## 2 The Joint Positioning Approach
### Joint TDOA Positioning Program  
The location of the interference source can also be estimated by the difference \(r_{1}-r_{2}=\Delta\tau_{12}c\) in the signal propagation time measured at the spatially displaced receiver. A common method is to calculate the cross-correlation function between the signals received at each receiver to determine the time difference between them. The inter-correlation function can be expressed as follows:  
\[\mathbb{R}_{x_{1}x_{2}}(\tau)=\mathbb{E}[x_{1}(\mathrm{t}+\tau)\cdot x_{2}^{*}( \mathrm{t})] \tag{17}\]  
where \(\mathbb{E}[\ ]\) is the expected value operator and \(()^{*}\) is the complex conjugate. However, due to the limited observation time, \(R_{x_{1}x_{2}}(\tau)\) can only be estimated. The estimation of traversing discrete samples without normalization can be expressed as follows:  
\[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)=\sum_{k=0}^{N-\tau-1}x_{1}(k+\tau)\cdot x _{2}^{*}(k) \tag{18}\]  
where N denotes the total number of digital samples in the original observation interval T; hence, \(\mathrm{N}=\mathrm{T}\cdot f_{s}\), where \(f_{s}\) is the sampling frequency. The value \(\tau\) that maximizes the cross-correlation function is the TDOA estimate:  
\[\hat{\tau}_{d}=\mathrm{argmax}[\hat{\mathbb{R}}_{x_{1}x_{2}}(\tau)] \tag{19}\]  
Using discrete-time cross-correlation, the time delay estimate \(\hat{\tau}_{d}\) will be an integer value because the delay is computed at an integer delay determined by the sampling rate of the received signal. If the sampling rate is \(f_{s}\), the inter-correlation time resolution is \(1/f_{s}\), but depending on the sampling rate, the inter-correlation peaks may not fall on the actual discrete samples and, therefore, the discretization of the inter-correlation function must be interpolated to determine the time delay more accurately. Thus, the time delay can be expressed in terms of two components:  
\[\tau_{0}=p+\delta \tag{20}\]  
where p is the integer delay associated with the sampling rate, and \(|\delta|\leq 0.5T_{s}\) is the fractional part of the delay. Parabolic interpolation is a simple interpolation method widely used to improve the accuracy of fitting the optimal peak of the cross-correlation function. The fractional part \(\delta\) of the delay estimate can be found by fitting a parabola to three samples of \(\widehat{R}_{x_{1}x_{2}}\) around the position of the integer cross-correlation peak, which can be expressed as follows:  
\[\delta=\frac{\hat{\mathcal{R}}_{x_{1}x_{2}}(p+1)-\hat{\mathcal{R}}_{x_{1}x_{2 }}(p-1)}{-\hat{\mathcal{R}}_{x_{1}x_{2}}(p+1)+2\hat{\mathcal{R}}_{x_{1}x_{2}} (p-)-\hat{\mathcal{R}}_{x_{1}x_{2}}(p-1)} \tag{21}\]  
where p is the integer cross-correlation peak position, \(\widehat{R}_{x_{1}x_{2}}(p)\) is the value of the cross-correlation function peak, and \(\widehat{R}_{x_{1}x_{2}}(p-1)\) and \(\widehat{R}_{x_{1}x_{2}}(p+1)\) are the values of the cross-correlation function to the left and right of the cross-correlation peak, respectively. Each TDOA estimation result produces a set of possible position equations. Each TDOA estimation result can be expressed as a set of two-dimensional hyperbolic equations with the coordinates of the interference source located at a point on the hyperbola, and the position equation can be expressed as [19]:  
\[\Delta\mathbf{t}_{i,j}\cdot c=\sqrt{\left(\mathrm{X_{i}-x}\right)^{2}+\left( \mathrm{Y_{i}-y}\right)^{2}}-\sqrt{\left(\mathrm{X_{j}-x}\right)^{2}+\left( \mathrm{Y_{j}-y}\right)^{2}} \tag{22}\]  
where (x, y) denotes the position of the transmitter, \((X_{i},Y_{i})\), \((X_{j},Y_{j})\) denotes the position of a pair of receivers, \(i\) and \(j\) satisfy \(\mathrm{i}=1,\ldots,N-1\) and \(\mathrm{j}=2,\ldots,N\), where N is the number of receivers, \(\mathrm{i}\
eq\mathrm{j}\), and c is the speed of light, and is the TDOA measurement between receivers. A monitoring system consists of N receivers, where there are N-1 linearly independent TDOA equations, such as \(\Delta\mathrm{t}_{1,2}\) and \(\Delta\mathrm{t}_{1,3}\).