The effectiveness of using the Kalman filtering technique can be observed from Fig. 6.1. It eliminates measurement noises and tracks the MS’s position and velocity in the longitude, latitude, and altitude directions. Fig. 6.2 - Fig. 6.4 shows the performance comparison between the proposed hybrid location estimation system, the satellite-based system, and the cellular-based system under urban, suburban, and rural environments. It can be seen that the RMS error of the MS’s position (i.e. ∆Pf = [(xf− x)2+ (yf− y)2+ (zf− z)2]12) obtained from the hybrid scheme is smaller comparing with that from the other two approaches. It is also noted that the estimation error acquired from the satellite-based system has the worst performance comparing with the other two systems in the urban area; while the cellular-based system causes degraded results in the rural area. The hybrid system is capable of adjusting itself to accommodate different situations, which provides consistent performance comparing with the satellite-based and the cellular-based systems.
0 20 40 60 80 100
Figure 6.1: Left Plots: Performance Comparison before (dots) and after (’+’ marks) using the Kalman Filtering Technique; Right Plots: The Velocity Tracking of the MS using the Kalman Filtering Technique
Average Position Error <= Abscissa
RMS Error (m) Environment − Urban
GPS Cell Hybrid
Figure 6.2: Performance Comparison between the Hybrid System, the Satellite-based System, and the Cellular-based system under Urban Environment
0 20 40 60 80 100 120
Average Position Error <= Abscissa
RMS Error (m) Environment − Suburban
GPS Cell Hybrid
Figure 6.3: Performance Comparison between the Hybrid System, the Satellite-based System, and the Cellular-based system under Suburban Environment
0 10 20 30 40 50 60 70 80 90
Average Position Error <= Abscissa
RMS Error (m) Environment − Rural
GPS Cell Hybrid
Figure 6.4: Performance Comparison between the Hybrid System, the Satellite-based System, and the Cellular-based system under Rural Environment
Part II
Enhanced Wireless Location
Estimation Algorithms
Chapter 7
Overview
A variety of wireless location techniques have been studied and investigated [1]- [3]. The network-based location estimation schemes have been widely exploited in the wireless com-munication system. These schemes locate the position of the MS based on the measured radio signals from its neighborhood BSs. The major time-based methods for the network-based lo-cation estimation techniques are the TOA, and the 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.
Since the equations accompanied with the network-based location estimation schemes are inherently nonlinear, it is required to adapt approximation techniques for location estimation.
In addition, the uncertainties induced by the measurement noises make it more difficult to acquire the estimated MS position with tolerable precision. Different approaches have been proposed to obtain an approximate location. The Taylor-Series Estimation (TSE) method was utilized in [24] to acquire the location estimation from the TDOA measurements. The method requires iterative processes to obtain the location estimate from a linearized system.
The major drawback of this method is that it may suffer from the convergence problem due to an incorrect initial guess of the MS’s position. The two-step LS method was adopted to solve the location estimation problem from the TOA [25], the TDOA [26], and the TDOA/AOA measurements [27]. It is an approximate realization of the Maximum Likelihood (ML)
estima-tor and does not require iterative processes. The Linear Line-of-Position (LLOP) [28] method presents a different interpretation of the TOA geometry to estimate the MS’s location com-paring with the conventional circular TOA methods. However, the algorithms as described above are primarily feasible for location estimation under Line-Of-Sight (LOS) environments.
The Non-Line-Of-Sight (NLOS) situations, which occur mostly under urban or suburban ar-eas, greatly affect the precision of these location estimation schemes. On the other hand, the algorithm proposed in [35] alleviates the NLOS errors by considering the cell layout between the MS and its associated BSs. A constrained nonlinear optimization is adopted to obtain improved location estimate for the MS. However, the approach proposed in [35] involves the requirement of solving an optimization problem based on a nonlinear objective function. The inefficiency incurred by the algorithm may not be feasible to be applied in practical systems.
In this thesis, an efficient Geometry-constrained Location Estimation (GLE) algorithm and a location estimation algorithm with the Virtual Base Stations (VBS) are proposed to obtain location estimation of the MS, especially for NLOS environments. The proposed GLE scheme integrates the geometric information from the cell layout into the conventional two-step LS algorithm. The MS’s position is obtained by confining the estimation based on the signal variations and the geometric layout between the MS and the BSs. Moreover, the proposed VBS scheme integrates the geometric information from the extended cell layout into the conventional two-step LS algorithm, especially not only NLOS environments but also poor Geometric Dilution of Precision (GDOP) [36]. The MS’s position is obtained from the time measurements by confining the estimation based on the signal variations and the geometric layout extended by the virtual base stations. The reasonable location estimations can be acquired from both the VBS and GLE within two computing iterations even with the existence of the NLOS errors. The numerical results via simulations shows that the VBS and GLE approaches can acquire higher accuracy for location estimation.
Chapter 8
The Proposed GLE and VBS Algorithms
The time-based algorithms, i.e. TSE, LLOP, and two-step LS as described in chapter 2, are primarily feasible for location estimation under Line-Of-Sight (LOS) environments. In order to preserve the computation efficiency and to obtain higher accuracy under Non-Line-Of-Sight (NLOS) environments, the Geometry-Constrained Location Estimation (GLE) and the location estimation with the Virtual Base stations (VBS) are designed to incorporate geometric constraints within the formulation of the two-step LS method with the consideration of the different geometric layouts between the MS and its associated BSs. The details of the proposed GLE and VBS algorithms are described in this chapter.
8.1 The GLE Algorithm
The proposed GLE algorithm associated with the applications within three different scenarios are described in this section. The measured distances r`, for ` = 1, 2, and 3, are illustrated as in Fig. 8.1. It is noted that the three circles which define the TOA measurements will intersect to a single point (i.e. the MS’s position) if the measurements are LOS and are free of the measurement noises. The concept of the GLE algorithm is to consider the geometric constraints between the MS and the BSs within the formulation of the two-step LS method. It
00 11
BS
BS BS
A B
C
r
r r
xe
1 1
3 3 2
2
Figure 8.1: The Schematic Diagram of the TOA-based Location Estimation for NLOS envi-ronments (Generic Case)
is recognized that the range measurements are in general corrupted by both the measurement noises and the NLOS errors. The conventional two-step LS algorithm obtains location estimate primarily by considering the measurement noises with gentle NLOS errors. In the TOA-based location estimation as shown in Fig. 8.1, the location estimation by using the two-step LS method may fall around the boundaries of the three arcs, AB, BC, and CA, i.e. either inside or outside of these arcs. Since the overlap region (i.e. constrained by the points A, B, and C) grows as the NLOS errors are increased, the location estimation of the MS acquired by the two-step LS method will result in deficient accuracy (i.e. the location estimate still falls around the boundaries of the enlarged arcs AB, BC, and CA).
The primary objective of the proposed GLE algorithm is to confined the location estimate within the overlap region by including the geometric constraints into the two-step LS method.
The following three different cases are considered based on the various cell layouts which may occur:
8.1.1 3 TOA Measurements – Generic Case
As illustrated in Fig. 8.1, three BSs associated with three TOA measurements are required for the location estimation of the MS. The overlap region (i.e. confined by the arcs AB, BC, and CA) is formed with the assumption that there is at least one NLOS error occurred from one of the three TOA range measurements. Since the objective of the proposed GLE scheme is to confine the estimated MS position within the region of ABC, the following constrained cost function is defined: represent the corresponding coordinates of the points A, B, and C. γ is defined as a virtual distance between the MS’s position and the three points A, B, and C. It is also noted that the value of γ varies as the three coordinates a, b, and c are changed. An expected MS’s position xe is chosen to locate within the triangular area ABC in order to fulfill the constraints from the geometric layout. The corresponding expected virtual distance γe can be obtained as
γe =
where nγ is the error induced by the computed deviation between γe and γ. The major functionality of the constrained cost function as in (8.2) is to minimize the deviation between the virtual distance γ and the expected virtual distance γe. The selection of the expected MS position xeis obtained by considering the signal variations from the three TOA measurements.
The coordinates of xe = (xe, ye) are chosen with different weights (w1, w2, w3) with respect to the A, B, and C points of the triangle as
xe= w1xa+ w2xb+ w3xc (8.3a) ye= w1ya+ w2yb+ w3yc (8.3b)
where
w` = σ`2
σ12+ σ22+ σ32 for ` = 1, 2, 3 (8.4)
σ1, σ2, and σ3 are the corresponding standard deviations obtained from the three TOA mea-surements, r1, r2, and r3. The decision of the weight w` is explained as follows. For the measurement r1 (as shown in Fig. 8.1), the MS position should be located around the bound-ary of the circle with the radius r1 without the existence of the NLOS errors. If the standard deviation σ1 of the measurement r1 is comparably large, it indicates that the true position of the MS should move toward inside of the circle boundary of the radius r1 due to the NLOS errors. Consequently, the weight w1 is assigned with a larger value, which specifies that the position of the MS should move toward the endpoint A of the triangle. The design concept is applied to the selection of the other two weights, w2 and w3, in the same manner. With the selection of the expected MS’s position xe, the expected virtual distance γe can be computed from (8.2).
The proposed GLE algorithm is formulated by solving the two-step LS problem with the additional geometric constraint. The solution is obtained by minimizing both (i) the errors coming from the three TOA measurements (as in (2.1)) and (ii) the deviation between the expected virtual distance and the virtual distance (as in (8.2)). By rearranging and combining (2.1) and (8.2) in matrix format, the following equation can be obtained:
Hx = J + ψ (8.5)
where
x =
·
x y β
¸T
H =
The corresponding coefficients are given by
β = x2+ y2
The noise matrix ψ in (8.5) can be obtained as
ψ = 2 c Bn + c2n2 (8.6)
Based on the two-step LS scheme, an intermediate location estimate after the first step can
be obtained as
ˆ
x = (HTΨ−1H)−1HTΨ−1J (8.7)
where
Ψ = E[ψψT] = 4 c2 BQB
It is noted that Ψ is obtained by neglecting the second term of (8.6). The matrix Q can be acquired as
Q = diag
½
σ12, σ22, σ23, σγ2e/c2
¾
It can be observed that Q represents the covariance matrix for both the TOA measurements and the expected virtual distance, where σ2γe/c corresponds to the standard deviation of γe/c.
The final location estimation after the second step of the two-step LS algorithm can be obtained by referring the approach as stated in [25].
8.1.2 3 TOA Measurements – MS Locates Closer to its Home BS
In certain situations, the MS may locate much closer to its home BS compared to the other BSs. Due to the fact that the NLOS errors grow as the distance between the MS and the BS is increased [37], there is high possibility to result in no geometric intersection formed by these TOA measurements. As is illustrated in Fig. 8.2, there is no intersection between the circles with the radiuses r1 and r2. Since the MS is located closer to its home BS (i.e.
BS1), the non-intersect scenario occurs while there is larger NLOS error induced by the TOA measurement from the BS2. The original proposal as stated in the previous generic case will not be applicable in this type of situation.
In order to employ the GLE algorithm within this circumstance, it is required to impose an additional constraint to appropriately formulate the problem. The inequality of r` > L1`+ r1, for ` = 2, 3, should be changed to ˆr` = L1`+ r1, where L1` corresponds the the distance between the `th BS to the home BS. As shown in the Fig. 8.2, the modified radius ˆr2 results in an intersection with the home BS (i.e. point E) in order to facilitate the formulation of
00
Figure 8.2: The Schematic Diagram of the TOA-based Location Estimation for NLOS envi-ronments (Special Case: MS Locates Closer to its Home BS)
proposed GLE scheme. It is noted that the assumption is applicable since the non-intersect situation is generally induced by the excessive amount of NLOS errors from the measurement r2. As a result, the GLE algorithm can be applied in this case by substituting the points A.
B, and C (as in Fig. 8.1) with the points E, F , and G. Similar procedures can be employed to solve for the two-step LS problem as mentioned in the previous case. The location estimate of the MS can therefore be constrained within the updated triangular area, which is enclosed by the points E, F , and G.
8.1.3 2 TOA and 1 AOA Measurements
Due to the weak incoming signals or the shortage of signal sources (e.g. at the rural area), there is great possibility that the MS may not be able to acquire enough signal sources from the environment, i.e. only two TOA measurements are available from the BS1 and the BS2. In order to employ the proposed GLE algorithm within this circumstance, an additional AOA measurement from the home BS is adopted. As mentioned in the 3GPP standard [33] [34], each BS should be equipped with the antenna arrays for adaptive beam steering in order to
00
Figure 8.3: The Schematic Diagram of the TOA/AOA-based Location Estimation for NLOS environments (Special Case: 2 TOA and 1 AOA Measurements)
facilitate the AOA measurements. It is also noted that only the AOA measurement from the home BS is applied to avoid the signal degradation due to the near-far effect. The geometric layout of the two TOA and the one AOA measurements is illustrated as in Fig. 8.3. The proposed GLE algorithm can be applied in this case with some modifications from the generic case. The region enclosed by the points A, B, and C (as in Fig. 8.1) is replaced by the triangular area defined by the points I, J, and K. The selection of σ3 within the expected MS’s position (i.e. in (8.4)) should be modified as σ3 = r1· σθ/c, where σθ corresponds to the standard deviation of the measurement noise nθ as in (2.4). It is noted that σ3 is referred as the signal variations coming from the AOA measurement. As the value of σ3 is increased, the expected MS’s position xe should move away from the line which connects the points I and J. The resulting location of xe is expected to move toward the point K as can be obtained from (8.4).
In additions, the matrices, H and J, within (8.5) should be reformulated as
H =
J =
The matrices B, n, and Q associated with the noise matrix ψ as in (8.6) can be obtained as
It is noted that value in the third diagonal term within the matrix B, i.e. the element ζ1, can be obtained as in [27] based on geometric approximation of the AOA measurement. The performance of the proposed GLE algorithm under the three different cases will be evaluated and compared via simulations in the next section.