Implementations of Geometry-Assisted Linearized Location (GALL) Algorithm(GALL) Algorithm

Figure 5.2: Implementations of proposed GALL algorithm.

As described in previous section, the main objective of the proposed GALL scheme is to acquire the positions of fictitiously movable BSs in order to provide better geometric layout for MS’s location estimation. Since the GALL scheme is designed based on the initial estimate of the MS, there can be different implementations to adopt the GALL scheme for location estimation. Fig. 5.2illustrates the schematic diagrams for the implementations of the proposed GALL algorithm. The GALL-TSLS scheme as shown in Fig. 5.2(a) is proposed to calculate both the initial and final estimates of the MS’s position based on the TSLS estimator [17]. On the other hand, a two-stage architecture named GALL-KF scheme as shown in Fig. 5.2(b), i.e., TSLS estimator with Kalman filter, is proposed to enhance the initial estimate with the historical information from Kalman filter. These two types of implementations of GALL scheme are explained in the following two subsections.

5.4. Performance Evaluation

5.3.1 GALL with TSLS estimator (GALL-TSLS)

As shown in Fig. 5.2(a), the MS’s initial estimate ˆx^o can be obtained by performing the TSLS method (i.e., ˆx^o= ˆx_T). The formulation and concept of the TSLS method can be found in Section 2.4. With the initial estimate, the fictitious BS set can be calculated based on the proposed GALL scheme targeting on achieving the minimum L-CRLB requirement. The TSLS is performed for the second time with the adjusted BSs and the received measurements to obtain the MS’s location estimate for the GALL-TSLS scheme.

5.3.2 GALL with Kalman Filter (GALL-KF)

In order to provide enhanced location estimate, the proposed GALL-KF scheme as shown in Fig.

5.2(b) is suggested to estimate the MS’s position using a two-stage estimator, i.e., a TSLS estimator with a Kalman filter. The measurements are collected in different time instants to obtain a better initial estimate, the notation in this section is considered with the time instant t in Section 2.1.1.

The Kalman filter is employed to estimate the MS’s position based on its previously estimated data.

Note that the formulation and concept can be referred to Section2.5. Note that for the two-stage location estimation, the input of the Kalman filter is obtained from the result of the GALL scheme as z^(t) = ˆx^(t)_T in Fig. 5.2(b). The estimated output/state ˆs^(t) is the 2-dimensional MS’s position.

The variables m^(t) and p^(t) denote the measurement and the process noises associated with the covariance matrices R and Q within the Kalman filtering formulation. Note that the matrix R can be determined by the FIM of L-CRLB and Q is set to be an identity matrix. Furthermore, the matrix E and the state transition matrix F in (2.16) and (2.17) respectively can be obtained as E = F = I_2×2.

As shown in Fig. 5.2(b), the execution process of the proposed GALL-KF scheme consists of two phases, including the transition period (Tp) and the stable period. During the transient period t < T_p, the GALL-KF scheme adopts the TSLS method to provide the initial estimate of the MS. After the tracking time is longer than T_p, the GALL-KF scheme starts to adopt the prediction from the output of the Kalman filter, i.e., ˆs^(t|t−1), which serves as the updated initial MS’s estimate for the GALL scheme. By adopting the Kalman filter, it can be observed that the GALL-KF scheme only requires to perform a single round of location estimation compared to the GALL-TSLS scheme after the system is executed in the stable state. The Kalman filter can refine the MS’s position estimation with the historical measurements based on the initial estimate, which should provide better estimation accuracy by adopting the GALL-KF scheme compared to the GALL-TSLS method.

0 20 40 60 80 100 120 140

GALL(1BS)−TSLS BS₁ moved GALL(1BS)−TSLS BS

2 moved GALL(1BS)−TSLS BS₃ moved GALL(1BS)−TSLS GALL−TSLS

(a) GALL-TSLS with original one fictitiously movable BS problem. GALL(1BS)−TSLS BS₂ moved GALL(1BS)−TSLS BS

3 moved GALL(1BS)−TSLS GALL−TSLS

(b) GALL-TSLS with approximate one fictitiously movable BS problem.

Figure 5.3: Validation on GALL-TSLS scheme with one fictitiously movable BS problem.

5.4 Performance Evaluation

Simulations are performed to show the effectiveness of the GALL algorithms (i.e., GALL-TSLS and GALL-KF) under different network topologies and the MS’s positions. The number of BSs is considered as three in the examples since three BSs is the minimum sufficient number for the localization problem. The model for the LOS measurement noise of the TOA signals is considered as in Section 2.2.1.1 as the Gaussian distribution with zero mean and standard deviation as σ_n meters in different cases. In the following examples 5.1 and 5.2, the GALL-TSLS algorithm is simulated to validate the effectiveness of the fictitiously movable BS schemes on the TSLS based estimation.

Example 5.1 (Validation on approximate one fictitiously movable BS problem). The purpose of this example is to compare and validate the difference between the original and approximate one fictitiously movable BS problems. Consider three sensors whose coordinates are b₁ = [50 cos 0^◦, 50 sin 0^◦]^T, b₂ = [30 cos α₂, 30 sin α₂]^T, and b₃ = [20 cos 140^◦, 20 sin 140^◦]^T where b₂ is function of α₂ with its range as indicated in the x-axis of Fig. 5.3. The MS’s true position is assumed to be placed at the origin, i.e., x = [0, 0]^T. Note that all the layouts formed by the three sensors with the change of α₂ are designed to be OPLs for validation purpose. More realistic network scenarios will be considered in the following examples. The standard deviation of the Gaussian noises is chosen as σ_n = 1 in this example. The root mean square error (RMSE) is defined as RMSE =

qPM

i=1kˆx − x k²/M where M denotes the number of trials as 1000. Fig. 5.3(a) shows the original one fictitiously movable BS problem obtained by exhaustively solving (5.2); while Fig.

5.3(b) illustrates the approximate one fictitiously movable BS problem acquired from (5.8) and

5.4. Performance Evaluation GALL(2BSs)−TSLS BS₂ fixed GALL(2BSs)−TSLS BS

3 fixed GALL(2BSs)−TSLS GALL−TSLS

(a) GALL-TSLS with original two fictitiously movable BSs problem.

GALL(2BSs)−TSLS BS₁ fixed GALL(2BSs)−TSLS BS

2 fixed GALL(2BSs)−TSLS BS₃ fixed GALL(2BSs)−TSLS GALL−TSLS

(b) GALL-TSLS with approximate two fictitiously movable BSs problem.

Figure 5.4: Validation on GALL-TSLS scheme with two fictitiously movable BSs problem.

(5.9), which are respectively denoted as “GALL(1BS)-TSLS BSi moved” in both plots, for i = 1, 2, and 3. In each of the three cases, i.e., i = 1, 2 and 3, BS_i is fictitiously moved for obtaining the optimal angle ˜α^m_i that can achieve the minimum value of L-CRLB. Moreover, the “GALL(1BS)-TSLS” curves in both plots respectively denote the GALL(1BS) problem as defined in (5.1) by selecting among different fictitiously movable angles from the original problem (5.2) in Fig. 5.3(a) and approximate problem (5.8) in Fig. 5.3(b). Since the “GALL(1BS)-TSLS” acquire the positions of the fictitious BSs such as to achieve the minimum L-CRLB value with respect to the MSs initial location instead of the MS’s true position, the RMSE performance is not necessarily the lowest compared to the “GALL(1BS)-TSLS BS_i moved” scheme for i = 1, 2, and 3. Note that both the CRLB and the conventional TSLS scheme are also illustrated in both plots for comparison purpose. It can be observed that even though the approximate problem will be differ from the original problem by individually moving one of the three BSs fictitiously, the resulting problem (5.1), i.e., the GALL(1BS)-TSLS scheme, obtained from (5.8) will be closely match with (5.2) as shown in both plots. Furthermore, it can be seen that the GALL(1BS)-TSLS scheme can provide better RMSE performance compared to the conventional TSLS scheme. ⋄ Example 5.2 (Validation on approximate two fictitiously movable BSs problem). This example is to compare and validate the difference between the original and approximate two fictitiously movable BSs problems in Fig. 5.4(a) and Fig. 5.4(b), respectively. Same network layout and noise variance as in example5.1are utilized in this example. The curves named “GALL(2BSs)-TSLS BS_i fixed” refer to the problems that the i-th BS is fixed while the other two BSs are movable, i.e., the angle set {˜α^m₂ , ˜α^m₃ } of the curve “GALL(2BSs)-TSLS BS1 fixed” are obtained via (5.12) and (5.18)

for the original (Fig. 5.4(a)) and approximate (Fig. 5.4(b)) problems respectively. Moreover, the curve “GALL(2BSs)-TSLS” denotes for the problem in (5.11) to select among different movable angles from the original problem (5.12) in Fig. 5.4(a) and the approximate problem (5.18) in Fig.

5.4(b). Similar to the previous example, the final GALL(2BSs)-TSLS scheme of both problems are observed to be consistent with each other from the simulation results. Meanwhile, the effectiveness of problem (5.22) for the GALL scheme is also validated by selecting among the GALL(1BS)-TSLS and the GALL(2BSs)-TSLS schemes. By observing both Figs. 5.3and5.4, the GALL-TSLS scheme with the problem (5.22) can achieve the lowest RMSE compared to the other methods. ⋄ It is intuitive that the closed form property of the approximate problem can provide efficiency in computational complexity compared to the original problem. Therefore, the approximate problem with the GALL scheme will be adopted in the rest of the examples for performance comparison. In order to provide better estimation precision for MS’s location estimate, the GALL-KF algorithm is simulated to compare with the existing TSLS [17], Beck [40], and SDR [41] algorithms, which are named as TSLS-KF, Beck-KF, and SDR-KF respectively. Note that these three algorithms are also cascaded with the Kalman filters to perform two-stage estimation in order to provide fair comparison.

0 200 400 600 800 1000

10⁻¹ 10⁰ 10¹

Simulation Time t

Average Position Error (m)

GALL−KF Beck−KF TSLS−KF SDR−KF

(a) Average position error versus simulation time under σn= 2.

1 2 3 4 5 6 7 8 9 10

10⁻¹ 10⁰ 10¹

Standard Deviation of Gaussian Noise (m)

RMSE (m)

GALL−KF Beck−KF TSLS−KF SDR−KF

(b) RMSE versus different σn.

Figure 5.5: Performance comparison of example5.3.

Example 5.3 (A special case of GALL-KF scheme). In this example, a special network scenario is simulated to provide performance comparison for the GALL-KF scheme. Consider an array of three sensors in the OPL whose coordinates are b₁= [20 cos 0^◦, 20 sin 0^◦]^T, b₂ = [30 cos 80^◦, 30 sin 80^◦]^T, and b₃ = [50 cos 140^◦, 50 sin 140^◦]^T; while the MS’s true position is fixed at x = [0, 0]^T. Fig.

5.5(a) demonstrates the performance comparison of average position error for the simulation time

5.4. Performance Evaluation

instant t = 1000, where each time instant is run with 1000 simulation samples. Note that the average position error at the time instant t is defined asPM

i=1kˆs^(t)− x^(t)k/M. The transient period T_p is chosen as 20 which means that the GALL-KF scheme starts to adopt the prediction from the Kalman filter at the time instant t = 21. The standard deviation of Gaussian noises σ_n is chosen as 2 in Fig. 5.5(a). Since the Kalman filter is effective in smoothing the estimation result, it can be observed that the average position error is decreased and converges with the increment of time instant t for all the schemes in Fig. 5.5(a). With the consideration of the L-CRLB in the algorithm design, the proposed GALL-KF implementation can achieve lowered average position error compared to the other schemes; while the SDR-KF has the worst performance among all the algorithms. Since the two-stage architecture does not provide smoothing gain to the SDR-KF method, the SDR-KF scheme will not be further considered in the rest of the simulation examples.

Fig. 5.5(b) illustrates the performance comparison of RMSE at the simulation time instant t = 1000 under different standard deviations of the Gaussian noises. It can be observed that the GALL-KF scheme can provide a significant gain over the other methods under different noise values. Note that the Beck-KF scheme is observed to be sensitive to the noise which results in higher RMSE compared to the TSLS-KF scheme under larger noise condition. For example, compared to the TSLS-KF and Beck-KF methods, the proposed GALL-KF scheme will result in 2.8 meter and 6 meter less of RMSE respectively under σ_n= 10 meter as shown in Fig. 5.5. ⋄

−50 −40 −30 −20 −10 0 10 20 30 40 50

−50

−40

−30

−20

−10 0 10 20 30 40 50

X Coordinate (m)

Y Coordinate (m)

BS position MS position

Figure 5.6: Network layout of example 5.4.

Example 5.4 (A general case of GALL-KF scheme under LOS environment). This example il-lustrates a general scenario of wireless sensor network as shown in Fig. 5.6 for performance comparison under LOS environment. The BSs’ coordinates are selected as b₁ = [50, −35.36]^T, b₂= [50, 35.36]^T, and b₃ = [−50, 35.36]^T, and there are 100 MSs randomly deployed in a 100×100

0 5 10 15 20

Standard Deviation of Gaussian Noise (m)

RMSE (m)

Standard Deviation of Gaussian Noise (m)

RMSE (m)

GALL−KF Beck−KF TSLS−KF

(b) MSs in the OPL.

Figure 5.7: RMSE performance verus different standard deviations of Gaussian noise with network layout in Fig. 5.6.

meter square space. Note that the number of MSs located in the OPL is larger than that in the IPL in this example in order to demonstrate that the OPL may frequently occur in a sensor network environment. The performance of the IPL and the OPL under the LOS condition are separately examined udner different noise standard deviation in Fig. 5.7(a) and (b), respectively. Since the difference between the L-CRLB and the CRLB is considered small in the IPL case, similar RMSE performance is observed among the three compared schemes as illustrated in Fig. 8(a). On the other hand, with the OPL scenario as shown in Fig. 8(b), the GALL-KF scheme can outperform the other two methods under different noise environments, e.g., the GALL-KF approach will result in 2.9 and 3.4 meters less of RMSE compared to the Beck-KF and TSLS-KF schemes respectively under σ_n = 20 meter in Fig. 5.7(b), which is considered the major contribution of the proposed

GALL-KF scheme. ⋄

Example 5.5 (A general case of GALL-KF scheme under realistic environment). In this example, the performance comparison is conducted for the GALL-KF scheme under the realistic environment.

The same network layout setting as example 5.4is adopted; while the noise setting is different by considering the NLOS in this example. In order to include the influence from the NLOS noise, the TOA model in cellular network as in Section2.2.1.2 is adopted. τ_m represents the median value of τ_i, which is utilized as the x-axis in Figs. 5.8(a) and 5.8(b).

Fig. 5.8(a) illustrates the performance comparison for the three schemes in the IPL under NLOS environment. It can be observed that the proposed GALL-KF approach can provide the smallest RMSE compared to the other two schemes, e.g., the GALL-KF scheme will result in 3.1 and 5.2 meters less of RMSE respectively under τm= 0.2 µs compared to the Beck-KF and

TSLS-5.5. Concluding Remarks

0.044 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 6

8 10 12 14 16 18

NLOS median value (µs)

RMSE (m)

GALL−KF Beck−KF TSLS−KF

(a) MSs in the IPL.

0.044 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 6

8 10 12 14 16 18

NLOS median value (µs)

RMSE (m)

GALL−KF Beck−KF TSLS−KF

(b) MSs in the OPL.

Figure 5.8: RMSE performance versus different NLOS median values with network layout in Fig. 5.6.

KF methods. The reason is that the GALL-KF scheme is designed based on the minimization of L-CRLB by fictitiously adjusting the locations of BS. As mentioned in Subsection II.C, the value of L-CRLB is affected by the distance between the BS and MS, which will be greatly influenced by the NLOS noises. Therefore, the effect from the NLOS noises has been implicitly considered in the design of GALL-KF scheme, which improves the location estimation performance. Moreover, Fig. 5.8(b) illustrates the performance comparison for the OPL scenario under NLOS environment.

The proposed GALL-KF scheme can still outperform the other two methods, e.g., around 1.9 and 6 meters less of RMSE in comparison with the Beck-KF and TSLS-KF schemes under τ_m= 0.2 µs in Fig. 5.7(b). The merits of proposed GALL-KF scheme can therefore be observed. ⋄

5.5 Concluding Remarks

The properties of linearized location estimation algorithms by introducing an additional vari-able are analyzed from the geometric point of view. By proposing the linearized location estimation problem based CRLB (L-CRLB), the linearization lost from the linearized location estimation algo-rithms can be observed. In order to minimize the linearization lost, the geometry-assisted linearized localization (GALL) algorithm is proposed in the chapter by fictitiously moving the base stations (BSs) in order to achieve the new geometric layout with minimum L-CRLB value. The GALL with two-step least squares (GALL-TSLS) implementation can enhance the estimation performance of the conventional TSLS estimator. By improving the initial estimation with the adoption of histor-ical information, the GALL with Kalman filter (GALL-KF) scheme further outperforms the other location estimators with similar two-stage estimation structure.

Hybrid Network/Satellite-Based