are \(k\) sources incident on the sensor array as shown in Figure 1, then the unit vectors that include the direction of \(\theta_{k}\) and \(\phi_{k}\) can be defined as follows:  
\[\mathbf{u}_{k}=\cos\phi_{k}\sin\theta_{k}\hat{a}_{x}+\sin\phi_{k}\sin\theta_{k}\hat{ a}_{y}+\cos\theta_{k}\hat{a}_{z} \tag{4}\]  
where \(\hat{a}_{x},\ \hat{a}_{y}\\) and \(\hat{a}_{z}\) are the unit vectors for Cartesian co-ordinates. With a ring array located on the x-y plane, where \(r\) is the array radius and phase angle (\(\varphi_{i}\)), the unit vector from the original reference point to the \(i\)th element is given as follows:  
\[\mathbf{v}_{i}=r_{i}\cos\varphi_{i}\hat{a}_{x}+r_{i}\sin\varphi_{i}\hat{a}_{y},\ i=\ 1,\,2,\ \ldots,\ M. \tag{5}\]  
\[\mathbf{\varphi}_{i}=\frac{2\pi}{M}(i-1) \tag{6}\]  
The angle (\(\gamma_{ik}\)) can be obtained from the dot product between unit vectors \(\mathbf{v}_{i}\) and \(\mathbf{u}_{k}\) as shown below:  
\[\gamma_{ik}=\cos^{-1}\biggl{(}\frac{\mathbf{v}_{i}.\mathbf{u}_{k}}{\parallel\mathbf{v}_{i }\parallel\cdot\parallel\mathbf{u}_{k}\parallel}\biggr{)}\]  
\[\gamma_{ik}=\cos^{-1}\biggl{(}\frac{(\cos\varphi_{i}\hat{a}_{x}+\sin\varphi_{ i}\hat{a}_{y})\cdot\bigl{(}\cos\phi_{k}\sin\theta_{k}\hat{a}_{x}+\sin\phi_{k} \sin\theta_{k}\hat{a}_{y}\bigr{)}}{\parallel\mathbf{v}_{i}\parallel\cdot\parallel \mathbf{u}_{k}\parallel}\biggr{)}\]  
\[\gamma_{ik}=\cos^{-1}\biggl{(}\frac{\cos\varphi_{i}\cos\phi_{k}\sin\theta_{k}+ \sin\varphi_{i}\sin\phi_{k}\sin\theta_{k}}{\parallel\mathbf{v}_{i}\parallel\cdot \parallel\mathbf{u}_{k}\parallel}\biggr{)}\]  
\[\mathbf{X}(t)\ =\mathbf{A}(\theta)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{1}\]0  
\[\mathbf{X}(t)\ =\mathbf{A}(\theta)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{1}\]1  
\[\mathbf{X}(t)\ =\mathbf{A}(\theta)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{1}\]2 M-Arbitrary element array with D arriving signals. \(\mathbf{\varphi}\) is vector with dimension (\(1\times M\)) as given below:  
\[\mathbf{X}(t)\ =\mathbf{A}(\theta)\ \mathbf{S}(t)+\mathbf{N}(t) \tag{1}\]3  
while \(\mathbf{\gamma}\) is the matrix with dimension (M\(\times\)D) given as follows:  
\[\mathbf{\gamma}=\left[