## 4 Test Validation
### Test for a Short-Distance Spoofing Signal Source  
Due to the simplification of the carrier motion in the computational model of this method, the increased influence of positional changes on the single-difference carrier phase between satellite navigation signal stations during close-range movements may affect the accuracy and reliability of the model. To analyse how the simplification of the position changes of the carrier affected the two-dimensional direction measurements for short-distance signal sources, we utilised a spoofing signal source to deceive the carrier. The position of the target carrier relative to the spoofing signal source is illustrated in Figure 8.  
Figure 7: BDS real signal two-dimensional signal source determining errors. (**a**) Two-dimensional direction-finding error of BeiDou-2 in this experiment; (**b**) two-dimensional direction-finding error of BeiDou-3 in this experiment; (**c**) yaw angle error and 3\(\sigma\) line of BeiDou-2 in this experiment; (**d**) yaw angle error and 3\(\sigma\) line of BeiDou-3 in this experiment; (**e**) pitch angle error and 3\(\sigma\) line of BeiDou-2 in this experiment; and (**f**) pitch angle error and 3\(\sigma\) line of BeiDou-3 in this experiment.  
Figure 9a depicts that the error in the two-dimensional heading angle of the signal source was basically less than 0.1 rad, which was in line with the expected positioning error for the signal source. When the difference between the distance of the signal source distance and the motion amplitude of the carrier was not obvious, the pitch angle error of the signal source rapidly increased, but the heading angle maintained high accuracy. When the system was applied to signal-source position search or fused with the zeroing antenna technology, it retained high reliability and effectiveness.  
Moreover, Figure 9b,c illustrates that the two-dimensional direction error of the signal source basically conformed to the envelope of the 3\(\sigma\) line, and the error gradually converged to approximately zero.  
Figure 8: Horizontal reference trajectories of the carrier.  
Figure 9: Near-range spoofing signal two-dimensional signal source determining errors. **(a)** Two-dimensional direction-finding error for the short-distance spoofing signal source; **(b)** yaw angle error and 3\(\sigma\) line for the short-distance spoofing signal source; and **(c)** pitch angle error and 3\(\sigma\) line for the short-distance spoofing signal source.  
Essentially, the sudden change of this error reflected the direction-finding error of the least squares model.  
A sudden change occurred in the computational error, carrier-phase compensation noise (error), and micro-inertial navigation error. The computational error was on the order of \(10^{-4}\) rad, which was not obvious under the influence of the carrier-phase compensation error. The carrier-phase compensation error was approximated as a normally distributed error with a mean value of 0, which was kept on the order of \(10^{-3}\) rad after the error was smoothed in the multi-epoch least squares algorithm. The micro-inertial navigation error could not be smoothed in the least squares algorithm because of the existence of the error dispersion, and the error experienced a sudden change on the order of \(10^{-2}\) rad. This implies that the micro-inertial navigation error in the experimental environment as depicted in Figure 10 was the main factor causing sudden changes in navigation errors.