## 2 Mathematical Modeling  
Suppose signals of \(L\) narrowband uncorrelated targets are impinging on CSA consisting of two sub ULAs. Sub-array 1 contains \(M\) antenna elements having \(Nd\) space among each antenna element as shown in Figure 1a while sub-array 2 has \(N\) antenna elements \(Md\) apart from each other as shown in Figure 1b. \(M\) and \(N\) are coprime integers and \(d\) is equal to the half of the wave length \((\lambda/2)\). By combining these two sub arrays, we get a generalized coprime antenna array having \(M+N-1\) elements, as seen in Figure 1c. The first antenna element is shared with both sub-ULAs and physical location of each antenna element of CSA is enclosed in Equation (1).  
\[I=\{0,Nd,Md,2 Nd,2 Md,...,(N-1)Md,(M-1)Nd\} \tag{1}\]  
Direction of arrival of \(L\) uncorrelated signals is summarized in vector \(\theta\)  
\[\theta=[\theta_{1},\theta_{2},\theta_{3},...,\theta_{L}] \tag{2}\]  
Hence, the received signal of L uncorrelated targets impinging on coprime array is given by  
\[x(t)=\sum_{l=1}^{L}a_{l}(\theta)s_{l}(t)+n(t)=A(\theta)s(t)+n(t) \tag{3}\]  
where  
\[A(\theta)=[a_{1}(\theta),a_{2}(\theta),...,a_{L}(\theta)] \tag{4}\]  
\[a_{l}(\theta)=[1,e^{\pi N\sin\theta_{l}},e^{\pi M\sin\theta_{l}},e^{\pi 2N \sin\theta_{l}},e^{\pi 2M\sin\theta_{l}},...,e^{\pi(N-1)Md\sin\theta_{l}},e^{ \pi(M-1)Nd\sin\theta_{l}}] \tag{5}\]  
\[s_{l}(t)=[s_{1}(t),s_{2}(t),...,s_{L}(t)]^{T} \tag{6}\]  
and \(n(t)\) is the iid and complex additive white Gaussian noise (AWGN) with variance \(\sigma I^{2}\) and zero mean.  
\[n(t)=[n_{1}(t),n_{2}(t),...n_{M+N-1}(t)] \tag{7}\]  
Based on this coprime array structure, we can be performed the computation of covariance matrix by taking expectation of the received signal as presented in following equation  
\[Rxx=E[x(t)x^{H}(t)] \tag{8}\]  
By solving this equation we will get  
\[Rxx=ARssA^{H}+\sigma^{2}I \tag{9}\]  
\(Rss\) is the amplitude covariance matrix of received signal as shown in following Eq (9)  
\[Rss=E[s(t)s^{H}(t)] \tag{10}\]  
Now virtual uniform linear array can be obtained by performing vectorization on covariance matrix \((Rxx)\) as expressed in the following equation  
\[z=vec(Rxx)=vec(ARssA^{H}+\sigma^{2}I)=D\sigma^{2}+\sigma^{2}I \tag{11}\]  
where \(D\) is the Kronecker product and the elements in D are expressed in the form of following equation  
\[e^{jK(I_{l}-I_{l})\sin\theta_{l}}\xrightarrow{where}\ i,j=0,1,2,...,M+N-1. \tag{12}\]In the process of vectorization, all columns of correlation matrix are placed one by one under the first column. After performing vectorization, the correlation matrix become a column vector \(z^{(M+N-1)^{2}\times 1}\). This column vector \(z\) is sorted out in accordance with exponent term of the received signal. After sorting, redundant terms of this sorted vector are being skipped. In [33], this can be assessed as continuous virtual uniform linear array (vULA)  
\[I_{c}=\{(Mnd-Nmd)\cup(Nmd-Mnd)\},\text{ \emph{where }}0<m<M-1\text{ \emph{and }}0<n<N-1 \tag{13}\]  
Locations of this vULA elements can be expressed as  
\[I_{p}=\{(I_{i}-I_{j})\text{ \emph{i}},j=0,1,2,...,M+N-1\} \tag{14}\]  
We concentrate on the continuous part of vULA of CSA for application of our proposed algorithms. Due to this continuous part of the vULA, we get a higher degree of freedom and this DOF is directly proportional to the length of continuous vULA.