## 2 System Description
### Direction Finding of ASK and MSK Modulated Signals  
In long-wave radio direction finding, different modulation methods may require distinct techniques. In the ASK modulation mode, which primarily modulates the signal amplitude without affecting the carrier phase, the arrival time difference can be directly calculated from the phase difference of the narrowband data measured by the two receivers:  
\[\tau_{d}(t)=\frac{\Delta\phi_{0}(t)}{2\pi f_{c}}, \tag{18}\]  
where \(\Delta\phi_{0}(t)\) represents the phase difference between the two receivers at time \(t\), and \(f_{c}\) is the carrier frequency of the wave source. By combining Equations (14) and (18), the angle of the wave source can be determined. However, it is important to note that during low-power periods of ASK modulation, noise can significantly affect the signal phase. Therefore, stable phase differences must be extracted during high-power periods for accurate direction finding. However, the MSK modulation method can affect the signal phase due to its frequency modulation and the path delay in receiving the two MSK signals. When calculating the phase difference between the two receivers, this difference includes not only the carrier phase shift but also the phase shift introduced by the MSK baseband signal [16]. If the  
Figure 12: Distribution of direction finding error \(\sigma_{\alpha}\) with incident angle \(\alpha\). phase of the narrowband signal is directly used for time difference estimation, errors will occur. The ideal MSK modulated signal is shown below:  
\[s(t)=\cos\left[2\pi f_{c}t+b(t)\frac{\pi t}{2T}+\theta(t)\right], \tag{19}\]  
where \(T\) is the symbol period, and \(b(t)\) represents the bit stream,  
\[b(t) = \begin{cases}+1,&a_{I}(t)\
eq a_{Q}(t),\\ \\ -1,&a_{I}(t)=a_{Q}(t),\end{cases}  \tag{20}\] \[\theta(t) = \begin{cases}0,&a_{I}(t)=1,\\ \\ \pi,&a_{I}(t)=-1.\end{cases} \]  
with \(\alpha_{I}\) and \(\alpha_{Q}\) representing in-phase and quadrature coding, respectively. The phase of the baseband signal is then  
\[\phi_{\text{MSK}}(t)=\frac{\pi}{2T}\int_{-\infty}^{t}b(t)dt=b(t)\frac{\pi t}{2T }+\theta(t), \tag{21}\]  
Since the signal sources and device parameters of the two receivers are identical, and the propagation path of one receiver is an extension of the other, the signals received by the two receivers can be expressed as follows:  
\[s_{1}(t) =\cos[2\pi f_{c}(t-\tau_{0}(t))+\phi_{\text{MSK}}(t-\tau_{1}(t))], \tag{22}\] \[s_{2}(t) =\cos[2\pi f_{c}(t-\tau_{0}(t)-\tau_{d}(t))+\phi_{\text{MSK}}(t- \tau_{1}(t)-\tau_{d}(t))],\]  
where \(\tau_{0}\) and \(\tau_{1}\) represent the propagation delays caused by the common path and device effects, respectively, at that moment, and \(\tau_{d}\) is the time difference when the signal reaches the two receivers. The instantaneous phase difference between the two signals at this moment can be expressed as follows:  
\[\Delta\phi(t) =\{2\pi f_{c}(t-\tau_{0}(t))+\phi_{\text{MSK}}(t-\tau_{1}(t))\} \tag{23}\] \[-\{2\pi f_{c}[(t-\tau_{0}(t))-\tau_{d}(t)]+\phi_{\text{MSK}}[(t- \tau_{1}(t))-\tau_{d}(t)]\}\] \[=\underbrace{2\pi f_{c}\tau_{d}(t)}_{\text{carrier phase difference}}+\underbrace{\left[\phi_{\text{MSK}}(t-\tau_{1}(t))-\phi_{\text{MSK}}((t-\tau_{1}(t))- \tau_{d}(t))\right]}_{\text{MSK baseband phase difference}}\] \[=2\pi f_{c}\tau_{d}(t)+\Delta\phi_{\text{MSK}}(t),\]  
hence,  
\[\tau_{d}(t)=\frac{\Delta\phi(t)-\Delta\phi_{\text{MSK}}(t)}{2\pi f_{c}}, \tag{24}\]  
where,  
\[\Delta\phi_{\text{MSK}}(t) =\frac{\pi}{2T}\int_{-\infty}^{t-\tau_{1}(t)}b(t)dt-\frac{\pi}{ 2T}\int_{-\infty}^{t-\tau_{1}(t)-\tau_{d}(t)}b(t)dt \tag{25}\] \[=\frac{\pi}{2T}\biggl{[}\int_{t-\tau_{1}(t)-\tau_{d}(t)}^{t-\tau _{1}(t)}b(t)dt\biggr{]}.\]  
To simplify the expression, assuming \(\tau_{d}\) is a small quantity (i.e.,\(\tau_{d}\ll T\)) and that \(\tau_{1}\) changes slowly, it can be approximated that \(b(t)\) remains constant within the interval \([t-\tau_{1}(t)-\tau_{d}(t),t-\tau_{1}(t)]\).