# DNCL: Hybrid DOA Estimation and NMDS Cooperative Multi-Target Localization for RFID
## 4 System Design
### Virtual Antenna Array Direction of Arrival Estimation  
The signal DOA was calculated by the phase difference between the reception of signals from the multiple antennas, and according to Equation (1), it is known that the phase of the signal rotates \(2\pi\) with each wavelength distance of the signal propagation. That is, there is an interval of multiple phase periods between the measured phase and the true phase. Specifically, as shown in Figure 4, the signal source is located at \(T\), and the respective distances of the tag and the two positions are \(d_{1}\) and \(d_{2}\). \(\varphi_{1}\) and \(\varphi_{2}\) are the received signal phases that antenna measured at two different locations, \(x_{1}\) and \(x_{2}\), respectively. The signal phase can be described as:  
\[\left\{ \begin{array}{l}\varphi_{1}+2k_{1}\pi=\frac{4\pi}{\lambda}d_{1}+ \varphi_{ini}+e_{1}\\ \varphi_{2}+2k_{2}\pi=\frac{4\pi}{\lambda}d_{2}+\varphi_{ini}+e_{2}\end{array} \right. \tag{12}\]  
Since the different hardware characteristics of different antennas can introduce different phase errors in the phase measurement, it is hard to determine \(\varphi_{A}\). Therefore, we chose to simulate an antenna array inspired by SAR techniques using a mobile antenna, and the measured value of the first phase was used as the calibration reference. In this way, the initial phase \(\varphi_{ini}\) could be considered as a constant independent of the distance and can be eliminated in the calculation of the phase difference. Figure 4: The direction of arrival of the tag with mobile scanning. Figure 3: Flow chart of hybrid DOA estimation and NMDS cooperative multi-target localization system. Therefore, the phase difference of the antenna at the two locations is \(\Delta\varphi_{1,2}=\varphi_{1}-\varphi_{2}\), and the distance difference is \(\Delta d_{1,2}=d_{1}-d_{2}\), and thus the Equation (12) can be transformed into  
\[\Delta\varphi_{1,2}+2k\pi=\frac{4\pi}{\lambda}\Delta d_{1,2} \tag{13}\]  
The antenna sampling point spacing cannot be arbitrarily selected. If the spacing is too large, the phase delay at adjacent sampling points will exceed \(2\pi\). At this time, the array direction vector cannot identify the specific phase delay in the value, and the problem of phase periodic ambiguity will occur. Therefore, the spacing of adjacent sampling points should not exceed \(\lambda/2\). In RFID-based localization, since the distance between reader antenna and the tag is a round-trip distance, the distance between adjacent sampling points should meet \(\Delta x<\lambda/4\). Due to the complexity of the indoor signal propagation environment, there are a large number of coherent multipath signals in the environment, which can lead to the rank deficiency of the signal covariance matrix, so that the signal eigenvectors diverge into the noise subspace. With the improved MUSIC algorithms, the rank of the signal covariance matrix can be restored. Let \(\mathbf{I}_{V}\) be the \(M\times M\) inverse unitary matrix, which can be expressed as  
\[\mathbf{I}_{V}=\left[ \begin{array}{cccc}0&0&\cdots&1\\ 0&\cdots&1&0\\ \vdots&\therefore&\vdots&\vdots\\ 1&0&\cdots&0\end{array} \right]_{M\times M} \tag{14}\]  
In combination with \(\mathbf{I}_{V}\), the correlation matrix is transformed into  
\[\mathbf{R}_{x}=\mathbf{R}+\mathbf{I}_{V}\mathbf{R}^{*}\mathbf{I}_{V} \tag{15}\]  
where \(\mathbf{R}^{*}\) is the conjugate of \(\mathbf{R}\), thus \(\mathbf{R}_{x}\) is a Toeplitz matrix of Hermite. In general, \(\mathbf{R}\) is just a Hermite matrix, not a Toeplitz matrix. Using the Toeplitz property, the estimated value \(\mathbf{R}_{x}\) of the Toeplitz covariance matrix was obtained by modifying \(\mathbf{R}\). Therefore, \(\mathbf{R}_{x}\) is an unbiased estimation of \(\mathbf{R}\). Then, the decomposition of \(\mathbf{R}_{x}\) was performed, and the noise eigenvectors produced during processing were replaced into the MUSIC algorithm, allowing the signal DOA to be accurately approximated.