## 2 The Joint Positioning Approach  
### Decoherence-Based AOA Positioning Scheme  
The MUSIC (multi-signal classification) algorithm is a subspace-based high-resolution method for AOA estimation. MUSIC exploits the properties of the covariance matrix of the received signal vector to estimate the AOA of input signals impinging on a receiver array. The MUSIC algorithm is unable to directionally measure coherent signals, due to the effects of multipath transmission, etc., multiple correlated narrowband interferences are simultaneously incident on the array antenna from different directions, at which time the covariance matrix \(R_{xx}\) of the data received by the array antenna will suffer from rank loss and the boundaries between the noise subspace and the signal subspace will become fuzzy, with the existence of array steering vector \(a^{H}(\phi)\
eq O^{T}\), where \(\phi=\phi_{1},\phi_{2},\cdots,\phi_{p}\) represent the AOAs of p far-field interference signals and, therefore, the MUSIC algorithm is not able to directly measure the coherent narrowband interference [17]. Assuming that the array antenna receives p far-field signals at the same time as an M-element direction-finding array antenna in the far-field. The autocorrelation matrix \(R_{xx}\) of the array antenna's output data under ideal conditions is as follows:  
\[ \begin{split}& R_{xx}=E\Big{\{}x[k]x^{H}[k]\Big{\}}\\ &=\text{AE}\Big{\{}\text{s}[k]\text{s}^{\text{H}}[k]\Big{\}}A^{ \text{H}}+\sigma^{2}\text{I}\\ &=\text{APA}^{\text{H}}+\sigma^{2}\text{I}\end{split}  \tag{1}\]  
where \(A=\big{[}a(\phi_{1}),a(\phi_{2}),\ldots,a(\phi_{p})\big{]}\) is the array flow matrix of the array antenna, \(\sigma^{2}\) is the Korst noise power, and I is the unit matrix. In particular, \(x[k]\) is the array antenna's output data vector for the k-th interference signal, \(\text{s}[k]\) is the k-th interference signal vector, and P represents the covariance matrix of \(s[k]\). However, when two of the signals are correlated, the rank of the signal subspace will be reduced to P\(-1\). In order to solve the rank loss, we introduce a nonlinearly correlated array vector into it, and for a uniform linear array, the \"inverse array vector\" can be chosen as the vector to be introduced. Therefore, for an M-element uniform linear antenna array, the inverse covariance matrix \(R_{B}\) of \(R_{xx}\) is as follows:  
\[ \begin{split}& R_{B}=JR_{xx}^{*}J\\ &=A\Phi^{-(M-I)}P\Phi^{-(M-I)}A^{H}+\sigma^{2}I\end{split}  \tag{2}\]  
where \(\Phi\) is the diagonal matrix whose diagonal elements are \(e^{im\phi}(m=1,2,\dots,M)\), and \(J\) is the substitution matrix of \(\mathrm{M}\times\mathrm{M}\). Averaging over \(R_{xx}\) and \(R_{B}\), the positive and negative array covariance matrices \(R_{FB}\) can be obtained as follows:  
\[ \begin{split}& R_{FB}=\frac{1}{2}(R_{xx}+R_{B})\\ &=\frac{1}{2}(R_{xx}+JR_{xx}^{*}J)\\ &=A\tilde{P}A^{H}+\sigma^{2}I\end{split}  \tag{3}\]  
Perform a feature decomposition of \(R_{FB}\) as follows:  
\[R_{FB}=U\times\sum\times U^{H} \tag{4}\]  
where \(\Sigma\) is the diagonal matrix consisting of M eigenvalues of \(R_{FB}\) and \(\mathrm{U}\) is the feature matrix of \(R_{FB}\). A is a column-full rank matrix, so that \(\mathrm{rank}(A\tilde{P}A)=\mathrm{rank}(\tilde{P})=\tilde{P}\), if \(\tilde{P}<\mathrm{M}\), we have the following:  
\[ \begin{split}&\mathrm{U}\times A\tilde{P}A^{H}\times\mathrm{U}^{H}= \sum\\ &=\mathrm{diag}\{a_{1}^{2},\dots,a_{\tilde{P}}^{2},0,\dots,0\} \end{split}  \tag{5}\]  
where \(a_{1}^{2}\) to \(a_{\tilde{P}}^{2}\) are the eigenvalues of the autocorrelation matrix \(A\tilde{P}A\). Multiply both sides of \(R_{FB}\) simultaneously by \(\mathrm{U}\), we can obtain the eigenvalues \(\lambda_{i}\) of \(R_{FB}\) as [18]:  
\[\mathrm{U}^{H}\mathrm{R}_{FB}\mathrm{U}=\mathrm{diag}\{a_{1}^{2},\dots,a_{ \tilde{P}}^{2},0,\dots,0\}+\sigma^{2}\mathrm{I} \tag{6}\]  
\[\lambda_{i}=\begin{cases}a_{i}  
Multiply G to the right of \(R_{FB}\), we can obtain the following:  
\[