# DNCL: Hybrid DOA Estimation and NMDS Cooperative Multi-Target Localization for RFID
## 2 Background
### Direction of Arrival  
When the model has \(N\) narrowband far-field signals incident on a spatial array with \(M\) array element numbers, the signal received by the \(m\)th antenna can be written as follows  
\[\begin{split} x_{m}(t)&=\sum_{i=1}^{N}g_{mi}s_{i}(t- \tau_{mi})+n_{m}(t)\\ &=\sum_{i=1}^{N}g_{mi}s_{i}(t)e^{-j\omega_{0}\tau_{mi}}+n_{m}(t), \ m=1,2,\ldots,M\end{split} \tag{2}\]  
where \(g_{mi}\) is the gain of the \(m\)th array element to the \(i\)th source signal, \(n_{m}(t)\) is the noise of the \(m\)th array element at moment \(t\), and \(\tau_{mi}\) is the relative time delay of the i-th source signal arriving at the m-th array element.  
As illustrated in Figure 2, a signal was received by the antennas in the direction of \(\theta_{p}\). The distance between adjacent antennas is denoted by the symbol \(d\). The difference in signal wave range between nearby antennas is \(dsin(\theta_{p})\). As a result, the phase difference of \(e^{-j2\pi\frac{d}{\lambda}\sin(\theta_{p})}\) was introduced for adjacent array elements. When described by a matrix, the array signal model could be simply represented as  
\[\mathbf{X}(t)=\mathbf{A}\mathbf{S}(t)+\mathbf{N}(t) \tag{3}\]  
where \(\mathbf{N}(t)\) is the noise vector, the antenna array receives signal vector \(\mathbf{X}(t)=\left[x_{1}(t),x_{2}(t),\ldots,x_{M}(t)\right]^{T}\) is an \(M\times 1\) snapshot data vector, \(\mathbf{A}=\left[\mathbf{a}(\theta_{1}),\mathbf{a}(\theta_{2}),\ldots, \mathbf{a}(\theta_{N})\right]\) is the orientation matrix of the \(M\times N\) dimensional spatial array, where the column vector \(\mathbf{a}\) is also an \(M\times 1\) dimensional direction vector  
\[\mathbf{a}(\theta_{i})=\left[1,e^{-j2\pi\frac{d}{\lambda}\sin(\theta_{i})}, \ldots,e^{-j2\pi\frac{d}{\lambda}(M-1)\sin(\theta_{i})}\right]^{T} \tag{4}\]  
where \(i=1,2,\ldots,N\) and \((\cdot)^{T}\) denotes the transpose operation.  
The super-resolution DOA algorithm is based on eigenstructure analysis of the correlation matrix of the array received data vector. Based on Equation (3), the correlation matrix is expressed as  
\[\mathbf{R}_{X}=\mathbf{A}\mathbf{R}_{S}\mathbf{A}^{H}+\sigma^{2}\mathbf{I} \tag{5}\]where \((\cdot)^{H}\) denotes the Hermitian transpose of the matrix, \(\mathbf{R}_{S}=E\Big{[}\mathbf{S}\mathbf{S}^{H}\Big{]}\) is the correlation matrix of the source signal vector. Based on the eigenvalue decomposition theorem for matrices, the eigenvalue decomposition of \(\mathbf{R}_{X}\) is performed as follows  
\[\mathbf{R}_{X}=\mathbf{U}_{S}\mathbf{\Sigma}_{S}\mathbf{U}_{S}^{H}+\mathbf{U}_{ N}\mathbf{\Sigma}_{N}\mathbf{U}_{N}^{H} \tag{6}\]  
where the matrices \(\mathbf{\Sigma}_{S}\) and \(\mathbf{\Sigma}_{N}\) are the diagonal matrix consisting of the largest N eigenvalues of \(\mathbf{R}_{X}\) and the diagonal matrix consisting of the remaining eigenvalues, respectively.  
\(\mathbf{R}_{X}\) has \(M\) eigenvalues. The eigenvector \(\mathbf{U}_{S}\) corresponding to the largest \(N\) eigenvalues constitutes the signal subspace, and the eigenvector \(\mathbf{U}_{N}\) corresponding to the other \(M-N\) eigenvalues constitute the noise subspace. The signal subspace and noise subspace are orthogonal, so the spatial spectral function enable to be constructed as  
\[P_{MUSIC}(\theta)=\frac{1}{\mathbf{a}^{H}(\theta)\mathbf{U}_{N}\mathbf{U}_{N} ^{H}\mathbf{a}(\theta)} \tag{7}\]  
then the sharp peak of the function represents the DOA of the incident signal.