## 3 Mathematical Model of Geolocation and Tracking Scenarios
### Geolocation and Target-Tracking Techniques  
#### 3.3.1 Cross Ambiguity Function (CAF) Approach  
CAF combines TDOA and FDOA to achieve more accuracy and reduce the requirements for geolocation and mapping measurements. The measurements of TDOA and FDOA are used for cross-correlation equations, as presented in (23). CAF is applied in order to compute the peaks that are assumed to be the sources of the emitter using the CAF function. Then, the evaluation will be performed to do selection of the most accurate position, such as the higher peak, as a precise interference source. \[\text{CAF}(\mathbf{\tau},\mathbf{f})=\int_{0}^{T}sig_{1}(t)sig_{2}^{*}(t-\mathbf{\tau})e^{ -j2\pi ft}dt\, \tag{23}\]  
where \(sig_{1}\) and \(sig_{2}\) are continuous-time signals; \"\(*\)\" means complex conjugate; \(T\) is the integration period in seconds; \(\mathbf{\tau}\) is the time delay in seconds, and \(\mathbf{f}\) is the frequency offset in Hertz. When the RFI emitter sends its signal towards the sensors, it reaches each sensor at a different time and with shifted frequency (due to their motion). The signals are captured for the selected period \(T\). Binary-phase-shift-keying (BPSK) signals takes two-phase reversals such as \(0^{\circ}\) and \(180^{\circ}\) in this model. This is widely used in satellite communications. A sinusoidal carrier wave is modulated by binary digits \"1\" and \"0\", so the data signal shifts the phase of the carrier waveform is either \(0^{\circ}\) or \(180^{\circ}\). From the trigonometric relationships,  
\[ \begin{cases}\sin(x+\pi)=-\sin(x),\\ \cos(x+\pi)=-\cos(x),\end{cases}  \tag{24}\]  
There are two possible states in a BPSK modulation, where the carrier is multiplied by \(\pm 1\). Therefore, general analytic expression of BPSK signal can be given as,  
\[sig_{i}(t)=A\cos\big{[}2\pi f_{c}t+\varphi_{i}(t)\big{]}\ \begin{cases}0\leq t\leq T_{sym}\,\\ i=1,2\end{cases}  \tag{25}\]  
where \(A\) is the amplitude, \(f_{c}\) is the carrier frequency, \(\varphi_{i}(t)\) is the data symbol period, which takes on the values of \"0\" or \"\(\pi\)\", and \(T_{sym}\) is the data symbol period that equal \((1/R_{sym})\). \(R_{sym}\) is a symbol rate for the generated BPSK signals. By substituting (24) in (25), it can achieve two waveforms transmitted in a BPSK of continuous-time or analog signals  
\[ \begin{cases}sig_{1}(t)=A\cos(2\pi f_{c}t)\,\\ sig_{2}(t)=-A\cos(2\pi f_{c}t)\,\end{cases}  \tag{26}\]  
The discrete time is represented as  
\[ \begin{cases}sig_{1}(n)=A\cos\left[2\pi f_{c}(nT_{s})\right]\,\\ sig_{2}(n)=-A\cos\left[2\pi f_{c}(nT_{s})\right]\,\end{cases}  \tag{27}\]  
Practically, (27) must be in discrete form to be easy for digital signal analysis and computation. For that, sampling frequency \((f_{s})\) must change to sample duration \(T_{s}=1/f_{s}\), Therefore let  
\[ \begin{cases}t=nT_{s}\,\\ \mathbf{f}=\frac{P^{D}f_{s}}{N}\,\end{cases}  \tag{28}\]  
where \(\mathbf{f}^{D}\) is the digital FDOA, \(N\) is the number of samples for each signal that equal \(\frac{T}{f_{s}}\). From that, we can produce the discrete CAF. where \(\varepsilon_{i}\) is the \(i\)th eigenvalue of \(\mathbf{J}\) that is supposed to be a symmetric tridiagonal matrix with zero diagonal elements given by \[\mathbf{J}_{i;i+1}=\sqrt{i/2}=0\,\] (33) (iii) Calculate the respective weights \(W_{i}\) of the quadrature points (\(\xi_{i}\)), where \(W_{i}\) is equal to the square of the first element of the \(i\)th normalized eigenvector of \(\mathbf{J}\). 2.