## 2 The Joint Positioning Approach
### Joint TDOA Positioning Program
The intersection of two or more hyperbolas generated from three or more TDOA measurements enables estimation of the two-dimensional position of the transmitter. According to the TDOA estimation, the interference position can be calculated by solving a set of nonlinear requirements. In matrix-vector form, the system of position estimation equations can be expressed as follows:  
\[\mathbf{\hat{d}}=f(\theta)+e \tag{23}\]  
where \(\mathbf{\hat{d}}\) is the vector of noise measurements, \(\mathrm{f}(\theta)\) is the vector of position equations derived on \(\mathbf{\theta}=\{\mathbf{x},\mathbf{y}\}\), and \(\mathbf{e}\) is the measurement noise. TDOA position estimation equations are inherently nonlinear and need to be solved iteratively. One way to solve the position equation is to minimize the least squares cost function, which can be defined as follows:  
\[\varepsilon=\sqrt{(\hat{d}-f(\theta))^{T}C^{-1}(\hat{d}-f(\theta))}\hat{\theta} =\mathrm{argmin}(\varepsilon) \tag{24}\]  
where \(\theta\) is the estimate of the parameter that minimizes the cost function and C is the covariance matrix. Finally, the final interference source localization is obtained by averaging the decoherence AOA localization results and the TDOA localization results as follows:  
\[(X,Y)=(\frac{x_{\mathrm{AOA}}+x_{\mathrm{TDOA}}}{2},\frac{y_{\mathrm{AOA}}+y _{\mathrm{TDOA}}}{2}) \tag{25}\]  
The estimation accuracy of TDOA is constrained by the bandwidth B of the interference signal to be positioned and the signal-to-noise ratio (SNR) [20]. And the Cramer-Rao lower bound (CRLB) can be approximately expressed as follows:  
\[\sigma_{\mathrm{TDOA}}^{2}\geq\frac{c^{2}}{8\pi^{2}B^{2}\cdot\mathrm{SNR}} \tag{26}\]  
Equation (26) shows that the TDOA positioning error becomes more pronounced in narrowband interference scenarios. Therefore, a joint decoherence AOA and TDOA localization algorithm is proposed. And the Cramer-Rao bound (CRB) for AOA estimation is given by the following [21]:  
\[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)=\sum_{k=0}^{N-\tau-1}x_{1}(k+\tau)\cdot x _{2}^{*}(k) \tag{18}\]0  
where d is the inter-element spacing and \(\lambda\) is the wavelength of the positioning interference signal. Inspired by [22; 23], the joint AOA and TDOA localization algorithm estimation accuracy can be approximately expressed as follows:  
\[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)=\sum_{k=0}^{N-\tau-1}x_{1}(k+\tau)\cdot x _{2}^{*}(k) \tag{18}\]1  
Equation (28) shows that the joint localization algorithm is better than the single localization algorithm with higher accuracy and better robustness. When \(\sigma_{\mathrm{TDOA}}^{2}\approx\sigma_{AOA}^{2}\), the joint positioning accuracy improves by approximately \(\sqrt{2}\) times. In practical applications, AOA and TDOA measurement errors may exhibit correlation due to shared base station clock synchronization errors or overlapping multipath effect regions, requiring correction of the positioning model through a covariance matrix, which refers to 3 GPP TR 38.855. When accounting for error correlation, the joint positioning error should be modified as follows:  
\[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)=\sum_{k=0}^{N-\tau-1}x_{1}(k+\tau)\cdot x _{2}^{*}(k) \tag{18}\]2  
where \(\rho\) denotes the error correlation coefficient and is defined as the ratio of the covariance of AOA and TDOA errors to the product of their respective standard deviations:  
\[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)=\sum_{k=0}^{N-\tau-1}x_{1}(k+\tau)\cdot x _{2}^{*}(k) \tag{18}\]3  
where N is the number of base stations, \(\Delta\theta_{i}\) denotes the AOA error of the \(i\)-th base station and \(\Delta\tau_{j}\) denotes the TDOA error of the \(j\)-th pair base station.