6.2.1 Validation with Angle Effect
As mentioned in Subsection V.A.(1), the primary objective of the proposed GPLT algorithm is to adjust the position of the virtual BS such that the predicted MS can be situated at a location
0 50 100 150 200 250 300 350
Relative Angle Between the MS and the Virtual BS ( Θ)
0 50 100 150 200 250 300 350 0
Relative Angle Between the MS and the Virtual BS (Θ)
0 50 100 150 200 250 300 350 0
Figure 6.2: Top Plot: the Average Position Error (Solid Line) and the GDOP Value (Dashed Line) vs the Relative Angle Between the MS and the Virtual BS (θ); Bottom Plot: the RMSE (Solid Line) and the GDOP Value (Dashed Line) vs the Relative Angle Between the MS and the Virtual BS (θ)
with minimal GDOP value. The design concept implicitly indicates that the estimation error can be reduced if the MS is possessed with a smaller GDOP value formed by its geometric layout. In this subsection, the relationship between the estimation errors and the GDOP values will be verified via simulations. As shown in Fig. 6.1, the two-BS case is considered associated with the locations of the BSs are BS1 = (505, 2957) and BS2 = (1520, 1234) in meters. The MS’s true position is located at x = (1020, 2100) m. The position of the virtual BS is assumed at xv,1(θ) = (1020 + 1500 cos θ, 2100 + 1500 sin θ) m with θ = 0 ∼ 359o. It can be seen that the potential positions of the virtual BS are considered to be located at a distance 1500 meters away from the MS’s true position along with different relative angles θ.
Fig. 6.2 illustrates the comparison between the average position error (left plot), the RMSE (right plot), and the GDOP value versus the relative angle (θ) between the true MS and the virtual BS. It is noted that the Average Position Error (∆x) and the RMSE are computed as: ∆x =hPN
i=1kx − ˆx(i)k i
/N and RMSE = hPN
i=1kx − ˆx(i)k2/N i1/2
where N
= 50 indicates the number of simulation runs. It is also noticed that the GDOP value (Gx) is evaluated at the MS’s true position; while the estimated MS’s position ˆx(i) is obtained by the two-step LS estimator employing the various positions of the virtual BS, i.e. xv,1(θ) for θ = 0 ∼ 359o. It can be observed from both plots in Fig. 6.2 that the average position error and the RMSE follow the similar trend as the computed GDOP value. Both the minimal mean estimation error (associated with the RMSE) and the minimal GDOP value occur at the locations of xv,1(30.2o) = (2316, 2855) m and xv,1(210.9o) = (−267.1, 1330) m. It is noted that the angle θkm for the minimal GDOP value can also be directly computed and verified from (5.6). Moreover, the maximal GDOP values and the maximal estimation errors (including both the average position error and the RMSE) happen around the locations of xv,1(120.5o)
= (258.7, 3392) m and xv,1(300.5o) = (1781, 807.6) m. The results can further be validated by observing the geometric layout as in Fig. 6.2. The minimal GDOP values of the true MS occur as the three BSs form a equilateral triangle; while the maximal GDOP values happen as the three BSs are situated along a straight line. The above observations validate the effectiveness of the proposed GPLT scheme by obtaining a position of the virtual BS with a
0 1 2 3 4 5 6 7 8 9 10
Distance from MS to Virtual BS (m)
Average Position Error (m)
Distance from MS to Virtual BS (m)
RMSE (m)
σnv1 = 10 σnv1 = 20 σnv1 = 30 σnv1 = 40
Figure 6.3: Top Plot: the Average Position Errors vs the Relative Distance Between the MS and the Virtual BS (rv1); Bottom Plot: the RMSE vs the Relative Distance Between the MS and the Virtual BS (rv,1) (with σnv1 = 10, 20, 30, 40)
smaller GDOP value, which consequently reduces the corresponding estimation error. On the other hand, the estimation errors can be severely augmented if the MS happens to be located at a position with the maximum GDOP value by adopting other schemes. It can therefore be concluded that the results obtained from the simulations comply with the design objectives of the GPLT algorithm.
6.2.2 Validation with Distance Effect
In this subsection, the results obtained from Lemma 1 will be validated via simulations.
It is stated in Lemma 1 that the expected value of the estimation error is independent to the distance between the MS and a specific BS by adopting the WLS location estimation algorithm. In order to validate Lemma 1 by the simulation data, the estimation errors induced by adopting the two-step LS estimator will be obtained for the evaluation of the distance effect.
Fig. 6.3 illustrates the average position error (left plot) and the RMSE (right plot) acquired from the two-step LS method under different relative distances between the MS and the virtual BS (i.e. rv,1). It is noted that the distance rv,1 is simulated from 1 to 106 m along the angle θ = 60o as shown in Fig. 6.1. The four simulated results are conducted under different signal variations (i.e. σnv1 = 10, 20, 30, 40) in order to exam the potential effect from the signal variances. As can be expected, the estimation errors are observed to be independent to the relative distance between the MS and the virtual BS, which are similar to the results as concluded from Lemma 1. Moreover, it is also reasonable to perceive that the increases on the signal variances σnv1 will induce proportional augmentation on the RMSE (in the right plot of Fig. 6.3); while the average position error is considered not related to the changes due to the signal variations (in the left plot of Fig. 6.3). From the above observations via the simulation data, the uncorrelated relationship between the distance rv,1 and the estimation error is found to be consistent with the results as acquired from Lemma 1.
0 30 60 90 120 150 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Time (sec)
Number of BSs
Figure 6.4: Total Number of Available BSs (Nk) vs. Simulation Time (sec)