GALE with One Movable Fictitious BS Scheme

The GALE(1BS) scheme is designed to fictitiously relocate the position of one BS accord-ing to the minimal GDOP criterion. Under this condition only one BS is defined to be fictitious movable and others are fixed.

4.1.1 GDOP-Assisted (GOLE) Location Estimation Scheme

Without lose of generality, it is considered that BS1 (i.e. x1,k) is the adjustable BS within the GALE(1BS) scheme. The position of the fictitious BS₁ is designed such that the

initial estimated MS (ˆx^o_k) will be located at a minimal GDOP position based on the existing geometric layout P_BS,k = {x_1,k, x_2,k, x_3,k}. In other words, based on the initial location estimate ˆx^o_k associated with the information coming from the BSs (i.e. r_k and PBS,k), the three relative angles α1,k, α2,k, and α3,k between the BSs w.r.t. the MS can be obtained. By adopting the results from Lemma 1, the minimal attainable GDOP G_xˆ^o_k w.r.t. the MS’s initial estimate ˆx^o_k occurs as the angle α_1,k is adjusted as

α^m_1,k = 1 2tan⁻¹

µ − sin(2α2,k) cos(2α_2,k) + 1

¶

(4.1)

It is noted that the angle α_2,k between BS₂ and BS₃ is considered a fixed value; while α_3,k is dependent to the variable angle α_1,k, i.e. α_3,k = (2π − α_2,k) − α_1,k. The following lemma generalizes the solution for the angle α^m_1,k that achieves minimal GDOP value.

Lemma 5. Considering the case that the MS is surrounded by three available BSs, i.e.

BS₁, BS₂, and BS₃. It is assumed that only the location of BS₁ is adjustable; while the positions of the other two BSs are considered fixed. The minimal GDOP occurs as BS1 is situated at the angle that equally bisects the angle formed by BS2 and BS3.

Proof. According to (4.1), α^m_1,krepresents the angle that achieves the minimal GDOP value w.r.t. the MS’s initial estimate. It is clear to conclude that (4.1) holds if α_2,k = 2π −2α^m_1,k. Consequently, the minimal GDOP occurs as BS₁ is positioned at the angle that equally bisects the angle formed by the other two BSs, i.e.

α^m_1,k = 2π − α_2,k

2 (4.2)

It is observed from Lemma 5 that the fictitiously movable BS₁ should be adjusted such that the angles α_1,k and α_3,k are equal. As a result, the new set of BSs for location

estimation is obtained as P⁽¹⁾_BSf,k = {x_{f 1,k}, x_2,k, x_3,k}, where x_{f 1,k} denotes the location of the fictitious BS as

xf 1,k = r1,kcos(θ2,k− α^m_1,k)

y_{f 1,k} = r_1,ksin(θ_2,k− α^m_1,k) (4.3)

where α^m_1,k is obtained from (4.2). The set of updated locations for the BSs P⁽¹⁾_BSf,k associated with the original TOA measurements r_k = {r_1,k, r_2,k, r_3,k} are exploited to conduct the second-phase two-step LS method as shown in Fig. 4.1. Consequently, the MS’s final location estimation ˆx^f_k = [ˆx^f_kyˆ^f_k] by adopting the proposed GALE(1BS) scheme can be obtained.

4.1.2 MOM-Assisted (MOLE) Location Estimation Scheme

By using the property of MOM we can design some scheme to implement the system as well. For the mathematical model derived from the chapter 3 . The fictitious position of the adjustable BS can be obtained in order to fit the lowest MOM. Assume that BS₁ is adjustable, the fictitious BS1 will be relocated at x1,k in order to make the MS (ˆx^o_k) locate at the lowest MOM.

In other words, whenever we get the MS’s initial coordinate ˆx^o_k and the angel α_2,k is considered a fixed value,we can get new fictitious BS set P_BS,kf. By adopting the results from Lemma 3, the minimal attainable MOM G_xˆ^o_k w.r.t. the MS’s initial estimate ˆx^o_k occurs as the angle α_1,k is adjusted as

α^m_1,k = 1 2tan⁻¹

µ − sin(2α_2,k) cos(2α2,k) + N

¶

(4.4)

the notation N is equal to (^r_r^3,k

2,k)², and α_3,k is dependent to the variable angle α_1,k, i.e.

α_3,k = (2π − α_2,k) − α_1,k. The following lemma generalizes the solution for the angle α_1,k^m that achieves minimal MOM value.P⁽¹⁾_BSf,k = {x_{f 1,k}, x_2,k, x_3,k}, where x_{f 1,k} denotes the location of the fictitious BS as

x_{f 1,k} = r_1,kcos(θ_2,k− α^m_1,k)

yf 1,k = r1,ksin(θ2,k− α^m_1,k) (4.5)

where α^m_1,k is obtained from (4.4). The set of updated locations for the BSs P⁽¹⁾_BSf,k associated with the original TOA measurements r_k = {r_1,k, r_2,k, r_3,k} are exploited to conduct the second-phase two-step LS method as shown in Fig. 4.1. Consequently, the MS’s final location estimation ˆx^f_k = [ˆx^f_kyˆ^f_k] by adopting the proposed GALE(1BS) scheme can be obtained.

Lemma 6. We can implement the result derived from MOM to the GALE as well. As-sumed that only the location of BS₁ is adjustable. If r_2,k = r_3,k, the minimal MOM occurs as BS₁ is situated at the angle that equally bisects the angle formed by BS₂ and BS₃ . Therefore, BS₁ ,BS₂ and BS₃ forms a isosceles triangle. From this Lemma ,it indicates that minimum MOM value occurs when all BSs composing a symmetric layout.

Proof. According to (4.4), α^m_1,k represents the angle that achieves the minimal MOM value w.r.t. the MS’s initial estimation. If r2,k = r3,k, α^m_1,k will have the same value as (4.1). It also holds the solution α_2,k = 2π − 2α_1,k^m .

It indicates that BS₁ will forms a isosceles triangle with BS₂ and BS₃ if the distance from MS to BS₂ is equal to distance from MS to BS₃ at this time.

4.1.3 Coverage-Maximize (CMLE) Location Estimation Scheme

The objective of the proposed GALE algorithms is to utilize the initial location informa-tion acquired from the BS to serve as the assisted measurement inputs. Besides, We design

BS1

BS2

BS3 MS

ζ

ζ

ζ

θ θ

1

12

,k 21

Figure 4.2: The location information for TOA signal

another algorithm to implement the location algorithm. For TOA system as fig.4.2, the error depend on the undetermine region (the area surrounded by dotted line). Because GDOP and MOM are designed in a noise-free environment, we can decrease the region to colored section. The object is to minimize the region and make the total coverage area maximize at the same time.This paper we call it the Coverage Maximum Location Estimation(CMLE).

Analogous to the GALE algorithm by adding geometric constraints within the con-ventional two-step LS method, the CMLE algorithm extends the concept of ”virtual”

assistances in the GALE algorithm to add the geometric constraints from the assisted information. The same scheme as GALE, Area model also have 1BS and 2BS scheme.

We set PBSf,k in order to make the area of colored region have a minimum value. The value is with respect to both related distance and angle. For example, the cross-section

dorm by BS₁ and BS₂ can be expressed as

Area = 1

2ζ_1,k² (θ_a1− sin θ_a1) (4.6)

and θ_i,1 is w.r.t α_i,k. Therefore the total area will be related to α_i,k.

Corollary 3. Following by the GALE 1BS algorithm, each scheme can design its own relocated BS set P⁽²⁾_BSf,k = {xf 1,k, xf 2,k, xf 3,k} as above scheme shows. For the circumstance that ζ_2,k= ζ_3,k, the proposed BS set P⁽²⁾_BSf,k will have the same position.

Proof. Assume all the relative distance are equal, we can take this data into each scheme.

When ζ_2,k = ζ_3,k, the proposed answer α_i,k^m will have same solution of α_1,k = ^2π−α₂ ^m2,k. As we have defined before,

xf 1,k = r1,kcos(θ2,k− α^m_1,k)

y_{f 1,k} = r_1,ksin(θ_2,k− α^m_1,k) (4.7)

This equation shows that the BS₁ position will be the same.