Proposed Geometry-Assisted Linearized Localization (GALL) AlgorithmAlgorithm

are drawn in Section5.5.

5.2 Proposed Geometry-Assisted Linearized Localization (GALL) Algorithm

The main objective of the proposed GALL scheme is to enhance the LLS-based algorithms by considering the geometric effect to the location estimation accuracy. The signal model for the TOA measurements in a synchronous network is utilized for two-dimension (2-D) location estimation,

5.2. Proposed Geometry-Assisted Linearized Localization (GALL) Algorithm

which can be referred to Section 2.1.1. The core component of the GALL scheme is to acquire the positions of the fictitious BSs such as to achieve the minimum L-CRLB value with respect to the MS’s initial location estimate ˆx^o. Note that the position of the i-th fictitious BS with respect to the MS’s initial location estimate can be represented based on the measurement distance r_i and the BS’s orientation α_i. Since r_iis available as the measured information, the determination of fictitious BS’s position corresponds to the adjustment of the BS’s orientation α_i. The position information of these fictitious BSs will be utilized to replace that of the original BSs in order to achieve better geometric layout for location estimation, which will be discussed as the implementation of the GALL scheme in Section5.3. In this section, the core mechanism of the GALL algorithm to identify which BSs should be fictitiously rotated will be demonstrated. The sub-schemes of the GALL algorithm with different numbers of fictitious BSs will be stated, i.e., the GALL with one fictitiously movable BS scheme (GALL(1BS)) and the GALL with two fictitiously movable BSs scheme (GALL(2BS)) in Subsections 5.2.1and5.2.2 respectively. Subsection5.2.3 describes the combined schemes of the GALL algorithm by selecting among different numbers of fictitious BSs based on the minimum L-CRLB requirement.

5.2.1 GALL with One Fictitiously Movable BS (GALL(1BS)) scheme

(x, y)

(a) An example of one fictitiously movable BS centered at the true MS’s position.

r₃

(b) An example of one fictitiously movable BS centered at the estimated MS’s position.

Figure 5.1: Schematic diagrams of GALL with one fictitiously movable BS scheme.

The GALL(1BS) scheme is designed to fictitiously relocate the position of one BS according to the criterion for achieving the optimal geometric layout, i.e., the minimum L-CRLB. Note that

only one BS is allowed to be fictitiously movable and the others remain fixed in this case. The objective of the GALL(1BS) scheme is to decide which BS should be fictitiously moved in order to obtain the minimum L-CRLB. The GALL(1BS) problem is defined as

j^m = arg min

C_L(˜α^m_j)<CL

j=1,...,N

C_L(˜α^m_j ),

(5.1)

where ˜α^m_j represents the orientation of the j^m-th fictitiously moveable BS that achieves the minimum L-CRLB. Providing that all the BSs within GALL(1BS) scheme cannot result in lowered L-CRLB values compared to the original CL, the constraint defined in (5.1) will not be satisfied. In other words, the original L-CRLB has already been the lowest under the given measurement conditions, where none of the BSs is required to be fictitiously moved and the initial MS’s location estimate will become the final estimate. By observing from the problem defined in (5.1), the optimal rotated angle ˜α^m_j of a single BS should be determined first. The one fictitiously movable BS problem is defined to obtain the optimal rotated angle of the j-th BS as

˜

α^m_j = arg min

∀˜αj

C_L(˜αj), ∀˜αj = [0^◦, 360^◦). (5.2) Note that the original H matrix in (4.15) for the computation of C_L cannot be obtained owing to the required true MS’s position and noiseless relative distances. The estimated matrix ˆH can be calculated based on the initial estimate ˆx^o and the measurement distance r as

H =ˆ

where ˆα_iin (5.3) represents the i-th BS’s estimated orientation based on the initial estimate ˆx^o. The parameters ˜b_j= (˜x_j, ˜y_j) and ˜α_j in (5.3) denote the position and orientation of the j-th fictitiously moveable BS, respectively. It is noticed that the design of proposed GALL scheme also considers the effect coming from the approximation of matrix ˆH. As shown in Fig. 5.1(b), the orientation of the fictitiously moved BS is considered to center at the MS’s estimated position instead of the true MS’s position as in Fig. 5.1(a). It can be observed that the selection of fictitiously moveable BS may induce additional error since it is designed based on the imperfect initial estimate even though it can provide better geometry for location estimation. This demonstrates the situation that the fictitiously moveable BS may not always result in lowered L-CRLB value than the original network layout as stated in the constraint of problem (5.1). Note that the initial estimation error

5.2. Proposed Geometry-Assisted Linearized Localization (GALL) Algorithm

can be approximated as a Gaussian noise distribution with the standard deviation of the initial estimation error σ_re, i.e., N(0, σ²_re) where σ_re depends on the precision of the initial estimate ˆ

x^o. Providing that the LLS-based estimation is utilized for the initial estimation, the matrix Iζ

for the computation of C_L can be derived based on the precision of MS’s initial estimate σ_re as I_ζ = diag{[σr⁻²1 , . . . , σ⁻²_ri , . . . , (σ_r²_j + σ²_re)⁻¹, . . . , σ_r⁻²_N]}. The L-CRLB CL(˜α_j) in (5.2) can thus be where D_θ denotes for the determinant of FIM matrix I_θ. By neglecting the terms in (5.4) that are not related to the parameter ˜α_j, the equivalent one fictitiously movable BS problem as presented in problem (5.2) can be obtained as

˜

where f₁(˜α_j) can be regarded as the cost function of the considered problem. It can be observed that the solution of problem (5.5) can be acquired if the following conditions on the first and second derivatives of f₁(˜α_j) are satisfied, i.e.,

Due to the complex formulation of (5.5) - (5.7), there does not exist closed form for obtaining the optimal value of ˜α^m_j . In order to solve the optimum rotated angle ˜α^m_j , root-finding algorithms can be utilized to find suitable solution candidates between [0, 360^◦) in (5.6), and these solutions will further be examined to satisfy the requirement of (5.7). If there are still multiple candidates that fits all the requirements, i.e., there are multiple local minimums for problem (5.2), the angle

˜

α^m_j that possesses with the global minimum L-CRLB value will be chosen from (5.4) among those solution candidates.

Furthermore, in order to reduce the computation complexity, a cost function g1(˜αj) is defined

to simplify the original problem (5.2) without the consideration of D_θ in (5.5). An approximate one fictitiously movable BS problem can therefore be obtained as

˜

It is interesting to notice that the cost function g1(˜αj) can be closely related to the conditions stated in (4.18) for Corollary4.1. Providing that the minimum value of g₁(˜α_j) approaches zero, the two conditions in (4.18) can be satisfied by solving problem (5.8). In other words, by fictitiously moving the j-th BS via angle ˜α^m_j with the consideration of σre for initial estimate, the layout with the smallest linearization lost can possibly be achieved where the L-CRLB is equivalent to the CRLB. Therefore, based on the design of fictitious movable BS, the two square terms within g₁(˜α_j) can be treated as the extension of the two error terms ε1 and ε2 as described after Corollary 4.1 that affect the precision of linearized location estimators. By considering the first derivative of g₁(˜α_j) equal to zero, the rotated angle ˜α^m_j for this problem can be derived as

Note that the angle ˜α^m_j does not depend on any information from the j-th BS, i.e., the measurement of the j-th BS. The angle ˜α^m_j lies in the domain of arc tangent function between (−90^◦, 90^◦) which is half of the domain of [0, 360^◦). Since both angles ˜α^m_j and ˜α^m_j + 180^◦ can be the local minimum of the subproblem (5.8), these two angles will be further substituted into (5.4) to choose the angle with smaller L-CRLB value. Following the procedures as stated above, all the BSs can be fictitiously moved and the associated rotated angle ˜α^m_j can be obtained. The j^m-th BS with the minimum L-CRLB value will be selected to be the fictitiously moveable BS for the GALL(1BS) scheme in problem (5.1). For example as shown in Fig. 5.1, only the third BS, i.e., j^m = 3, is fictitiously adjusted and the other two BSs remain at the same position. The j^m-th fictitiously moved BS will be relocated to the coordinate as

( ˜x_j^m = r_j^mcos(˜α^m_j )

˜

y_j^m= r_j^msin(˜α^m_j ) . (5.10) Note that the measurement of the j^m-th BS remains the same as rj^m. The noise variance of this measurement is recalculated as σ_r²m

j + σ_r²_e. Based on the new set of BSs adjusted by the proposed GALL(1BS) scheme, the LLS-based estimation can be adopted to obtain the final estimation of

5.2. Proposed Geometry-Assisted Linearized Localization (GALL) Algorithm

MS’s position.

5.2.2 GALL with Two Fictitiously Movable BSs (GALL(2BSs)) Scheme

The GALL(2BSs) scheme is designed to fictitiously relocate the position of two BSs according to the minimal L-CRLB layout criterion. Under this condition, two BSs are defined to be fictitiously movable and the others are fixed. The objective of the GALL(2BSs) scheme is to select the specific two BSs that should be fictitiously moved in order to achieve the layout with the minimum L-CRLB.

The GALL(2BSs) problem is defined as

{j^m, k^m} = arg min

C_L(˜α^m_j , ˜α^m_k)<CL

j=1,...,N,k=1,...,N,j6=k

C_L(˜α^m_j , ˜α^m_k),

(5.11)

where ˜α^m_j and ˜α^m_k represent the orientation of two fictitiously moveable BSs. The constraint in (5.11) is to verify if the original L-CRLB has already been the lowest under the given measurement conditions. Before solving problem (5.11), the optimum rotated angles ˜α^m_j and ˜α^m_k of the two fictitiously movable BSs should be decided first. The two fictitiously movable BSs problem is defined to find the optimum rotated angles as

{˜α^m_j , ˜α^m_k} = arg min

∀˜αj, ˜αk,j6=k

C_L(˜αj, ˜α_k), ∀˜αj, ˜α_k= [0^◦, 360^◦). (5.12)

Note that the matrix Iζ in C_L(˜α_j, ˜α_k) can be obtained as Iζ = diag{[σ⁻²r1 , . . ., (σ²_rj+ σ_r²_e)⁻¹, . . ., (σ²_rk+ σ_r²_e)⁻¹, . . ., σ⁻²_rN]}, where the standard deviation σ^re of MS’s initial estimate is considered in both the j-th and k-th fictitiously movable BSs. Therefore, the L-CRLB can be derived as

C_L(˜α_j, ˜α_k) = 1

Similar to the GALL(1BS) scheme, the equivalent two fictitiously movable BSs problem for problem (5.12) can also be acquired as

{˜α^m_j , ˜α^m_k} = arg min

∀˜αj, ˜αk,j6=kf₂(˜α_j, ˜α_k), (5.14)

where f₂(˜α_j, ˜α_k) is regarded as the cost function of problem (5.14) as

It can be observed that problem (5.14) can be solved if the following conditions on the first deriva-tives of f₂(˜α_j, ˜α_k) are satisfied, i.e.,

Since the closed form solution of ˜α^m_j and ˜α^m_k can not be obtained in this case, numerical methods can be utilized to acquire the optimal angles ˜α^m_j and ˜α^m_k for achieving minimum L-CRLB for problem (5.14). To reduce the computation complexity, a cost function g2(˜αj, ˜α_k) is defined to simplify the original problem (5.12) into an approximate two fictitiously movable BSs problem

{˜α^m_j , ˜α^m_k} = arg min

The closed form solution of (5.18) for the N = 3 case is illustrated as follows. For ease of computation, the three BSs’ orientation can be represented by their adjacent included angles as

ˆ

α= {ˆα₁ = ˆα₂− ˆβ₁, ˆα₂, ˆα₃ = ˆα₂+ ˆβ₂} as shown in Fig. 5.1. The GALL(2BSs) scheme fictitiously re-locates the positions of two BSs among the three which can be denoted as ˜α= {ˆα₂− ˜β₁, ˆα₂, ˆα₂− ˜β₂}.

5.2. Proposed Geometry-Assisted Linearized Localization (GALL) Algorithm

Furthermore, according to the rotation property, the orientation of the fictitiously moved BSs can be transformed as ˜α = {− ˜β₁, 0^◦, − ˜β₂}. By considering the first derivative equation in (5.16), the BSs’ adjacent included angles for achieving the minimum CRLB are calculated as

β˜₁^m= cos⁻¹ σ_r⁴₁r₁²σ_r⁴₂r²₂− σr⁴2r₂²σ_r⁴₃r₃²− σr⁴1r₁²σ_r⁴₃r₃²

2σ_r²₁r1· σ²r2r2· σr⁴3r₃² , (5.20) β˜₂^m= cos⁻¹ σ_r⁴₂r₂²σ_r⁴₃r₃²− σr⁴1r₁²σ_r⁴₃r₃²− σr⁴1r₁²σ_r⁴₂r₂²

2σ_r⁴₁r₁²· σr²2r2· σr²3r3

, (5.21)

where the angle ˜β_j^m for j = 1 and 2 lies in the domain of arc cosine function from [0, 180^◦] which is half of the domain [0, 360^◦). Both angles ˜β_j^m and ˜β_j^m+ 180^◦ are considered the local minimums of subproblem (5.18). Therefore, these angles will be substituted into (5.13) to determine the angle with smaller L-CRLB value. After the adjacent included angles are calculated, the BSs’ orientation of the GALL(2BSs) scheme can be acquired as ˜α^m = {˜α^m₁ = ˆα2 − ˜β^m₁ , ˆα2, ˜α^m₃ = ˆα2 + ˜β₂^m}.

Accordingly, all the BSs can be fictitiously moved and the associated rotated angle ˜α^m_j and ˜α^m_k will be obtained. The j^m-th and k^m-th BSs with the minimum L-CRLB value are decided to be the fictitiously moveable BS for the GALL(2BSs) scheme in problem (5.11). The positions of j^m-th and k^m-th BSs can be fictitiously relocated and computed based on (5.10). Note that the measurement remains the same while the noise variance of the measurement is recalculated by considering the initial estimation error. With the new set of BSs obtained from the GALL(2BSs) scheme, the LLS-based estimation algorithms can be adopted to obtain the final MS’s location estimation. Based on the derivation of proposed GALL(2BSs) scheme, the number of fictitiously movable BSs can also be increased by extending the GALL scheme with multivariable optimization, i.e., GALL(3BSs), GALL(4BSs).

5.2.3 GALL Scheme

Based on Subsections 5.2.1 and 5.2.2, it can be observed that the GALL(2BSs) scheme pro-vides one more degree of freedom compared to the GALL(1BS) scheme, which should increase the precision of location estimation owing to the enhancement from the geometric effect. However, the two fictitiously moveable BSs associate with the initial MS’s estimation error may degrade the performance for location estimation. Therefore, the GALL scheme is designed to select between the two sub-schemes, i.e., GALL(1BS) and GALL(2BSs), in order to achieve minimum L-CRLB value among all different cases. The GALL problem can be defined as

˜

α^m = arg min

˜

α^m_1BS, ˜α^m_2BS C_L( ˜α^m_1BS),CL( ˜α^m_2BS)<CL

[C_L( ˜α^m_1BS), C_L( ˜α^m_2BS)],

(5.22)

where ˜α^m_1BS and ˜α^m_2BS represent the sets of BS’s orientation with the lowest L-CRLB by adopting the GALL(1BS) and GALL(2BSs) schemes, respectively. Each of the fictitiously moved BS set

is selected according to the minimum L-CRLB criteria. Note that if both the GALL(1BS) and GALL(2BSs) schemes cannot provide a lower L-CRLB scenario compared to the original unmoved version, the GALL scheme will choose the original BS’s positions for MS’s location estimation according to the constraint in (5.22).

5.3 Implementations of Geometry-Assisted Linearized Location