The Computation of the Angle θ k

The main objective of the GPLT scheme is to acquire the angle θ_k of x^{GP LT}_v1,k such that the predicted MS (ˆx_k|k−1) will possess a minimal GDOP value within its network topology for location estimation. As illustrated in Fig. 4.1, the following equality can be obtained based on the geometric relationship:

ˆ

x_k|k−1− x^{GP LT}_v1,k = (r^{GP LT}_v1,k · cos θ_k, r^{GP LT}_v1,k · sin θ_k) (5.3)

As mentioned above, the position of the virtual BS (x^{GP LT}_v1,k ) is designed such that the predicted MS (i.e. ˆx_k|k−1) will be located at a minimal GDOP position based on the extended geometric set P^e_BS,k = {x_1,k, x_2,k, x^{GP LT}_v1,k }. By incorporating (1) into (5.1) and (5.2), the GDOP value (i.e. G_xˆk|k−1) computed at the predicted MS’s position ˆx_k|k−1 = (ˆx_k|k−1, ˆy_k|k−1) can be obtained. The associated matrix H_xˆk|k−1 becomes

H_xˆk|k−1 =

It is noted that the noiseless relative distance ζ_i,k in (5.1) are approximately replaced by r_i,k in (5.4) since ζ_i,k are considered unattainable. It can be observed from (5.4) that the matrix H_xˆk|k−1 associated with the resulting G_xˆk|k−1 value are regarded as functions of the angle θ_k, i.e. H_xˆk|k−1(θ_k) and G_xˆk|k−1(θ_k). Based on the objective of the GPLT scheme, the angle θ^m_k which results in the minimal GDOP value can therefore be acquired as

θ^m_k = arg

By substituting (5.4) and (5.1) into (5.5), the angle θ^m_k can be computed as

θ^m_k = tan⁻¹

where

Γ = 2[r²_2,k(ˆx_k|k−1− x_1,k)(ˆy_k|k−1− y_1,k) + r²_1,k(ˆx_k|k−1− x_2,k)(ˆy_k|k−1− y_2,k)]

r_2,k² (ˆx_k|k−1− x_1,k)²− r²_2,k(ˆy_k|k−1− y_1,k)²+ r_1,k² (ˆx_k|k−1− x_2,k)²− r²_1,k(ˆy_k|k−1− y_2,k)²(5.7) It is noted that the noiseless relative distance ζ_i,k in (5.7) are replaced by r_i,k for the com-putation of Γ since ζ_i,k are in general considered unattainable. At each time instant k, the relative angle θ_k^m between ˆx_k|k−1 and x^{GP LT}_v1,k can therefore be obtained such that ˆx_k|k−1 is located at the position with a minimal GDOP value based on its current network layout.

5.2.2 The Selection of the Distance r^{GP LT}_v1,k

In this subsection, the virtual measurement r^{GP LT}_v1,k will be determined, which can be utilized for acquiring the position of the virtual BS x^{GP LT}_v1,k . It is observed in (5.4) that the GDOP value at the predicted MS’s position is primarily dominated by the relative angle (i.e. θ_k) between the MS and the BSs; while the distance information (i.e. r^{GP LT}_v1,k ) is considered uninfluential to the GDOP value. This uncorrelated relationship between the GDOP value and the relative distance has also been observed as in [22]. The following Lemma shows that the selection of the distance r^{GP LT}_v1,k becomes insignificant for the WLS-based location estimation.

Lemma 1 A time-based location estimation problem is considered for the MS using the Weighted Least Square (WLS) algorithm. Assuming that a measurement input from a specific BS is associated with zero mean random noises, the expected value of the location estimation error is independent to the distance between the specific BS and the MS.

Proof : Considering three TOA measurements are available for estimating the MS’s position (as described in (2.1) with N_k = 3), it is assumed that the third TOA measurement r_3,k is only contaminated with random noises with zero mean value, i.e. E[n_3,k] = 0 and e_3,k= 0 in (2.1). The target of this proof is to illustrate that the expected value of the estimation error resulting from the WLS method is independent to the magnitude of the measurement input r_3,k. By combining (2.1) and (2.2), the following matrix format can be obtained:

A_kb_k= J_k (5.8)

where

L represents the covariance matrix of measured noise. The primary concern of this proof is to acquire the expected value of the estimation error ∆ˆx_k= [∆ˆx_k, ∆ˆy_k]^T, which can be obtained by rewriting (5.9) as

∆ˆx_k= C(A^T_kΨ⁻¹A_k)⁻¹A^T_kΨ⁻¹∆J_k (5.12)

It is noted that (5.12) indicates that the estimation error vector ∆ˆx_k is incurred by the variation within the vector J_k. The value of ∆J_k is obtained by considering the variations

from the measurement inputs as (i.e. r_i,k = ζ_i,k+ n_i,k+ e_i,k in (2.1))

where e_3,kis considered zero as mentioned at the beginning of this proof. The approximation is valid by considering that the noiseless distance ζ_i,k is in general larger than the combined noise effect (n_i,k+e_i,k). For simplicity and without lose of generality, coordinate transformation can be adopted within (5.12) such that (x_1,k, y_1,k) = (0, 0). The expected value of the estimation error (i.e. ∆ˆx_k= [∆ˆx_k, ∆ˆy_k]^T) can therefore be acquired by expanding (5.12) as

It is noted that the second equalities for both (5.14) and (5.15) are attained based on the assumption that E[n_3,k] = 0. From (5.14) and (5.15), it can clearly be observed that the expected value of the estimation error (i.e. E[∆ˆx_k] = [E[∆ˆx_k], E[∆ˆy_k]]^T) is independent to the measured distance r_3,k under the assumption that its associated measurement noise n_3,k is considered a zero mean random variable, i.e. E[r_3,k] = E[ζ_3,k] + E[n_3,k] = E[ζ_3,k]. This completes the proof.

This lemma states that the expected value of the location estimation error is independent to the distance between a specific BS to the MS if the noises associated with the measure-ment inputs are statistically distributed with a zero mean value. In generic time-based location estimation, the phenomenon stated in Lemma 1 does not usually exist since most of the mea-surement inputs are contaminated with NLOS noises, i.e. e_i,k in (2.1) is randomly distributed

with positive mean value. The NLOS error is augmented as the distance between the specific BS and the MS is increased, which causes the corresponding measurement input to become unreliable comparing with the other signal sources. This result is consistent with the intuition that BSs with closer distances to the MS are always selected for location estimation. In the proposed GPLT scheme, the virtual measurement r^{GP LT}_v1,k is considered as a designed distance which is infected by its corresponding zero mean virtual noise n_v1,k as in (4.10). Based on Lemma 1, the selection of the distance r_v^{GP LT}_1,k becomes uninfluential to the estimation error while exploiting the WLS algorithm for location estimation. This result is similar to the derived GDOP value that is unrelated to the distance information between the BSs and the MS (as can be observed from (5.4)). In the simulation section, the uncorrelated relationship between r_v^{GP LT}_1,k and the estimation error will further be validated by exploiting the two-step LS estimator, which is considered one of the the WLS-based algorithms for location estima-tion. It will be demonstrated via the simulation results that the influence from the length of the virtual measurement to the estimation error is considered insignificant.

The procedures of the proposed GPLT scheme under the two-BSs case is explained as follows. The target is to obtain the position of the MS at the k^th time step (i.e. ˆx_k|k) based on the available information, including the measurement and location information acquired from both BS₁ and BS₂ along with the predicted position of the MS (i.e. ˆx_k|k−1). Two steps are involved within the proposed GPLT scheme: (i) the determination of the virtual BS’s position and the virtual measurement; and (ii) the estimation and tracking of the MS’s position. As shown in Fig. 4.1, the orientation of the virtual BS (θ_k^m) relative to the the predicted MS’s position ˆx_k|k−1 is determined based on the criterion of minimizing the GDOP value on ˆx_k|k−1(as obtained from (5.5) and (5.6)). As was indicated by Lemma 1 in Subsection V.A.(2), the selection of the virtual distance r^{GP LT}_v1,k w.r.t. the predicted MS’s position ˆx_k|k−1 is considered insignificant to the estimation errors. Therefore, the distance is selected the same value as was designed in the PLT algorithm, i.e. r_v^{GP LT}_1,k = r_v^{P LT}_1,k as in (4.5). The location of the virtual BS (x^{GP LT}_v1,k ) and the length of the virtual measurement (r_v^{GP LT}_1,k ) can consequently be acquired. It is also noticed that the design of the virtual noise can therefore be selected

the same as that in the PLT scheme, i.e. zero mean random distributed with variance σ_n²

v1,k

= Var(r^{P LT}_v1,k ) = Var(kˆx_k|k−1− ˆx_k−1|k−1k).

After acquiring the information of the virtual BS as the additional signal source, the ex-tended sets of the BSs and the measurement inputs can be established as P^e_BS,k= {x_1,k, x_2,k, x^{GP LT}_v1,k } and r^e_k = {r_1,k, r_2,k, r^{GP LT}_v1,k }. As illustrated in Fig. 3.3, the extended set of signal sources are utilized as the inputs to the two-step LS estimator. The estimated MS’s position ˆx_k|k can therefore be obtained by adopting the correcting phase of the Kalman filter, which completes the location estimation and tracking processes at the k^th time step.

5.3 The Single-BS Case

As illustrated in Fig. 4.2, only one BS (x_1,k) associated with the measurement input r_1,k is available at the considered k^th time instant. Additional two virtual BSs associated with their virtual measurements are required as the inputs for the two-step LS estimator, i.e.

P^{GP LT}_BSv,k = {x^{GP LT}_v1,k , x^{GP LT}_v2,k } and r^{GP LT}_v,k = {r^{GP LT}_v1,k , r^{GP LT}_v2,k }. By adopting the design from the PLT scheme with the single-BS case, the first virtual BS is designed to be located at x^{GP LT}_v1,k = ˆx_k−1|k−1 associated with the first virtual measurement r^{GP LT}_v1,k as defined in (4.5).

The second virtual measurement r_v^{GP LT}_2,k is also designed to be the same as in the PLT scheme (in (4.9)), which considers the averaged prediction error from the previous time steps.

As shown in Fig. 4.2, the position of the second virtual BS (x^{GP LT}_v2,k ) is designed at a location with distance r_v^{GP LT}_2,k relative to the predicted MS’s position ˆx_k|k−1. The relative angle θ_k^m be-tween x^{GP LT}_v2,k and ˆx_k|k−1is determined by minimizing the GDOP value based on the predicted MS’s position ˆx_k|k−1. Both of the information from BS₁ and BS_v1 alone with the predicted MS’s position ˆx_k|k−1are utilized for the computation of the angle θ_k^m(as in (5.5) and (5.6)). It is noticed that instead of altering the position of BS_v1, the BS_v2’s location is adjusted in order to acquire a better GDOP value for the predicted MS ˆx_k|k−1. The design concept is primarily owing to the fact that the average prediction error is in general smaller than the length of each prediction within the Kalman filtering formulation, i.e. r_v^{GP LT}_1,k > r^{GP LT}_v2,k . The expected MS’s position ˆx_k|k−1is considered more sensitive to r^{GP LT}_v2,k due to its smaller value comparing

with r_1,k and r^{GP LT}_v1,k . It will be beneficial to adjust the location of BS_v2 (by rotating the angle θ_k^m) such that a smaller GDOP value can be achieved at the predicted location of the MS (ˆx_k|k−1).

As indicated by Lemma 1, the selection of the virtual measurement r_v^{GP LT}_2,k is considered insignificant on the precision for location estimation. Nevertheless, the distance r_v^{GP LT}_2,k is chosen as in (4.9) in order to facilitate the design of the weighting coefficient associated with the two-step LS estimator. Similar to the design within the PLT scheme, the virtual noise associated with the second virtual measurement r^{GP LT}_v2,k can be regarded as zero mean with variance σ²_n

v2,k = Var(r_v^{GP LT}_2,k ). Therefore, the information from the additional two virtual measurements r_v^{GP LT}_1,k and r^{GP LT}_v2,k can be acquired such as to provide sufficient signal sources for the two-step LS location estimator. The precision for location estimation and tracking of the MS can consequently be enhanced.

Chapter 6

Performance Evaluation

Simulations are performed to show the effectiveness of the proposed PLT and GPLT schemes under different numbers of BSs, including the scenarios with deficient signal sources. The noise models and the simulation parameters are illustrated in Subsection A. Subsection B validates the GPLT scheme according to the variations from the relative angle and the distance between the MS and the designed virtual BS. The performance comparison between the proposed PLT and GPLT algorithms with the other existing location tracking schemes, i.e. the Kalman Tracking (KT) and the Cascade Location Tracking (CLT) techniques, are conducted in Subsection C.

6.1 The Noise Models and the Simulation Parameters

Different noise models [28] [47] for the the TOA measurements are considered in the simula-tions. The model for the measurement noise of the TOA signals is selected as the Gaussian distribution with zero mean and 10 meters of standard deviation, i.e. n_i,k ∼ N (0, 100) . On the other hand, an exponential distribution p_ei,k(τ ) is assumed for the NLOS noise model of the TOA measurements as

p_ei,k(υ) =







λi,k1 exp

³

−_λ^υ

i,k

´

υ > 0

0 otherwise

(6.1)

−500 0 500 1000 1500 2000 2500 3000

Figure 6.1: An Exemplify Diagram for the Scenarios with the Two-BSs Layout. Stars (x_v,1(30.2^o) and x_v,1(210.9^o)): the Positions of the Virtual BS Cause the Minimal GDOP Value of the MS; Squares (x_v,1(120.5^o) and x_v,1(300.5^o)): the Positions of the Virtual BS Cause the Maximal GDOP Value of the MS

where λ_i,k = c · τ_i,k = c · τ_m(ζ_i,k)^ερ. The parameter τ_i,k is the RMS delay spread between the i^th BS to the MS. τ_m represents the median value of τ_i,k, which is selected as 0.1 in the simulations. ε is the path loss exponent which is assumed to be 0.5, and the factor for shadow fading ρ is set to 1 in the simulations. The parameters for the noise models as listed in this subsection primarily fulfill the environment while the MS is located within the rural area. It is noticed that the reason for selecting the rural area as the simulation scenario is due to its higher probability to suffer from deficiency of signal sources. Moreover, the sampling time ∆t is chosen as 1 sec in the simulations.