# Dynamic Phase Comparison Planar Direction-Finding Algorithm on Satellite Radio Receiver
## 2 Description of the Algorithm  
### Dynamic Phase Comparison Algorithm  
The dynamic phase comparison algorithm is based on the phase interferometer algorithm. In the interferometer DF system, as shown in Figure 1, it is accomplished using array antennas consisting of at least two antennas. Assume that single-tone plane EM waves emitted from a far-field radiation source propagate to a single-baseline array antennas. Due to the time difference between the signals received by the two antennas, there is a phase difference between them. In this way, the azimuth angle of the EM waves, defined as the angle between the direction of the EM waves and the baseline of array antennas, can determine the phase difference  
\[\varphi=\frac{2\pi l}{\lambda}\cos\alpha=\frac{2\pi fl}{c}\cos\alpha, \tag{1}\]  
where \(c\) is the speed of light; \(l\) is the distance between the antennas, as known as the baseline length of the interferometer; and \(\lambda\) and \(f\) are the wavelength and frequency of the  
Figure 1: Principle of the phase interferometer algorithm. EM waves. If \(l\) and \(f\) are known, the azimuth angle of the radiation source signal can be determined as  
\[\alpha=\arccos\left(\frac{c\varphi}{2\pi fl}\right). \tag{2}\]  
Compared with the phase interferometer algorithm, the dynamic phase comparison algorithm does not use array antennas. Instead, it employs a single antenna element receiving signals under high-speed uniform motion. As shown in Figure 2, if the radiation source continuously emits EM waves periodically for a certain period, it can be considered as two (or more) virtual antenna elements during the high-speed uniform motion on the satellite, thereby achieving signal reception and sampling at different positions. During the process where a far-field radiation source emits single-tone EM waves periodically for a certain period, a satellite receiver equipped with a dipole antenna and a single-channel receiver move at a high velocity \(v\). Considering that the plane waves are emitted by the far-field radiation source, the two receptions of EM waves are treated as parallel waves with constant frequency. Respectively, the signals sampled by the satellite receiver are \(s^{\prime}(t)\) and \(s(t)\) at two different times, \(T^{\prime}\) and \(T\). From Equation (1), the phase difference \(\Delta\varphi\) caused by the angle between the direction of the EM waves can be known. Assume the position of the radiation source remains constant and ignore the short movement of the receiver during the brief signal sampling period. Then, the signals received at two different positions can be approximately represented as  
\[ \begin{cases}s^{\prime}(t)\approx K\cos(2\pi ft+\varphi_{0})\\ s(t)\approx K\cos[2\pi f(t+\Delta t)+\varphi_{0}+\Delta\varphi],\end{cases}  \tag{3}\]  
where \(\Delta t\) is the time difference experienced by the moving satellite and \(K\) is the antenna gain. Hence, the total phase difference between the received signals \(s^{\prime}(t)\) and \(s(t)\) consists of two parts: one part is due to the time difference \(\Delta t\) between the two receptions of the signal by the satellite receiver; and the other part is due to the angle between the direction of the satellite's motion and the incoming direction of the EM waves, which also causes a phase difference in the signals. Thus, the total phase difference can be approximately obtained as  
\[\varphi=s(t)-s^{\prime}(t)\approx 2\pi f\Delta t+\Delta\varphi. \tag{4}\]  
Based on the principles of the phase interferometer algorithm from Equation (2), the azimuth angle of the radiation source signal can be obtained as  
\[\alpha=\arccos\left(\frac{c\varphi}{2\pi fd}\right), \tag{5}\]  
Figure 2: Principle of the dynamic phase comparison algorithm. where \(d\) is the distance covered by the satellite during its high-speed motion between the two signal receptions, which can be obtained from the satellite's velocity \(v\) and \(\Delta t\).