This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or the copyright holders . All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. This work may not be re-posted without the explicit permission of the copyright holder.
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
641
A Nonline-of-Sight Error-Mitigation Method for TOA Measurements Changlin Ma, Richard Klukas, and Gérard Lachapelle, Member, IEEE
Abstract—In wireless location, nonline-of-sight (NLOS) errors are a major error source. To improve positional performance, these errors should be identified and mitigated before measurements are used in position computations. A novel NLOS mitigation algorithm is proposed in this paper. Employing system redundancy, the algorithm derives from the distribution function of hyperbolic intersections, which is an intermediate mobile location. Cost functions based on the intermediate position are then created for each time-of-arrival measurement. NLOS errors are detected and mitigated via hypothesis tests performed on the cost functions. Simulation tests are presented in this paper, and the results verify the effectiveness of this NLOS-error-mitigation method. This paper also demonstrates that algorithm performance depends closely on receiver noise. Reducing receiver noise or providing the intermediate mobile location via external aiding can improve performance. Index Terms—Distribution function (DF), hyperbola intersection, nonline-of-sight (NLOS) error, wireless location.
I. I NTRODUCTION
T
HE BASIC problem of wireless location is to estimate the geographic position of a wireless device. In recent years, position estimation of cellular phones has, in particular, received considerable attention. The impetus for this research stems mainly from a series of regulations passed in 1996 by the United States Federal Communications Commission. The intent of these regulations is to encourage cellular service providers to improve the quality of Enhanced 911 (E-911) service for cellular phone users. The mandate was deemed necessary due to the rising number of emergency calls made from cellular phones. The accuracy requirement of the E-911 mandate for handset-based solutions is 50 m or better 67% of the time and 150 m or better 95% of the time. For network-based solutions, the respective accuracies are 100 and 300 m [1].
Currently, the most often used techniques for wireless location are signal strength, angle of arrival, time-of-arrival (TOA), and time-difference-of-arrival (TDOA) methods. All of them require line-of-sight (LOS) communication links. Unfortunately, such direct links rarely exist because of the intrinsic complexity of the mobile channel. Quite often, the signal received is either a combination of the LOS signal and multipath signals or consists of only multipath signals. In either case, there exist so-called non-LOS (NLOS) propagation errors, which are actually the dominant error compared to receiver noise [2]. Actual field testing shows that the average NLOS range error can be as large as 0.589 km in an IS-95 codedivision multiple-access (CDMA) system [3]. NLOS-error-identification and mitigation techniques must be applied to prevent observations from being seriously corrupted and to yield satisfactory positioning accuracies. In [3], NLOS errors are identified by comparing the standard deviations of range measurements with a detection threshold. In [4], a timehistory-based hypothesis test is proposed to identify and then remove NLOS errors. In [5], a decision framework for NLOS identification is formulated, which can process both Gaussian and non-Gaussian NLOS errors. In [6], a residual-weighting algorithm proposed for a TOA location system is also able to identify NLOS errors with unknown distribution. Additional efforts are currently being made in this area, and a substantial number of NLOS-mitigation algorithms [7], [8] have recently been proposed. In this paper, the NLOS errors in a TDOA-based wirelesslocation system are investigated, and an algorithm is proposed to identify and mitigate these errors. Simulation results are presented to demonstrate the performance improvement due to the NLOS-error-mitigation method proposed. II. NLOS-E RROR I SSUE
Manuscript received July 5, 2004; revised April 2, 2005, October 1, 2005, and December 13, 2005. This work was supported in part by the Defence Reseach and Development Canada, Department of National Defence. The review of this paper was coordinated by Prof. T. Lok. C. Ma was with the Department of Geomatics Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. He is now with the Centrality Communications, Santa Clara, CA 95054 USA (e-mail:
[email protected]). R. Klukas is with University of British Columbia Okanagan, Kelowna, BC VIV 1V7, Canada (e-mail:
[email protected]). G. Lachapelle is with the Department of Geomatics Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: lachapel@ geomatics.ucalgary.ca). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2007.891439
Fig. 1 illustrates a typical seven cell cluster with base stations (BS) and a mobile station (MS) currently located in cell one. As illustrated by the signal transmission between BS7 and the MS in Fig. 1, an NLOS error results from the blockage of the direct signal and the reflection of multipath signals. The NLOS error is the extra distance that a signal travels from transmitter to receiver and, as such, always has a nonnegative value. Normally, an NLOS error can be described as a deterministic error, a Gaussian error, or an exponentially distributed error. However, at any given time instant, it can be treated as a constant [9].
0018-9545/$25.00 © 2007 IEEE
642
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
Fig. 1. NLOS error. Fig. 2.
Taking the TOA method as an example, the performance degradation due to NLOS errors can be analytically demonstrated as follows. The least squares (LS) estimator used for MS location estimation is x ˆ = arg min x
(ri − x − Xi )2
(1)
i∈S
where · denotes the norm operation over a vector, x repˆ represents the estimate of the MS resents the MS position, x position, Xi is the position of the ith BS, x − Xi is the distance between x and Xi , S is the set of BSs used, and ri is the measured distance from the MS to the ith BS, i ∈ S. With receiver noise and NLOS errors, the range measurements, as derived from TOA measurements of an MS with respect to N BSs, can be expressed as ri = Li + ni + NLOSi ,
i = 1, . . . , N
(2)
where Li is the true distance, ni is the receiver noise, and NLOSi is the NLOS error. The receiver noise ni is assumed to be a zero-mean Gaussian random variable with a standard deviation of about 60–100 m for an IS-95 CDMA system [9]. It can be smaller if better signal-receiving techniques are applied. As will be shown in the following section, NLOS errors are dominant error sources. Expressed in matrix–vector form, the range measurements in a TOA wireless-location scheme are r = L + n + NLOS.
(3)
Taking the true MS location as the initial point, the range measurements can be expressed via a Taylor-series expansion as r≈L+G
∆x ∆y
(4)
where G is the design matrix, and [∆x ∆y]T is the MS location-estimation error. Obviously, the final solution of the
Intersection of hyperbolas.
problem is ∆x = (GT G)−1 G · n + (GT G)−1 G · NLOS. ∆y
(5)
Because NLOS errors are much larger than the receiver noise, positioning errors result mainly from the NLOS errors, should they exist. III. NLOS-E RROR -M ITIGATION A LGORITHM A. General Algorithm Description The algorithm proposed employs system redundancy. Without losing generality, the TDOA-location scheme is used to demonstrate the algorithm. As illustrated in Fig. 2, each TDOA measurement determines a hyperbola between two BSs, and two of these hyperbolas determine an intersection, which is a candidate for the MS location to be estimated. A number of hearability-improvement techniques [10], [11] have been proposed to receive signals from a greater number of BS and thereby achieve redundant hyperbolas. The idle-period-downlink technique is applied to mitigate interference, whereas longer integration times may be employed to increase processing gain and improve receiver sensitivity. According to Bartlett [10], approximately 8–9 BSs can be received with a probability greater than 50%. Therefore, a set of intersections can be achieved to form an intersection distribution. The intersection distribution has the following properties. The area of uncertainty is small, if there are no NLOS errors. In other words, the intersections are concentrated near the true MS location, as shown in Fig. 3(a). In contrast, the area of uncertainty is large if any BSs suffer from NLOS errors. When the LOS signal of a BS is blocked, the TDOA measurements related to this BS will have a bias equal to the NLOS error, and the associated hyperbolas will move away from the true MS location. Consequently, the intersections between these shifted hyperbolas and other hyperbolas will also move away from the true MS location, enlarging the area of uncertainty. In Fig. 3(b), hyperbola H4 contains an NLOS error. As a result, H4 and all
MA et al.: NONLINE-OF-SIGHT ERROR-MITIGATION METHOD FOR TOA MEASUREMENTS
643
nonlinear. The following optimization problem is solved to calculate the intersection of two hyperbolas: x ˆ = arg min x
x − X1 − x − X2 − TDOA1
2
2 + x − X3 − x − X4 − TDOA2
(7)
since the intersection satisfying (6) will also minimize the cost function in (7). Fig. 3. Intersections offset by NLOS errors. (a) NLOS-error-free case. (b) NLOS-error-corrupted case.
C. Construction of the Distribution Function (DF) The intersection DF is constructed to quantify intersection density and, thereby, arrive at an intermediate MS location estimate for succeeding NLOS-error identification and mitigation. The DF is defined as
Ip (x − xi )2 + (y − yi )2 DF(x, y) = exp − (8) ε2 i=1
Fig. 4.
Steps in NLOS mitigation algorithm.
of its intersections with the other hyperbolas move away from the true MS location. The proposed NLOS-error-mitigation algorithm explores the different distributions of NLOS-free intersections (clear intersections) and NLOS-corrupted intersections (biased intersections). A clear intersection is calculated from two NLOS-free TDOA measurements, and a biased intersection is calculated from two TDOA measurements, at least one of which is NLOS error corrupted. If the system redundancy is sufficiently high and only a small number of observations contain NLOS errors, one can expect that a significant number of clear intersections will exist near the true MS position. Therefore, a higher intersection density should exist near the true MS position. By seeking the maximum of the intersection distribution, an intermediate estimate of the MS position can be obtained, from which NLOS errors can be identified and mitigated. The procedure to mitigate NLOS errors is summarized in Fig. 4 and discussed in the following subsections.
B. Hyperbola-Intersection Calculation The intersection of two hyperbolas is the solution of the following two equations:
TDOA1 = TDOA2 =
(x−x1 )2 +(y− y1 )2 − (x−x3 )2 +(y− y3 )2 −
(x−x2 )2 +(y − y2 )2 (x−x4 )2 +(y− y4 )2 (6)
where X1 = (x1 , y1 )T , X2 = (x2 , y2 )T , X3 = (x3 , y3 )T , and X4 = (x4 , y4 )T are the coordinates of four BSs, and X = (x, y)T represents the intersection to be solved for. It is difficult to obtain a closed-form solution, since these equations are
where Ip is the total number of intersections, (xi , yi ) are the coordinates of one intersection computed via the method proposed above, and ε2 is a parameter to adjust the contribution of an intersection to the final DF. The value of ε must be well selected because it affects the estimation accuracy of the intermediate MS location. In Fig. 5(a), ε is selected as 0.1 times the standard deviation of receiver noise. In this case, the final DF has only discrete spikes that prevent the successful estimation of the MS location because of the lack of a dominant peak. In Fig. 5(b), ε is chosen to be ten times the standard deviation of the receiver noise, and the final DF has only one flat peak that may result in poor accuracy. In Fig. 5(c), ε is equal to 1.5 times the standard deviation of the measurement noise. In that case, the final DF has a much better shaped peak and yields an accurate estimate of the MS position. Normally, ε should be chosen as 1–2 times the standard deviation of the receiver noise. This is because ε is used to describe the intersection inaccuracy due to such receiver noise, and the position uncertainty is related to measurement noise by the following equation: σPOS = DOP · σmeasurement_noise .
(9)
DOP is a function of the MS’s and BS’s geometry and varies with the number of BSs received, together with their locations. In practice, DOP generally varies from 1 to 2 when six or seven BSs are received [12]; therefore, σPOS is approximately 1–2 times σmeasurement_noise . From (8), it is √ evident that the normal DF with a standard deviation of ε/ 2 is used in the derivation of the intersection DF. Thus, if ε is selected between σmeasurement_noise and 2σmeasurement_noise , the position uncertainty region will be a circle centered at the calculated intersection with radius 0.7–1.5 times σmeasurement_noise . This approximately reflects the position uncertainty due to receiver noise. Selecting ε within the above range will prevent the DF from being overly sensitive [Fig. 5(a)] or too smooth [Fig. 5(b)].
644
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
E. NLOS-Error Identification For each BS, a cost function is formed to determine whether that BS’s measurements are NLOS error corrupted. Although not true in practice, it is initially assumed that the true distance differences between the MS and the BSs are known in order to simplify the analysis. Performance of the cost function without knowledge of the true distance differences is discussed below. When the true distance differences between the MS and the BSs are known, the cost function used in this paper is M
0 TDOAm i,k − TDOAi,k ,
L(BSi ) =
i = 1, 2,. . . , M
k=1k=i
(11) TDOAm i,k
where M is the number of BSs used, and and 0 TDOAi,k are the measured and the true distance difference between MS-BSi and MS-BSk . A TDOA measurement can be separated into the true value, the NLOS error, and the receiver noise parts, as in the following equation: m m TDOAm i,k = TOAi − TOAk
= TOA0i + NLOSi + ni − TOA0k − NLOSk − nk = (NLOSi − NLOSk ) + (ni − nk ) + TDOA0i,k . (12) Substituting (12) into (11), one easily finds that the cost function consists of two components: the NLOS-error component and receiver-noise component. M
L(BSi ) =
M
(NLOSi − NLOSk ) +
k=1k=i
(ni − nk )
k=1k=i
= M · NLOSi −
M
NLOSk + M · ni −
k=1
NLOS part
M k=1
noise part
nk . (13)
Fig. 5. Selection of ε for DF construction. (a) ε = 0.1 · STDTDOA . (b) ε = 10 · STDTDOA . (c) ε = 1.5 · STDTDOA .
Receiver noise is assumed to have a zero-mean Gaussian distribution N (0, σ 2 ), and consequently, the noise portion—a combination of receiver noise—is also zero-mean Gaussian distributed but with a different variance N (0, M (M − 1)σ 2 ). As a result, the cost function is of a Gaussian distribution but with a nonzero-mean value:
M 2 NLOSk , M (M − 1)σ . L(BSi ) ∼ N M · NLOSi −
D. Intermediate-MS-Location Calculation
k=1
In this paper, the following strategy is used to estimate the intermediate MS position:
(x, y) = arg max (x,y)
Ip
(x − xi )2 + (y − yi )2 exp − ε2 i=1
. (10)
(14)
The cost function of an NLOS-free BS is of the following distribution:
L(BSc ) ∼ N
−
M k=1
NLOSk , M (M − 1)σ
2
.
(15)
MA et al.: NONLINE-OF-SIGHT ERROR-MITIGATION METHOD FOR TOA MEASUREMENTS
645
TABLE I MINIMUM DETECTABLE NLOS ERROR
Fig. 6.
Determination of minimum detectable NLOS error.
To identify if a BS is NLOS corrupted, hypothesis tests are conducted. To this end, γci , the difference between the cost function of the BSi to be evaluated and that of an NLOS-free BSc , is evaluated: N 1 q ˆ L(BS L (BSi ) i) = N q=1 N M 1 (M − 1)NLOSqi − = NLOSqk N q=1 k=1k=i
NLOS part
N M 1 (M − 1)nqi − + nqk N q=1 k=1k=i
N 1 q ˆ L (BSi ) L(BSi ) = N q=1
noise part
≈ M · NLOSi −
M
N M 1 (M − 1)NLOSqi − NLOSqk = N q=1 k=1k=i
NLOSk
k=1
NLOS part
N M 1 q q M · ni − + nk N q=1 k=1
measurements, the minimum NLOS error that can be identified is approximately 250–320 m. This value is so large that positioning accuracy cannot be improved to a satisfactory level. A method, namely the cost-function-smoothing method, can be applied to mitigate receiver noise. This method is based on the phenomenon that NLOS errors are low-frequency components while receiver noise consists of high-frequency components, especially in static situations. In this case, the cost functions of several consecutive time instants, in which NLOS errors are generally unchanged, can be combined together in order to mitigate receiver noise as follows:
NLOS part
N M 1 q q (M − 1)ni − + nk N q=1 k=1k=i
(16)
noise part
noise part
γci
where is a Gaussian random variable with a mean of NLOSi and a variance of 2σ 2 . Obviously, the hypothesis test decides whether or not γci is zero mean. BSi is an NLOS-free BS if γci is zero mean; otherwise it is NLOS corrupted.
H1 : γci ∼ N(0, 2σ 2 ),
NLOS-free BS
(17)
H2 : γci ∼ N(NLOSi , 2σ 2 ),
NLOS-corrupted BS.
(18)
Techniques of quality control or reliability analysis [13] can be used to identify whether γci is zero mean. If α/2 is chosen as the false-alarm probability of recognizing an NLOSfree γci as being NLOS corrupted and β is the miss-detection probability of accepting an NLOS-corrupted γci as NLOS free, the minimum detectable NLOS error can be calculated from Fig. 6, and the results are presented in Table I. Supposing that α is 5% and β is 20%, the minimum detectable NLOS error is 3.96σ. If σ is 60–80 m for TOA
≈ M · NLOSi −
M k=1
NLOS part
NLOSk
N M 1 q q M · ni − + nk N q=1 k=1
(19)
noise part
where N is the number of consecutive cost functions. Obviously, the new cost function is of the following distribution:
ˆ L(BS i) ∼ N
M · NLOSi −
M k=1
M (M − 1)σ 2 NLOSk , N
. (20)
646
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
The variance decreases from M (M − 1)σ 2 to M (M − 1) σ /N · γˆci changes to
Therefore, the cost function becomes
2
ˆ ˆ L(BS i ) − L(BSc ) γˆci = M M N M 1 q q M ·NLOSi − NLOSk + ni M ·ni − N q=1 k=1 k=1 NLOS part
=
noise part
M M N M 1 q q − M · nc − NLOSk + nc N q=1 k=1 k=1 −
NLOS part
L(BSi ) =
e TDOAm i,k − TDOAi,k
k=1,k=i
− (rk − ri ) · [∆x = L(BSi ) +
M
(21)
(22)
k=1,k=i
where TDOAei,k is the estimated TDOAi,k based on the intermediate MS location. With a Taylor-series expansion around the true TDOA0i,k , TDOAei,k can be approximated as TDOAei,k = TDOA0i,k + (rk − ri ) · [∆x ∆y]T .
k=1,k=i
= L(BSi ) −
M
rk · [∆x
∆y]T
∆y]T
(23)
∆y]T
(24)
where [∆x ∆y]T is the estimated position error, and ri is the unit direction vector between the user and the ith BS. The approximation in the last equation holds under the assumption that BSs are evenly located around the MS. Substituting Le (BSi ) into (16), γci becomes γci = NLOSi + (ri − rc ) · [∆x
∆y]T
(ri − rk ) · [∆x ∆y]T
≈ L(BSi ) + M ri · [∆x
It has now the distribution of N (NLOSi , 2σ 2 /N ). Still using the above example, when 16 cost functions are combined together, the √ minimum detectable NLOS decreases to NLOSmin = 3.96σ/ 16 ≈ σ with the same probabilities of α = 5% and β = 20%. Obviously, γˆci or γci can be used as an NLOS-error estimate, with which TDOA measurements can be explicitly corrected to mitigate NLOS errors and to improve performance. It is worth noting that the NLOS-free BS BSc is actually unknown in advance, and thus, the algorithm herein selects the BS with the smallest cost function as BSc . This is reasonable since only distance difference makes sense in a TDOA-location scheme. If BSc also contains NLOS errors, (16) and (21) actually give the difference between two NLOS errors. When constructing TDOA measurements, the residual NLOS error will be cancelled out, and thus, the final solution will not be affected. However, the above discussion is based on the assumption that only receiver noise exists in the cost-function derivation. When other errors exist, such as when the true-distance differences are not available, the performance will deteriorate. If the true-distance differences are not available, (11) becomes M
0 TDOAm i,k − TDOAi,k
+ M ri · [∆x
noise part
N 1 (ni − nc ). N q=1
M
k=1
M
= NLOSi +
Le (BSi ) =
∆y]T + (ni − nc ).
(25)
It is evident that the NLOS error and the intermediate MS location error cannot be separated. As a result, NLOS-erroridentification and mitigation performance will deteriorate. F. NLOS-Error-Mitigation Capability The success of the proposed algorithm depends on the capability of accurate estimation of the intermediate MS location. In fact, the intermediate MS location can be correctly estimated only when there are multiple clear hyperbola intersections. This is because the biased intersections are generally randomly distributed. As a result, multiple clear intersections can provide a dominant intersection distribution peak near the true MS location. Therefore, NLOS-error-mitigation capability can be evaluated by the number of clear intersections and the total number of intersections. The total number of intersections can be calculated via 3 4 + 3∗ CM p = CM
(26)
where p is the total number of intersections, M is the number of BSs, and Ctv stands for the combination operation. The first item is the number of intersections derived from three distinct BSs, and the second item is the number of intersections derived from four distinct BSs. Similarly, the number of clear intersections when there are n NLOS errors is 3 ∗ 4 q = CM −n + 3 CM −n .
(27)
The first item is the number of intersections calculated from three distinct NLOS-free BSs, and the second item is the number of intersections calculated from four distinct NLOSfree BSs.
MA et al.: NONLINE-OF-SIGHT ERROR-MITIGATION METHOD FOR TOA MEASUREMENTS
647
TABLE II CLEAR INTERSECTIONS VERSUS TOTAL INTERSECTIONS IN TDOA WIRELESS LOCATION
Table II is a summary of the number of clear intersections and the number of total intersections. Obviously, one cannot mitigate NLOS errors if the number of BSs is less than or equal to four because of the lack of redundancy. With five BSs heard, up to one NLOS error can be identified and mitigated. With six BSs heard, up to two NLOS errors can be identified. From the above discussion, it is clear that the algorithm proposed consists of essentially two steps. The first is the derivation of the intermediate MS location from the intersection DF. The second is the identification and mitigation of NLOS errors in order to improve positioning accuracy. Note that these two steps are in fact independent. Although the second step requires a satisfactory intermediate MS location to construct the cost functions for effective NLOS-error mitigation, the intermediate locations proposed in this paper are not the only ones that may be used. Two very possible solutions are a global-navigation satellite system (GNSS) and a low-cost inertial-navigation system (INS), from which independent-MSpositioning information may be obtained. The integration of the cellular network, GNSS and INS may be a more effective means of providing a more accurate intermediate MS location to the NLOS-error-mitigation algorithm. This is especially true when none of the individual systems are independently reliable. Although the algorithm here also uses system redundancy to identify and mitigate NLOS errors, its computational burden is not particularly heavy, especially when the intermediate MS location is provided by a GNSS/INS method rather than a DFbased method. This is because system redundancy is only explored in the derivation of the intermediate MS location, which is the first step of this method. In addition, only a small amount of computation needs to be completed in the second step—the cost function derivation—and NLOS-error mitigation, as shown in (11)–(21). In the first step, two types of computation effort need to be made: One is the hyperbola-intersection calculation, and the other is the determination of the maximum point of the DF. As shown in Table II, the more BSs used the greater the number of intersections and, therefore, the higher the computational complexity. To decrease the computational load, simple solutions to derive the intersection of two hyperbolas need to be adopted. However, the values of the DF at all intersections must be calculated since the maximum point of the distribution needs to be found. IV. S IMULATION R ESULTS Monte Carlo simulation tests were conducted to verify the performance improvement due to the proposed NLOS-error-
Fig. 7. Successful NLOS-error-detection probability with two NLOS errors (250, 450 m), (350, 550 m), and (450, 750 m).
identification and mitigation algorithm. In the simulation, the seven-cell 2-D cellular system, shown in Fig. 1, was used, and the MS to be located is in the central hexagonal cell surrounded by six adjacent hexagonal cells of the same size. The following subsections present the results. A. Successful NLOS-Error-Detection Probability With Respect to the Number of BSs Used Two types of receiver noise were simulated to evaluate NLOS-error-detection capability. The first type of receiver noise added to the TOA measurements was assumed to have a standard deviation of 70 m. This value comes from the work of Wylie and Holtzman [4] and is currently thought to be pessimistic. The second type of receiver noise was assumed to have a standard deviation of 25 m, which is obtainable via advanced receiver techniques such as those used in modern GPS receivers [14]. The cell radius was 3 km, the static MS to be located was at (700, 1200 m), and each Monte Carlo test contained 500 independent runs. Fig. 7 shows the successful NLOS-error detection probabilities when two NLOS errors exist at BS2 and BS4. A successful NLOS-error detection is defined here as a correct identification of NLOS-free BSs and NLOS-corrupted BSs. The horizontal axis represents the number of BSs used, and the vertical axis represents the successful detection probability. The six scenarios summarized in Table III were studied. It is obvious that the identification of NLOS errors is easier when the receiver noise is smaller and that the larger the NLOS
648
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
TABLE III SIX SCENARIOS IN THE TEST OF SUCCESSFUL NLOS-ERROR-DETECTION PROBABILITY
Fig. 9.
Positioning accuracy with three NLOS errors (250, 350, 450 m).
Fig. 8. MS-position-estimation accuracy with one NLOS error of 400 m.
errors, the higher the detection probability. However, it cannot be guaranteed that successful detection probability increases with an increase in the number of BSs used, although higher redundancy is preferred. This conclusion is supported by (25), which shows that the NLOS-error-identification variable γi , while a function of NLOS errors, receiver noise, and errors resulting from the intermediate MS location error, is in fact independent of the number of BSs used. However, a greater number of BSs is still preferred since this will improve geometry and as a result positioning accuracy. B. MS Location Accuracy The positioning accuracies of three methods are compared here. The first is the LS method with the proposed NLOS-errordetection and correction method and is denoted as NLOS-LS. The second is the LS method without the proposed NLOSerror-detection and correction method and is denoted as RAW-LS. The third, which is denoted as DF, is the DF method in which the intermediate MS location derived from an intersection DF is directly used as the MS location solution. Fig. 8 demonstrates the performance of these three methods when there is only one 400-m NLOS error at BS2. Fig. 9 demonstrates the positioning accuracy when there are three NLOS errors at BS2 (250 m), BS4 (350 m), and BS7 (450 m). Note that better NLOS-error mitigation is obtained when re-
Fig. 10.
Positioning accuracy with σTOA = 100 m.
ceiver noise is small. The RAW-LS method has the worst positioning accuracy since it does not mitigate NLOS errors, whereas the NLOS-LS and DF methods produce better results, especially when receiver noise is small. C. Positioning-Accuracy Improvement in Multipath-Propagation Environments To evaluate the benefits of this NLOS-error-mitigation method in real-world situations, the method was applied to a multipath-propagation channel, where NLOS errors are assumed to be time-variant and have exponential distribution characteristics in urban areas, as suggested by Yacoub [15] and Greenstein et al. [16]. The seven-cell system was again used, and the cell size was 3 km in radius. The exponentially distributed NLOS errors were simulated by the inversion method [17]. The location accuracy is evaluated with respect to the distance between the MS and the MS-serving BS (i.e., BS1). Fig. 10 illustrates the location accuracies of the different algorithms when the receiver noise is σTOA = 100 m.
MA et al.: NONLINE-OF-SIGHT ERROR-MITIGATION METHOD FOR TOA MEASUREMENTS
649
Fig. 11. Positioning accuracy with σTOA = 35 m.
The horizontal axis represents the MS-serving BS separation, and the vertical axis represents rms values of positioning errors. Note that positioning accuracy may degrade with the use of NLOS-error mitigation when the receiver noise is large. This is because large receiver errors make it difficult to correctly estimate NLOS errors and, thus, may introduce larger errors in MSlocation computation. However, if a cost-function-smoothing technique is applied, the receiver noise can be decreased. As shown in the figures, when the cost functions of 30 consecutive epochs are averaged, better NLOS mitigation and superior location performance is achieved. Fig. 11 illustrates the location accuracies of the different algorithms when the receiver noise is σTOA = 35 m. Due to the smaller receiver noise in these two cases, the performance of the NLOS-error-mitigation method is much better than that of a normal LS method without NLOS mitigation. In a real multipath-propagation environment, NLOS errors are timevariant, and all of the BSs may have NLOS errors, so the number of NLOS-corrupted BSs is beyond the capability of the proposed method. Even so, the positioning accuracy can still be significantly improved. D. Performance Improvement via GPS-Assisted NLOS Mitigation This section demonstrates the effectiveness of this NLOSmitigation algorithm when external GPS aiding is available to provide the intermediate MS location. An automobile kinematic test was conducted in an urban area near the University of Calgary. In the test, the GPS receiver used was a NovAtel OEM4 GPS sensor, and the cellular network assumed was a seven-cell system with a cell radius of 2 km. The system layout is shown in Fig. 12, where red triangles represent BSs, and the thick blue plot represents the benchmark automobile trajectory as derived from GPS measurements. The GPS derived trajectory is accurate enough to act as the benchmark, since its rms error is generally less than 5 m, while that of cellular-networkderived location can reach 100 m. In the test, the GPS data
Fig. 12. System layout for simulation. TABLE IV PERFORMANCE IMPROVEMENT DUE TO NLOS-ERROR MITIGATION
used was the real GPS data, whereas the cellular-network TOA data used was simulated data. The simulated data was generated by first calculating the true TOA value from the benchmark trajectory and, then, corrupting the true value with measurement noise and NLOS errors. Similar to the previous tests, the receiver noise was assumed to be zero-mean Gaussian noise with a standard deviation of 10 m, and the NLOS errors were assumed to be of the urban exponential distribution, as discussed previously. Three methods were evaluated: TDOA solution without NLOS-error mitigation; TDOA solution with DF-based NLOSerror mitigation; and TDOA solution with GPS-assisted NLOSerror mitigation. Simulation results in Table IV and Fig. 13 demonstrate the performance improvement resulting from the two NLOS mitigation methods. The benchmark trajectory (indicated in red) in each of the plots was again derived from GPS measurements. The results demonstrate that positioning accuracy is poor if the NLOS errors are not removed from the TOA/TDOA measurements since NLOS errors are the dominant error sources and can reach several hundred meters. DF-based NLOS-error
650
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 2007
only 7–15 m. Of course, providing accurate intermediate MS location by standalone GPS is not very feasible in real-world applications, especially in serious satellite blockage areas, such as city core. In these cases, the integration of multiple positioning systems to improve performance is always preferred. The performance superiority of the GPS-assisted NLOSerror-mitigation method can be explained as follows. In the DFbased NLOS-error-mitigation solution, the intermediate MS location is derived exclusively from TOA measurements. The accuracy is poor because large errors exist in measurements and the number of NLOS errors exceeds the algorithm capability. On the contrary, in the GPS-assisted NLOS-error-mitigation solution, a better MS-location estimation may be directly available from a GPS receiver. Taking the GPS-derived position as the required intermediate MS-location estimation, the capability of the proposed algorithm may be significantly improved, and thus, NLOS errors may be effectively identified and mitigated. The result will be improved positioning accuracy.
V. C ONCLUSION The presence of NLOS errors is a serious issue in wireless location, because these errors tend to be very large and, hence, dramatically degrade positioning accuracy. As a result, they must be mitigated or removed before fixing the position. The novel NLOS-error-mitigation method proposed in this paper depends on system redundancy or high mobile-BS hearability such that the greater the number of BSs used, the higher the number of NLOS errors that can be detected and mitigated. Simulation results verified that with this NLOS-errormitigation method, the positioning error could be decreased from several hundred meters to tens of meters, depending on NLOS errors and receiver noise. The smaller the receiver noise, the better the NLOS-error mitigation. The receiver noise can be reduced by a cost-function-smoothing technique, since receiver noise is a high-frequency component, and the NLOS error is a relatively low-frequency component. External adding is another possible method to improve the performance. In this paper, GPS is used as an example to provide the intermediate MS location and thereby improve the performance of NLOS-error mitigation. The simulation results demonstrate the effectiveness of this innovative method. R EFERENCES
Fig. 13. Performance improvement due to NLOS-error mitigation. (a) Seven TOAs without NLOS mitigation. (b) Seven TOAs with DF-based NLOS mitigation. (c) Seven TOAs with GPS-assisted NLOS mitigation.
mitigation can significantly improve positioning accuracy. The horizontal positioning error decreases from around 70 m for the case of no LOS error mitigation to around 40 m for the DFbased NLOS-error-mitigation case. The GPS-assisted NLOSerror-mitigation solution produces the best performance. The rms of the final horizontal location error is in the range of
[1] FCC, Fact sheet: FCC Wireless 911 Requirements, 2001. [Online]. Available: http://www.fcc.gov/911/enhanced/releases/factsheet_ requirements_012001.pdf [2] J. Caffery, Jr. and G. Stüber, “Overview of radiolocation in CDMA cellular systems,” IEEE Commun. Mag., vol. 36, no. 4, pp. 38–45, Apr. 1998. [3] S.-S. Woo, H.-R. You, and J.-S. Koh, “The NLOS mitigation technique for position location using IS-95 CDMA networks,” in Proc. IEEE VTC—Fall, Sep. 2000, vol. 4, pp. 2556–2560. [4] M. P. Wylie and J. Holtzman, “The non-line of sight problem in mobile location estimation,” in Proc. IEEE Int. Conf. Univer. Pers. Commun., 1996, vol. 2, pp. 827–831. [5] J. Borrás, P. Hatrack, and N. B. Mandayam, “Decision theoretical framework for NLOS identification,” in Proc. IEEE VTC, 1998, vol. 2, pp. 1583–1587. [6] P.-C. Chen, “A non-line-of-sight error mitigation algorithm in location estimation,” in Proc. IEEE Wireless Commun. Netw. Conf., 1999, vol. 1, pp. 316–320.
MA et al.: NONLINE-OF-SIGHT ERROR-MITIGATION METHOD FOR TOA MEASUREMENTS
[7] S. Venkatraman and J. Caffery, “A statistical approach to non-lineof-sight BS identification,” in Proc. IEEE WPMC Conf., Honolulu, HI, 2002, pp. 296–300. [Online]. Available: http://www.ececs.uc.edu/ ~jcaffery/nlos_i.pdf [8] S. Venkatraman, J. J. Caffery, and H.-R. You, “Location using LOS range estimation in NLOS environments,” in Proc. IEEE VTC—Spring, Birmingham, AL, May 2002, pp. 856–860. [9] L. Cong and W. Zhuang, “Non-line-of-sight error mitigation in TDOA mobile location,” in Proc. IEEE Globecom, 2001, pp. 680–684. [10] D. Bartlett, presented at the Hearability Analysis for OTDOA Positioning in 3G UMTS Networks, CPS White Paper CTN-2002-4-V 1.0. [11] Ericsson, Recapitulation of the IPDL Positioning Method, 1999. TSGR1, 4(99)346. [12] C. Ma, “Techniques to improve ground-based wireless location performance using a cellular telephone network,” Ph.D. dissertation, Dept. Geo. Eng., Univ. Calgary, Calgary, AB, Canada, Jun. 2003. Rep. 20177. [Online]. Available: http://www.geomatics.ucalgary.ca/links/ GradTheses.html [13] Y. Gao, “A robust quality control system for GPS navigation and kinematic positioning,” Ph.D. dissertation, Dept. Geomatics Eng., Univ. Calgary, Calgary, AB, Canada, Jun. 1993. Rep. 20075. [14] E. D. Kaplan, Understanding GPS: Principles and Applications. Norwood, MA: Artech House, 1996. [15] M. D. Yacoub, Foundations of Mobile Radio Engineering. Boca Raton, FL: CRC, 1993. [16] L. J. Greenstein, V. Erceg, Y. S. Yeh, and M. V. Clark, “A new path gain/delay spread propagation model for digital cellular channels,” IEEE Trans. Veh. Technol., vol. 46, no. 2, pp. 477–484, May 1997. [17] L. Devroye, Non-Uniform Random Variate Generation. New York: Springer-Verlag, 1986.
Changlin Ma received the B.S. and M.S. degrees from Northwestern Polytechnical University, Xi’an, China, in 1992 and 1995, respectively, and the Ph.D. degree from Tsinghua University, Beijing, China, in 1998, all in electrical engineering. He received the Ph.D. degree in geomatics engineering from the University of Calgary, Calgary, AB, Canada, in 2003. From 1998 to 2000, he was with the Telecommunications Ltd., focusing on network communications. From 2003 to 2005, he worked as a Senior Research Associate with the Department of Geomatics Engineer, University of Calgary, where his interests focused on GPS receiver design and wireless location. He is currently with the Centrality Communications Ltd., Santa Clara, CA, focusing on high-sensitivity global navigation satellite systems receiver design.
651
Richard Klukas received the B.Sc. and M.Sc. degrees in electrical engineering and the Ph.D. degree in geomatics engineering from the University of Calgary, AB, Canada. From 1997 to 2001, he was with Cell-Loc Inc., where he coinvented Cell-Loc’s patented wirelesslocation technology. He also has industrial experience with Northern Telecom Canada (now Nortel Networks). He is currently an Assistant Professor with the School of Engineering, University of British Columbia Okanagan, Kelowna, BC, Canada. He is the author of a number of papers on wireless positioning.
Gérard Lachapelle (M’81) received the degrees in surveying and geodesy from Laval University, Quebec City, QC, Canada, the University of Oxford, Oxford, U.K., the University of Helsinki, Helsinki, Finland, and the Technical University at Graz, Graz, Austria. He is currently the CRC/iCORE Chair in Wireless Location with the Department of Geomatics Engineering, the University of Calgary, Calgary, AB, Canada, where he has been a Professor since 1988. He has been involved in a multitude of global navigation satellite systems R&D projects since 1980, ranging from Real-Time Kinematic (RTK) positioning to indoor location.
