## 3 Method
### The Construction of Virtual Satellite  
As shown in Figure 6, in environments with dense obstacles or buildings, NLOS wireless propagation between the satellite and the user is common, which can cause significant degradation in localization accuracy. The virtual satellite method addresses this issue by first identifying the reflection point in the NLOS environment. The extension line between the user and the reflection point is then treated as the signal's incident direction, and the pseudo-distance measured by the user is taken as the distance between the virtual site and the user's location. This effectively transforms the NLOS localization problem into a LOS problem. By fully leveraging the additional information provided by the reflection points, this method significantly improves localization performance in environments with obstacles.  
In any virtual satellite method, accurately localizing reflection points is crucial. The virtual base station method achieves this by constructing virtual satellites through mirroring NLOS satellites on reflective surfaces, such as building walls. These virtual satellites can be treated as LOS satellites for user terminals, ensuring that a sufficient number of satellites are available for accurate positioning.  
Signal measurements can be classified into LOS and NLOS categories based on the received signal strength. For multiple NLOS signal peaks, the proposed virtual satellite inversion method only considers perfectly reflecting surfaces (i.e., those perpendicular to the ground where the signal's angle of incidence matches the elevation angle of the satellite, based on the user's ephemeris information). By leveraging pseudo-ranges derived from NLOS reflection errors, this method eliminates the need to separately correct the extended pseudo-ranges caused by NLOS propagation. As a result, the virtual satellite's position can be directly established. The detailed process is as follows:  
* NLOS Reflected Signal Recognition: Signal-to-noise ratio (SNR) calculations are used to identify reflected NLOS signals based on a predefined threshold, distinguishing them from LOS (line-of-sight) signals.
* DOA Estimation Preparation: The covariance matrix (\(R_{x}\)) of the received signal data are calculated. Then, a feature decomposition is performed, separating the signal and noise subspaces.  
Figure 6: NLOS virtual satellite inversion diagram: Red lines represent NLOS paths, blue lines represent unavailable LOS paths, and dashed lines represent direct paths of NLOS inverted virtual satellites.  
* Spatial Spectrum Construction: A spatial scanning spectrum is constructed by exploiting the orthogonality between the signal direction vector and the noise subspace.
* Ephemeris Information Acquisition: Satellite ephemeris data are received and analyzed to extract important satellite elevation angle information.
* Two-Dimensional Spectrum Downscaling: The satellite ephemeris data are used to downscale the 2D spatial spectrum to a 1D spectrum by matching the elevation angle of the perfect reflection.
* Azimuth Search: A search is conducted in the 1D spectrum for spectral peaks to obtain azimuth angle information.
* Incidence Direction Inversion: The matched elevation and azimuth information is combined to calculate the signal's direction of incidence.
* Virtual Satellite Construction: A virtual satellite is generated using the signal's pseudo-range and the calculated incidence direction, treating it as if it were an LOS satellite.
* Positioning Calculation: The final step involves using the information from the virtual satellite to perform the positioning algorithm and output the positioning result.