# Dynamic Phase Comparison Planar Direction-Finding Algorithm on Satellite Radio Receiver
## 2 Description of the Algorithm
### Accuracy Analysis  
The dynamic comparison phase algorithm, based on the phase interferometer algorithm, treats the distance traveled by the satellite as equivalent to the baseline length. To perform an accuracy analysis of the dynamic comparison phase algorithm, a total differential of Equation (7) is carried out, which can be obtained as  
\[\mathrm{d}\alpha=\frac{\partial\alpha}{\partial\varphi}\mathrm{d}\varphi+ \frac{\partial\alpha}{\partial f}\mathrm{d}f+\frac{\partial\alpha}{\partial d }\mathrm{d}d. \tag{9}\]  
By applying the chain rule to solve for the three partial derivatives, the results can be obtained as  
\[\frac{\partial\alpha}{\partial\varphi}=\frac{-1}{\sqrt{1-u^{2}}} \cdot\left(\frac{c}{2\pi fd}\right) \tag{10}\] \[\frac{\partial\alpha}{\partial f}=\frac{-1}{\sqrt{1-u^{2}}} \cdot\left(\frac{-c\varphi}{2\pi f^{2}d}\right)\] (11) \[\frac{\partial\alpha}{\partial d}=\frac{-1}{\sqrt{1-u^{2}}} \cdot\left(\frac{-c\varphi}{2\pi fd^{2}}\right), \tag{12}\]  
where  
\[u=\frac{c\varphi}{2\pi fd}. \tag{13}\]  
Hence, the total differential can be obtained as  
\[\mathrm{d}\alpha=\frac{-c}{2\pi fd\sqrt{1-u^{2}}}\mathrm{d}\varphi+\frac{c \varphi}{2\pi f^{2}d\sqrt{1-u^{2}}}\mathrm{d}f+\frac{c\varphi}{2\pi fd^{2} \sqrt{1-u^{2}}}\mathrm{d}d. \tag{14}\]  
Analyzing the total differential equation reveals:  
1. From Equation (10), when the frequency of the EM waves or the satellite motion distance is relatively small, the azimuth accuracy is significantly influenced by the measurement accuracy of the phase difference. 2. From Equation (11), the azimuth accuracy is more greatly affected by the measurement accuracy of the frequency of the EM waves, especially when the absolute value of the phase difference is large (i.e., when \(\alpha\) is close to \(0^{\circ}\) or \(180^{\circ}\)) or when the satellite motion distance is small. The frequency appears in the form of \(f^{2}\), which impacts the azimuth accuracy more. 3. From Equation (12), the azimuth accuracy is more significantly influenced by the measurement accuracy of the satellite motion distance, particularly when the absolute value of the phase difference is large (i.e., when \(\alpha\) is close to \(0^{\circ}\) or \(180^{\circ}\)) or when the frequency of the EM waves is low. The distance appears in the form of \(d^{2}\), which also impacts the azimuth accuracy more. Given that the measurement error of phase difference significantly impacts the DF error, and because obtaining the azimuth angle from the phase difference is a nonlinear transformation, the effect of different EM waves azimuth angles on the phase difference cannot be ignored. The following analysis of the measurement accuracy of phase difference involves performing a total differential of the equation, which can be obtained as  
\[\mathrm{d}\varphi=\frac{2\pi l\cos\alpha}{c}\mathrm{d}f+\frac{2\pi f\cos \alpha}{c}\mathrm{d}l. \tag{15}\]  
The analysis of the equation reveals that the measurement error in phase difference originates from the measurement errors in the frequency of the EM waves and baseline length. Specifically, the impact of errors is more significant when the azimuth angle of the EM waves is close to the orientation of the baseline (i.e., when \(\alpha\) is close to \(0^{\circ}\) or \(180^{\circ}\)). In summary, when designing and applying the dynamic comparison phase DF system, special attention should be paid to the following three core elements to ensure that the DF accuracy meets requirements:  
1. Improving the measurement accuracy of the phase difference is key to enhancing the performance of the phase comparison algorithm. Accurate measurement of the phase difference can effectively reduce DF errors. 2. The satellite motion distance (i.e., the baseline length) should be sufficiently large to provide higher DF accuracy when direction finding is performed on VLF signals. 3.