GDOP-Assisted Location Estimation Algorithms in
Wireless Location Systems
Lin-Chih Chu, Po-Hsuan Tseng, and Kai-Ten Feng
Department of Communication Engineering National Chiao Tung University
Hsinchu, Taiwan
{iwill.cm95g, walker.cm90}@nctu.edu.tw and ktfeng@mail.nctu.edu.tw Abstract—In recent years, wireless location estimation has
attracted a significant amount of attention in different areas. The network-based location estimation schemes have been widely adopted based on the radio signals between the mobile station (MS) and the base stations (BSs). The two-step Least Square (LS) method has been studied in related research to provide efficient location estimation of the MS. However, the algorithm results in inaccurate location estimation under the circumstances with poor geometric dilution of precision (GDOP). In this paper, the GDOP-assisted location estimation (GOLE) schemes are proposed by considering the geometric relationships between the MS and its associated BSs. According to the minimal GDOP criterion, the BSs are fictitiously repositioned and are served as a new set of BSs within the formulation of the two-step LS algorithm. The proposed GOLE schemes can both preserve the computational efficiency from the two-step LS method and obtain precise location estimation under poor GDOP environments. Comparing with other existing schemes, numerical results demonstrate that the proposed GOLE algorithms can achieve better accuracy in wireless location estimation.
Index Terms—Wireless location estimation, geometric dilution of precision (GDOP), time-of-arrival (TOA).
I. INTRODUCTION
Wireless location technologies, which are designated to estimate the position of a Mobile Station (MS), have drawn a lot of attention over the past few decades. Different types of location-based services (LBSs) [1] have been proposed and studied, including the emergency 911 (E-911) subscriber safety services, the navigation system, and applications for the wireless sensor networks (WSNs) [2]. Due to the emergent interests in the LBSs, it is required to provide enhanced precision in the location estimation of a MS under different environments.
A variety of wireless location techniques have been stud-ied and investigated. The network-based location estimation schemes have been widely proposed and employed in the wireless communication system. These algorithms locate the position of the MS based on the measured radio signals either from its neighborhood base stations (BSs) in the cellular-based networks or the sensor nodes (SNs) in the WSNs. The major time-based methods for the network-based location
1This work was in part funded by the MOE ATU Program 95W803C, NSC
96-2221-E-009-016, MOEA 96-EC-17-A-01-S1-048, the MediaTek research center at National Chiao Tung University, and the Universal Scientific Indus-trial (USI) Co., Taiwan.
estimation techniques are the time-of-arrival (TOA) and the time difference-of-arrival (TDOA). The TOA scheme measures the arrival time of the radio signals coming from different wireless BSs; while the TDOA scheme measures the time difference between the radio signals.
The equations associated with the network-based location estimation schemes are inherently nonlinear. The uncertainties induced by the measurement noises make it more difficult to acquire the MS’s estimated position with tolerable precision. Different approaches have been proposed to obtain an approx-imate location estimation in the previous studies [3] [4]. The Taylor series expansion (TSE) method was utilized in [3] to acquire the location estimation from the time measurements. The scheme requires iterative processes to obtain the location estimate from a linearized system. The major drawback of the TSE method is that it may suffer from the convergence problem due to an incorrect initial guess of the MS’s position. The two-step least square (LS) method [4] was adopted as an approximate realization of the maximum likelihood estimator, which does not require iterative processes. It is observed that feasible location estimation can be achieved by adopting these algorithms while the time measurements are symmetrically located w.r.t. the MS. However, asymmetrical measurement inputs to the MS in general result in degraded precision for location estimation. It is especially noticed that this type of situation can frequently occur under non-regular shapes of geometric layouts, e.g. under the randomly distributed sensor networks.
In order to consider the geometric effect to the accuracy of location estimation, the well-known geometric dilution of precision (GDOP) [5] metric can be adopted to facilitate the design of location estimation algorithms. The GDOP is utilized as an index for observing the location precision of the MS under different geometric positions within the networks, e.g. the cellular, the satellite, or the sensor networks. The work in [6] describes the effect and cost resulting from the network topology with significant GDOP values. However, the method for mitigating the GDOP effect has not been extensively addressed in previous studies. The ridge regression signal processing [7] is proposed for reducing the effects of GDOP in the position-fixed navigation systems. Nevertheless, a pre-filtered set of initial range measurements is required before the proposed estimation method can be activated.
In this paper, two GDOP-assisted location estimation (GOLE) algorithms are proposed to enhance the estimation precision by incorporating the GDOP information within the conventional two-step LS algorithm. Based on an initial esti-mate of the MS’s location, the proposed GOLE schemes deter-mine the fictitious locations of the BSs such that the estimated MS will be relocated at a position with the minimal GDOP value. According to the GDOP criterion, the GOLE(1BS) and GOLE(2BS) algorithms are proposed to consider the cases by fictitiously rotating (i.e. not physically relocate) one and two BS’s locations respectively. Reasonable location estimation can be acquired within the GOLE algorithms, especially feasi-ble for the cases with poor GDOP circumstances. Simulation results illustrate that the proposed GOLE schemes can achieve higher accuracy for the MS’s estimated location compared to the other existing methods.
The remainder of this paper is organized as follows. Section II describes the measurement models and the GDOP metric. The proposed GOLE algorithms are explained in Section III; while Section IV shows the performance evaluation of the proposed schemes. Section V draws the conclusions.
II. PRELIMINARIES A. Measurement Models
The signal model for the TOA measurements is uti-lized in this paper. The set rk contains all the available measured relative distance at the kth time step, i.e. rk = {r1,k, . . . , ri,k, . . . , rNk,k}, where Nk denotes the number
of available BSs at the time step k. The measured relative distance (ri,k) between the MS and the ith BS (obtained at the kth time step) can be represented as
ri,k= c · ti,k = ζi,k+ ni,k+ ei,k i = 1, 2, ..., Nk (1)
where ti,k denotes the TOA measurement obtained from the ith BS at the kth time step, and c is the speed of light. ri,k
is contaminated with the TOA measurement noise ni,k and the non-line-of-sight (NLOS) error ei,k. It is noted that the measurement noiseni,kis in general considered as zero mean with Gaussian distribution. On the other hand, the NLOS error ei,k is modeled as exponentially-distributed for representing
the positive bias due to the non-line-of-sight effect [8]. The noiseless relative distance ζi,k (in (1)) between the MS’s true position and the ith BS can be obtained as
ζi,k= [(xk− xi,k)2+ (yk− yi,k)2]12 (2) where xk = [xk yk] represents the MS’s true position and xi,k = [xi,k yi,k] is the location of the ith BS for i = 1 to Nk. Therefore, the set of all the available BSs at thekth time step can be obtained as PBS,k = {x1,k, . . . , xi,k, . . . , xNk,k}. B. GDOP Metric
The GDOP metric is utilized to describe the geometric effect on the relationship between the measurement error and the position determination error [5]. Fig. 1 illustrates the schematic diagram of the network layout for the GDOP computation. In general, a larger GDOP value corresponds
x y x y x y k k k k k k 1 1 2 2 , , , , , , , ( ) ( ) ( ) BS1 BS2 BS BS x3,k,y3,k ( ) BS3 ζ3,k ζ2,k θ1,k θ2,k θ3,k α1,k α3-l ζ1,k xN kk,,yN kk, ( ) MS x y θ α α ζ ζ N k N k N k k k N k N k k k k k k N -1 N , , , , , − − 1 1 Nk k Nk k x −,,y −,
(
1 1)
α24-l =α 12,k =α 23,kFig. 1. Schematic diagram of the network layout for GDOP computation.
to a comparably worse geometric layout (established by the MS and its associated BSs), which consequently results in augmented errors for location estimation. On the other hand, as the GDOP value becomes smaller, the effect from the geometric relationship to the location estimation accuracy will turn out to be insignificant. Considering the MS’s location under the two-dimensional coordinate, the GDOP value (G) obtained at the MS’s true position xk can be represented as
Gxk= trace(HTxkHxk)−1 1 2 (3) where Hxk = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ xk−x1,k ζ1,k yk−y1,k ζ1,k . . . . . . xk−xi,k ζi,k yk−yi,k ζi,k . . . . . . xk−xNk,k ζNk,k yk−yNk,k ζNk,k ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4)
The elements within the matrix Hxk can be acquired from
(2). It is noted that (3) associated with (4) are utilized for representing the GDOP metric in most of the research studies. In order to facilitate the design of the proposed GOLE schemes, coordinate transformation is conducted as shown in Fig. 1, i.e. xk− xi,k= [ζi,kcos θi,k, ζi,ksin θi,k] for i = 1 to
Nk, where θi,k represents the angle formed by the vector of
xk − xi,k w.r.t. the positive x-axis. Furthermore, the relative anglesαi,k between the neighboring BSs are defined as
αi,k=
θi+1,k− θi,k 1 ≤ i ≤ Nk− 1
2π + θ1,k− θNk,k i = 0, Nk
(5) It is noted thati = 0 is utilized for circular counting in order to facilitate the notations that will be utilized in the paper.
TOA signal input
If number of signal more than 3 Location Estimation (Two-step LS method) Location Estimation (Two-step LS Method) MS’s Final Estimation r Pk, BS k, GOLE(1BS) / GOLE(2BS) Scheme MS’s initial location Estimatexk0 r Pk, BS kf, Fictitious BS set xkf r Pk, BS k, rk
Fig. 2. Schematic diagram of the proposed GOLE algorithms.
Furthermore, the relative anglesαpq,k between each arbitrary pth and qth BSs are further defined as
αpq,k= q−1
∀p<q
αp,k (6)
where 1 ≤ p < Nk, 1 ≤ q ≤ Nk and αp,k is defined as in (5). Consequently, the GDOP value with the matrix form in (3) can be reformulated as a function of αpq,k, which is also a function of αi,k, as Gxk= Nk Nk q=2 q−1 p=1sin2(αpq,k) 1 2 (7) In the next section, the results obtained from (7) will be utilized for the design of the proposed GOLE algorithms.
III. PROPOSEDGDOP-ASSISTEDLOCATIONESTIMATION (GOLE) ALGORITHMS
The main objective of the proposed GOLE schemes is to enhance the conventional two-step LS algorithm [4] by consid-ering the geometric effect to the location estimation accuracy. Fig. 2 illustrates the schematic diagram of the proposed GOLE algorithms. In order to facilitate the location estimation for the MS, three TOA measurements and the location information of the corresponding BSs are considered available to the MS at the time instant k, i.e. rk = {r1,k, r2,k, r3,k} and PBS,k = {x1,k, x2,k, x3,k}. With the available information,
the two-step LS method can acquire the MS’s initial location estimate ˆxok= [ˆxokˆyko] within two computing iterations.
The GOLE algorithms are proposed to further enhance the precision of the initial location estimation of the MS. Based on the available measurement information from the BSs, the concept of the proposed GOLE schemes is to acquire the loca-tions of the fictitious BSs such as to attain the minimal GDOP value w.r.t the MS’s initial location estimate. At the second phase of the GOLE schemes, 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. Two GOLE algorithms, i.e. the GOLE(1BS) and GOLE(2BS) schemes, are stated as follows.
A. GOLE(1BS) Scheme
In order to facilitate the design of the proposed GOLE(1BS) scheme, the property obtained from the GDOP metric is observed and derived. The minimal GDOP value is determined by adjusting one BS’s location in Lemma 1 (i.e. with one degree-of-freedom) as follows.
Lemma 1. The MS located at xk is surrounded by three BSs at xi,k (fori =1 to 3). The angles between every two adjacent BSs to the MS are defined as αi,k. It is assumed that only theth BS’s location is adjustable; while the positions for the other two BSs are considered fixed. The minimal attainable GDOP occurs as the angleα,k is adjusted to be
αm ,k=
1 2tan−1
− sin(23i=1,i=−1,αi,k)
cos(23i=1,i=−1,αi,k) + 1
(8) Therefore, the minimal attainable GDOP value w.r.t. xk be-comes Gm xk = 3 2 sin2(αm ,k) + 3
i=1,i=−1,sin2(αi,k)
1 2
(9) Proof: According to (7), it is observed that the GDOP value Gxk w.r.t. xk is regarded as a function of the anglesαi,k
for all i = 1 to 3. Since only the th BS (for 1 ≤ ≤ 3) is considered adjustable, there is merely one degree-of-freedom that is considered tunable (i.e.α,k) among all the anglesαi,k for i = 1 to 3. It is noted that the other angle α−1,k, which is also modified due to the movement of theth BS, can be represented as a function of α,k, i.e. α−1,k = 2π − α,k− 3
i=1,i=−1,αi,k. Consequently, the GDOP value as denoted
in (7) will only be dependent to the angleα,k as Gxk(α,k).
The angleα,km which results in the minimal GDOP value can therefore be acquired as αm ,k= arg min ∀α,k Gxk(α,k) (10) It can be observed that (10) can be achieved if the following conditions on the first and second derivatives of Gxk are
satisfied, i.e. ∂Gxk(α,k) ∂α,k α,k=αm,k = 0 (11) ∂2G xk(α,k) ∂2α,k α,k=αm,k > 0 (12)
By solving (11) and (12), the angleαm,k can be computed as in (8). The minimal GDOP value w.r.t. xk can consequently be obtained as in (9).
The GOLE(1BS) scheme is designed to fictitiously relocate the position of one BS according to the minimal GDOP criterion. Without lose of generality, it is considered that
BS1 (i.e. x1,k) is the adjustable BS within the GOLE(1BS) scheme. The position of the fictitious BS1 is designed such that the initial estimated MS (ˆxok) will be located at a min-imal GDOP position based on the existing geometric layout
PBS,k= {x1,k, x2,k, x3,k}. In other words, based on the initial
location estimate ˆxok associated with the information coming from the BSs (i.e. rkand PBS,k), the three relative anglesα1,k, α2,k, andα3,kbetween the BSs w.r.t. the MS can be obtained. By adopting the results from Lemma 1, the minimal attainable GDOP Gˆxo
k w.r.t. the MS’s initial estimate ˆx o k occurs as the angle α1,k is adjusted as αm 1,k= 1 2tan−1 − sin(2α2,k) cos(2α2,k) + 1 (13) It is noted that the angle α2,k between BS2 and BS3 is considered a fixed value; while α3,k is dependent to the variable angle α1,k, i.e. α3,k= (2π − α2,k) − α1,k.
As a result, the new set of BSs for location estimation is obtained as P(1)BSf,k= {xf1,k, x2,k, x3,k}, where xf1,kdenotes the location of the fictitious BS as
xf1,k= r1,kcos(θ2,k− αm1,k)
yf1,k= r1,ksin(θ2,k− αm1,k) (14) where αm1,k is obtained from (13). The set of updated loca-tions for the BSs P(1)BS
f,k associated with the original TOA
measurements rk = {r1,k, r2,k, r3,k} are exploited to conduct the second-phase two-step LS method as shown in Fig. 2. Con-sequently, the MS’s final location estimation ˆxfk = [ˆxfkˆykf] by
adopting the proposed GOLE(1BS) scheme can be obtained. B. GOLE(2BS) Scheme
For the design of the GOLE(2BS) scheme, the statement and proof for Lemma 2 that regulates all the locations of the BSs (i.e. with three degree-of-freedom) is described as follows.
Lemma 2. The MS located at xk is surrounded by three BSs at xi,k(fori = 1 to 3). The angles between every two adjacent BSs to the MS are defined asαi,k. Considering the case that the locations of all the three BSs are adjustable. The minimal GDOP value w.r.t. xk is obtained as Gmxk = 2/
√
3, which occurs as the angles αi,k are regulated to be equivalent with each other as αmi,k= 2π/3 for all i =1 to 3.
Proof: It can be observed from (7) that the GDOP value is regarded as a function of the angles αi,k for all i = 1 to 3, i.e. Gxk(α1,k, α2,k, α3,k). By defining αk = [α1,k α2,k α3,k],
the anglesαmi,k which result in the minimal GDOP value can therefore be acquired as αm i,k = arg min ∀αi,k Gxk(αk) (15) for i = 1 to 3. Similar to the proof as in Lemma 1, αmi,k in (15) can be acquired if ∂Gxk(αk) ∂αi,k αk=αmk = 0 (16) ∂2Gx k(αk) ∂2αi,k αk=αmk > 0 (17)
for i = 1 to 3. It is noted that αmk [αm1,k α2,km αm3,k]. By solving the set of equations obtained from (16) and (17), the anglesαmi,k can be computed as
αm
i,k=2π3 (18)
which are considered equivalent for all i = 1 to 3. By sub-stituting (18) into (7), the minimal GDOP value can therefore be obtained as Gm xk(αmk) = 2 √ 3 (19)
In order to achieve the minimal GDOP for the existing geometric layout, the GOLE(2BS) scheme is designed by con-sidering the case while all the BSs are fictitiously movable. By adopting the results from Lemma 2, it can be observed that the three BSs are fictitiously adjusted such that equally partitioned angles are acquired, i.e.αmi,k= 2π/3 for i = 1 to 3. Therefore, the set of fictitious BSs by exploiting the GOLE(2BS) scheme is represented as P(2)BSf,k = {xf1,k, xf2,k, xf3,k}, where xfi,k
(fori = 1 to 3) indicates the locations of the fictitious BSs. It is noted that xf1,kis selected to be x1,kas the rotation reference. Consequently, the locations of the other two fictitious BSs can be acquired as
xfi,k = ri,kcos(θ1,k+
i−1
s=1
αm s,k)
yfi,k = ri,ksin(θ1,k+
i−1
s=1
αm
s,k) (20)
for i = 2 and 3. Similar to the GOLE(1BS) scheme, the fictitious locations of the BSs P(2)BSf,k associated with the TOA measurements rk are utilized to serve as the new set of measurement inputs for the two-step LS method at the second stage. As a result, the final location estimate of the MS (i.e. ˆxf
k) can be acquired.
IV. PERFORMANCEEVALUATION
Simulations are performed to show the effectiveness of the GOLE algorithms under different network topologies and the MS’s positions. The proposed GOLE(1BS) and GOLE(2BS) schemes are compared with the exiting two-step LS and the TSE algorithms. As shown in Fig. 3, two different types of geometric layouts are designed to validate the effectiveness of the proposed GOLE algorithms. The left plot illustrates the case while the MS is located at a symmetrical geometric layout, i.e. with smaller GDOP value as Gbxk = 1.44. On the
other hand, a comparable worse geometric layout is designed as shown in the right plot, which results in the GDOP value as Gwxk = 11.08. It is noted that the minimal GDOP value for
a three-BS geometry is obtained as Gmxk = 2/
√
0 200 400 600 8001000 0 200 400 600 800 1000 1200 1400 1600 1800
Worse Geometric Layout
0 500 1000 1500 2000 0 200 400 600 800 1000 1200 1400 1600 1800
Better Geometric Layout BS1(1000,1732) MS(1000,520) BS2(0,0) BS3(2000,0) BS1(1000,1732) MS(500,800) BS2(0,100) BS3(0,0)
Fig. 3. Network topologies for performance evaluation (left plot: better geometric layout withGgxk = 1.44; right plot: worse geometric layout with Gb
xk= 11.08).
A. Noise Models
Different noise models [8] are considered in the simulations in order to represent the environments with both the LOS and the NLOS signals. 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. ni,k ∼
N (0, 100) . On the other hand, an exponential distribution pei,k(τ) is assumed for the NLOS noise model of the TOA
measurements as pei,k(υ) = 1 λi,kexp − υ λi,k υ > 0 0 otherwise (21)
where λi,k = c · τi,k = c · τm(ζi,k)ερ. The parameter τi,k is the RMS delay spread between the ith BS to the MS, and τm
represents the median value ofτi,k.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. It is noted that the parameters for the noise models as listed in this subsection primarily fulfill the environment while the MS is located within the rural area. B. Simulation Results
Figs. 4 and 5 illustrates the case for performance compar-ison under the NLOS environment. It is noted that Fig. 5 is illustrated by observing the RMS errors versus the median values of the NLOS noises (i.e. τm). It can be observed that the proposed GOLE(2BS) scheme outperforms the other algorithms even under the existence of the NLOS errors, i.e. around 250 m less in RMS error compared to the two-step LS method with τm = 0.4. Moreover, the benefits by
fictitiously adjusting the locations of two BSs compared to that for one BS can also be observed in both the LOS and the NLOS environments. With the incorporation of the geometric information into the location estimation, the merits of the proposed GOLE schemes can be observed.
V. CONCLUSION
The location estimation algorithms with the assistance from the geometric dilution of precision (GDOP) are presented in this paper. Two GDOP-assisted location estimation (GOLE) algorithms are proposed by considering the geometric layouts
GDOP=1.44 GDOP=11.082 0 100 200 300 400 500 600 700 800 900 RMS Error (m) TSE Two−step GOLE(1BS) GOLE(2BS)
Fig. 4. Performance comparison under the NLOS environment with both better (left:Gbxk= 1.44) and worse (right: Gwxk= 11.08) geometric layouts (τm= 0.3). 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 200 300 400 500 600 700 800 900 1000 1100 1200
Median Value of NLOS Noise (τm)
RMS Error (m)
TSE Two−step GOLE(1BS) GOLE(2BS)
Fig. 5. Performance comparison under the NLOS environment with worse geometric layout (Gwxk = 11.08): RMS error v.s. median value of NLOS noise.
between the mobile station and its associated base stations (BSs). The GDOP information is utilize to fictitiously relocate the positions of the BSs in order to obtain a better geometric layout for location estimation. It is shown in the simulation results that the proposed GOLE schemes can provide con-sistent accuracy for location estimation, especially under the environments with poor GDOP values.
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