## 4 Simulation Results and Discussion
### Feasibility Test  
The feasibility of the proposed algorithm is tested using five simulated satellites and five corresponding pseudolites. Figure 9 shows the procedure of the feasibility test. First of all, true values are provided for the initial setting module. The positions of simulated GPS satellites and pseudolites are shown in Figure 10. For convenience, the receiving point is set at (0,0,0), and satellite positions are set as they were on the sphere of radius of 22,000 km centered at the receiving point. Local user positions were set at (0,0,0) and (1,1,0). PDOPs for GPS satellites and pseudolites were then calculated to be 1.7489 and 2.0468, respectively. Figure 9: Simulation procedure. The initial setting module stores each true value to the proper variables. Using these values, pseudorange measurements are generated according to Equation (2). Tropospheric and ionospheric delays are assumed to be eliminated by the model or by other methods. Additional Gaussian random noise is added to the measurements. Variance of the measurement noise is set to 1 meter for all pseudolites. Figure 11 shows the generated pseudorange measurements for each pseudolite received by the User #2. The GPS receiver calculates the _a priori_ position using generated measurements as if it were the general legacy GPS receiver. The post-calculation module uses this _a priori_ position, the true GPS satellite position, and the true pseudolite position to re-estimate the final position. Instead of the data server, necessary information are provided to the post-calculation module directly in this  
Figure 11: Generated Pseudorange Measurements. Figure 10: (**a**) Skyplot of simulated GPS satellites viewed from the receiving point (**b**) Local positions of pseudolites and user. simulation. Figure 12a shows that the _a priori_ position of User #1 calculated by the algorithm stated in Section 3.3.1 is not biased. In this case, the distances from all pseudolites to the user are the same. Therefore, all pseudolite signals are subject to the same delay, which affects only the clock error, not the position measured. The horizontal and vertical drums errors are calculated to be 1.2977 m and 2.2927 m, respectively. Unlike User #1, User #2 got biased _a priori_ positions from the GPS receiver, as shown in Figure 12b. The distances from the pseudolites to the user are different from each other in this case, and they affect the position results. However, using the proposed algorithm, the final position is estimated with reasonable accuracy. The horizontal and vertical drums errors are calculated to be 1.2977 m and 2.2927 m, respectively. The _a priori_ position of User #2 has much bias error than that of User #1. However the error variance is calculated as the same value. As described in Section 3.3.2, though the GPS receiver output does not give the correct position, it contains some information about the original measurements. The measurements from the position output can be estimated via Equation (10). Figure 13 shows the difference between the originally generated and re-estimated pseudorange measurements. The residual values are affected by the geometry of the simulated GPS satellites and pseudolites. Figure 12: **(a)**_a priori_ position for user #1 (**b**)**_a priori_ position for user #2. Figure 13: Estimated Pseudorange Measurements. Using the _a priori_ positions and the estimated pseudorange measurements, the final estimation of user position can be calculated as described in Section 3.3.3. Figure 14 shows the _a priori_ and final position of User #1 and User #2. In the final estimation, the bias error is eliminated for not only the User #1, but also for User #2. The values of the horizontal and vertical drums and bias errors became smaller than those of the _a priori_ estimates. The results show that using the proposed system, user position can be estimated despite reliance on pseudolites combined with an unmodified legacy GPS receiver. The algorithm works regardless of the existence of bias in the output of the GPS receiver.