Non-Line-of-Sight Error Mitigation in TDOA Mobile Location
Li Cong and Weihua Zhuang
Centre for Wireless Communications (CWC)
Department of Electrical and Computer Engineering
University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Abstract— In this paper, we investigate the non-line-of-sight (NLOS)
propagation identification and correction for time difference of arrival
(TDOA) based mobile user location in wireless communication systems.
Based on the defined TDOA residual, an NLOS base station identification
algorithm is proposed. Different choices of the reference location for the
residual calculation are compared via simulation. To correct the NLOS
error with a certain distribution, we propose a maximum likelihood (ML)
estimator for TDOA location systems. Simulation results demonstrate
that the proposed NLOS recovering algorithm performs better than that
using only LOS measurements, especially when the number of available
base stations is small and/or the LOS base stations have an undesirable
geometric layout.
I. I NTRODUCTION
Mobile location is to determine the position of a mobile station (MS) in a wireless network. It has received considerable
attention over the past few years. In addition to the emergency
911 (E-911) subscriber safety services, wireless location information can be used for location-sensitive billing, intelligent
transport system (ITS), resource management and performance
enhancement in mobile cellular networks.
The major sources of error in time difference of arrival
(TDOA) mobile location are measurement noise and non-lineof-sight (NLOS) propagation error. Measurement noise is usually modeled as a zero-mean Gaussian random variable, while
NLOS error usually has an unknown distribution with a positive mean. NLOS error is the dominant error in location estimation [1]. A field test shows that the average NLOS range
error can be as large as 0.589 km in an IS-95 code-division
multiple access (CDMA) system [2].
To protect location estimates from NLOS error corruption, NLOS identification and reconstruction techniques have
been investigated. Generally NLOS range measurements have
a larger variance than LOS range measurements, especially
when the MS is moving. In [2], an NLOS base station (BS)
is identified by calculating the standard deviation of a series of
range measurements and comparing that with a certain threshold. A Kalman filter is then used with an extra unknown constant, the NLOS range error, to reconstruct the NLOS error and
obtain the location estimate. A time-history based hypothesis
test is proposed in [3] to identify and then to reconstruct the
NLOS error. In [4], a decision theoretic framework for NLOS
identification is formulated, where the NLOS error is modeled
This work was supported by research grants from the Canadian Institute for
Telecommunications Research (CITR) and the Geomatics for Informed Decisions (GEOIDE).
as a non-zero mean Gaussian random variable. If the stochastic model for the NLOS error is not Gaussian, the approach in
[4] relies only on the fact that the variance of the NLOS errors
is greater than that of the LOS errors. For an unknown NLOS
error distribution, a residual weighting algorithm is proposed
in [5] for a time of arrival (TOA) location system to identify
the BS which suffers from NLOS propagation, based on the
weighted residuals for all possible BS combinations. A similar residual algorithm for an angle of arrival (AOA) location
system is proposed in [6].
Other approaches have been proposed in the literature to alleviate the NLOS problem without the knowledge of which
BS(s) suffers from NLOS propagation. A mapping method using dynamic location database is proposed in [7] to improve the
global positioning system (GPS) accuracy in an NLOS propagation situation. Using a local building information database,
the ray-launch technique gives an improved estimate of the mobile location. In [8], robust estimation functions are proposed
for mobile location based on path loss measurements. Since
the NLOS error is non-symmetric, a pseudo-median estimator
is used to work with a small number of samples. Simulation results show that the proposed robust estimator maintains a low
estimation error when several of the measurements are contaminated by NLOS propagation.
In this paper, we investigate the NLOS error in TDOA mobile location, and propose an algorithm for NLOS BS identification and NLOS error mitigation. The algorithm is not
time-history based, and thus is not limited to a low MS mobility situation. The proposed algorithm can be extended to
TOA, AOA and hybrid mobile location. Simulation results are
presented to demonstrate the location accuracy improvement
when applying the proposed algorithm in NLOS situations.
II. S YSTEM M ODEL
The system model under consideration is a wideband
CDMA cellular system. We focus on the case of macrocells
and two-dimensional (2-D) mobile location. The BS serving
the target MS (to be located) is referred to as home BS for the
MS and is denoted by BS1 . All the neighboring BSs can be involved in an MS location process, provided the signal-to-noise
ratio of the signal from each BS is above a certain threshold at
the MS. Those that do not have a line of sight (LOS) propagation path to the MS are referred to as NLOS BSs. All the BSs
are precisely synchronized based on the GPS time reference.
The strict synchronization requirement among the BSs facilitates the accurate TDOA measurements in the system. At all
0-7803-7206-9/01/$17.00 © 2001 IEEE
680
times the MS keeps monitoring the forward Pilot Channel signal levels from the neighboring BSs, and reports to the network
those that cross a given set of thresholds. The cross-correlators
at the MS receiver are capable of measuring the TDOA between the signal from BS 1 and that from any other BS.
When there are N ( 3) BSs available for the MS location, we have a set of nonlinear location equations. Because
of measurement noise and NLOS errors, the solution to the
over-determined equations is not unique. The equations that
incorporate the measurement noise and NLOS error are
Cti = Di
D1 + ni + ei
(i = 2;
; N)
(1)
where C is the speed of light, t i is the measured TDOA between BSi and BS1, Di is the distance between the MS and
BSi , D1 is the distance between the MS and BS1 , ni and ei are
the TDOA measurement noise and NLOS error, respectively.
We assume that ni is a Gaussian random variable with zero
mean and variance i. If BSi has an LOS path to the MS,
then ei = 0. For NLOS BSs, ei is a positive random variable
with mean nlos and variance nlos . We further assume that
Ænlos > i , which is consistent with field test results [2].
III. NLOS I DENTIFICATION
Consider the NLOS error for TDOA location. As illustrated
in Fig. 1, each TDOA measurement determines that the MS
must lie on a hyperbola with a constant range difference between the two BSs. The intersection points of these hyperbolas define an area of uncertainty where the final estimate of
MS location lies in. For a given set S k of BSs (in addition to
BS1) and a reference MS location (^
x; y
^), we define the TDOA
residual as the difference between the measured data and the
calculated data using the reference location
Rk =
X
BSi 2Sk
^
[(D
i
^ )
D
1
Cti]2
(2)
H3'
BS1
H3
p
(xi
x
^)2 + (yi
y^)2
(3)
and (xi ; yi ) is the BSi location.
If the measurement noise can be assumed to be zero mean
Gaussian noise with small variance and each BS has an LOS
path to the MS, the area of uncertainty will be small; in other
words, the hyperbolas will intersect at almost the same point –
the true location. If the reference location is close to the true
MS location, the TDOA residual will be close to zero.
As illustrated in Fig. 1, if one BS (BS 3) suffers from NLOS
propagation, it will push the corresponding hyperbola H 3 away
from itself since the signal from this NLOS BS has to travel an
extra distance to reach the MS. A new hyperbola H30 is formed
depending on the value of the NLOS error. Therefore, the intersection points of the new hyperbola H 30 with hyperbolas H 2
and H4, A and B , move further away from the true location,
forming a larger area of uncertainty. The TDOA residual will
generally become much greater than that of the LOS case.
As a result, it is reasonable to assume that if a large NLOS
error exists for a particular BS, on the average the residual will
be large in magnitude. In the following, we propose an algorithm to identify the NLOS BS. Suppose we have N BSs
available for the location estimate of the target MS. We assume that only one BS has NLOS propagation and this BS is
not the home BS. The minimum number of BSs, in addition to
BS1, required for a 2-D TDOA location estimation is 2. We
can thus group our measurements for various sizes (< N ) of
BS sets, each set containing BS1. For example, if N = 5, we
will have two groups of measurement data:
(a)measurement
4
data from BSs of size 4, with a total of
3
= 4
combi-
nations; and (b)
measurement data from BSs of size 3, with a
total of
(3
B
MS
^ =
D
i
4
2
= 6
combinations. For each BS set of size
M , we can calculate the intermediate MS location, and determine the TDOA residual for each combination. The residual
serves as a weighting factor to reflect the reliability of BSs in
the combination.
Our algorithm of identifying an NLOS BS in TDOA location has the following steps: (a)
For eachBS set of size M
H4
A
where
M < N ), form K =
N
M
1
1
TDOA measure-
; K , to denote the
ment combinations. Use Sk ; k = 1; 2;
kth combination of the BSs; (b) For each BS set S k , use a
reference MS location (^
xk ; y
^k ) to calculate the corresponding
TDOA residual Rk . Assign a weight of R k to the BS set Sk ;
(c) After evaluating all the K BS combinations, calculate the
H2
BS4
BS2
total weight of each BS by summing the weights of the BS sets
that the BS belongs to; (d) Rank each BS according to the total
weight, and pick up the one with the heaviest weight.
The choice of the reference MS location (^
xk ; y^k ) plays an
important role in the NLOS BS identification. Ideally we
should use the true location of the MS, but it is not achievable and can only be used as a performance benchmark. We
BS3
Fig. 1. NLOS effect on the TDOA location estimation.
681
can use measurement data from all the N BSs to determine an
overall location estimate, and use that as the reference location for all the combinations; alternatively we can use the BSs
in each combination Sk to obtain a current intermediate location estimate and use it as the reference for the combination.
The results of using different reference locations are compared
via computer simulation and are given in Section V.
methods to solve the maximization problem. As an example,
the exponential distribution delay profile model is widely used
to model the NLOS delay [11]. To derive the ML estimator for
the NLOS contaminated TDOA, we first derive the probability
density function (PDF) of the sum of the Gaussian noise n N
and the exponentially distributed NLOS error e N with mean
1=, given by
IV. NLOS E RROR C ORRECTION
Ideally the solution to the NLOS problem would be to detect which TDOA measurement had been contaminated by the
NLOS error and use only measurements with LOS propagation. However, in a practical cellular system the number of
available neighboring BSs for location purposes is always limited. Hence, we should use as much information as possible to
obtain the final location estimate. That is especially true when
the number of LOS BSs is small, or the geometric layout of
LOS BSs does not favor the location estimation.
fn+e (x) =
Cti = Di
CtN = DN
D1 + ni
(i = 2;
D1 + nN + eN :
;N
1)
(4)
(5)
This is a set of nonlinear equations with the unknown =
The measurement noise ni is a zero-mean Gaussian random variable, while the NLOS error eN is an unknown
constant. With the extra unknown element e N , the equations
can be solved using Taylor series linearization [9] or two-step
least square (LS) estimator [10].
(x; y; eN )T .
B. Gaussian NLOS Error
If we consider the NLOS error eN as a random variable, a
simple way to model it is to use the Gaussian distribution with
2
mean nlos and variance nlos
. An ML estimator can be used
to obtain the optimum location estimate. Assuming that the
Gaussian measurement noise nN has zero mean and variance
2
N
, the sum of eN and nN is also a Gaussian random variable
2
2
with mean nlos and variance nlos
+ N
. We can then obtain
the location estimate by using the same approach as in the case
of deterministic NLOS error, i.e., Taylor series linearization
[9] or two-step LS estimator [10]. The only difference is in the
noise mean and covariance matrix.
C. Other NLOS Distributions
For the non-Gaussian NLOS error with a known distribution, we can derive an ML location estimator and use numerical
682
2
e (x
2
N
)
2
[1 + erf (
2
x
p N )]
(6)
2N
where erf () denotes the error function. Rewrite (4) and (5) in
a matrix form
L = H() + V
(7)
where L = C (t ; t ; : : : ; tN )T , H( ) = (D
D ;D
D ; : : : ; DN D )T , V = (n ; n ; : : : ; nN + eN )T , and =
T
2
1
3
1
2
2
1
3
3
(x; y ) . The ML estimator aims at maximizing the conditional
joint PDF
A. Deterministic NLOS Error
The NLOS error depends on the propagation environment
and changes from time to time. However, at each time instance,
NLOS error can be treated as a constant. We can estimate the
value of the NLOS error when there are enough BSs available
to determine the MS location.
Let the NLOS BS be BSN . We rewrite the TDOA hyperbolic
equations as
f(
Lj) = fV (L H( )j):
(8)
Under the assumption that n i (i = 2; 3; : : : ; N ) and eN are
independent random variables, we have
L H()j) = 2 e
fV (
NY1
i=2
p1
2i
exp(
2
2
(x 2N ) [1 + erf( xp N )]
2N
(Cti
Di + D1 )2
2i2
):
(9)
This maximization problem is difficult to solve by analysis, so
numeric methods are used to find the ML estimate which
maximizes (9).
V. S IMULATION R ESULTS
This section presents simulation results to demonstrate the
performance of the proposed NLOS identification algorithm
and the ML location estimator with an exponentially distributed NLOS error. For simplicity, we assume that the variances of all the TDOA measurement noises are the same. For
the 2-D array BS layout, we consider a center hexagonal cell
(where the MS is located) with six adjacent hexagonal cells of
the same size, as shown in Fig. 2. The cell radius is set to be
5 km. The root mean square (RMS) location error is obtained
from the average of 10,000 independent runs.
A. NLOS BS Identification
Fig. 3 compares the NLOS BS false detection rates using
the proposed NLOS identification algorithm with the true,
overall, and current reference locations as described in
Section III. All the seven BSs are used to determine the MS location, and BS3 has the NLOS error of 0.2 km, 0.3 km, 0.4 km,
and 0.5 km, respectively. The standard deviation of the measurement noise is 0.1 km. The performance of the proposed algorithm is studied for different BS set sizes. To be consistent,
ence location does not change for each combination in the BS
set. Therefore, the location error due to the NLOS propagation
becomes less noticeable when the set size increases. The same
applies to the case of using the current reference location,
however the set size has to be large enough to provide enough
information for a reasonably accurate intermediate estimate.
From the simulation results, with the unknown true location,
the algorithm using the current reference location and 5-BS
set gives the best performance, followed by the one using the
overall reference location and 3-BS set.
BS6
BS5
BS7
Home BS1
MS
BS4
BS2
B. NLOS Error Correction
Consider N = 4. Among the 4 BSs (BS 1, BS2, BS3 and
BS4), TDOA measurements with BS2 suffer from an exponentially distributed NLOS error with a mean of NLOS = 0:1 km
and 0.5 km respectively. The TDOA measurement noise for
each BS pair is assumed to be zero mean Gaussian noise with
standard deviation of 0.1 km. Figs. 4-6 show the RMS location
error using the proposed ML location estimator with NLOS
correction, the ML estimator with the LOS measurement only from the 3 BSs, and the estimator without
NLOS correction with all the LOS and NLOS measurement data, respectively. It is observed that: (a) the proposed
NLOS error correction greatly reduces the location error, and
the location error increases negligibly with the NLOS error;
(b) without NLOS error correction, the location error increases
significantly as the NLOS error increases; (c) the proposed ML
estimator with NLOS correction performs consistently better
than that using only the LOS measurement data, since it uses
one more BS — the NLOS BS. The measurement from the
NLOS BS helps to improve the location accuracy. However,
when the NLOS error is large, the improvement is negligible.
Location estimation depends heavily on the BS/MS geometric layout when the number of the available BSs is small. If,
BS3
Fig. 2. The 7-cell system layout.
90
True
Overall
Current
0.2 km NLOS error
0.3 km NLOS error
0.4 km NLOS error
0.5 km NLOS error
80
False detection rate (%)
70
60
50
40
30
20
10
0
3
4
5
6
Number of BSs in each set
Fig. 3. Comparison of the NLOS BS false detection rates.
0.6
683
With NLOS correction
Without NLOS correction
Using LOS measurements only
0.5
0.4
RMS Error (km)
only Taylor-series method is used to calculate the final location
estimate. The initial position guess in the Taylor-series method
is chosen to be the true solution to allow for convergence.
It is clear that false detection rates decrease as the NLOS error increases. The identification algorithm using the true location as reference always gives the best performance, and the
false detection rate increases with the set size. This is because
the bias caused by the NLOS error is being averaged out when
more BSs are involved in the location estimation. The false
detection rates using the overall reference location change
with the BS set size in a similar pattern to that with the true
reference location. However, when using the current reference location, the false detection rate first decreases with the
set size, then starts to increase when the set size is larger than
5. This can be explained as follows. When using the true
or overall reference location, the smaller the set size, the
higher the resolution in the identification, because the refer-
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Probability of location error smaller than ordinate
0.8
0.9
1
Fig. 4. Comparison of 4 BSs location error, with BS 2 having an NLOS error
(NLOS = 0:1km).
for example, BS3 suffers from the NLOS propagation instead
of BS2 , the result will be quite different, as shown in Fig. 5.
Because the LOS BSs (BS1 , BS2 and BS4 ) are located almost
linearly on the plane, using only the LOS measurements will
introduce a large error in the final location estimate. In this
case, the proposed ML estimator with NLOS error correction
performs significantly better than that using only the LOS measurement data. Table I shows the results of the RMS location
error (in km) for different numbers of the available BSs, among
which BS3 is the NLOS BS with NLOS (in km). The NLOS
recovering algorithm always gives the best performance. However, as the number of BSs increases, the difference between
the NLOS recovering algorithm and the algorithm using only
LOS measurements becomes negligible.
0.7
With NLOS correction
Without NLOS correction
Using LOS measurements only
0.1 km NLOS
0.5 km NLOS
0.6
RMS Error (km)
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Probability of location error smaller than ordinate
0.8
0.9
TABLE I
RMS LOCATION ERROR ( KM ) FOR DIFFERENT NUMBERS OF BS S .
no NLOS
correction
with NLOS
correction
LOS Only
With NLOS correction
Without NLOS correction
Using LOS measurements only
0.5
RMS Error (km)
0.4
0.3
0.2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Probability of location error smaller than ordinate
0.8
0.9
N =6
N =7
0.101
0.208
0.095
0.099
0.100
0.081
0.160
0.075
0.078
0.081
0.064
0.110
0.061
0.062
0.064
R EFERENCES
0.6
0.1
N =5
0.230
0.565
0.192
0.248
0.400
We have proposed the NLOS identification algorithm and
the ML location estimator with NLOS error correction for
TDOA mobile location in a wideband CDMA cellular system.
The NLOS identification is based on the TDOA residual calculation. We group all the available BSs into sets of various
sizes, and use either the overall or the current reference location to obtain a residual for each set. The ranking
of the total residual for each BS determines the NLOS BS. Using the overall reference location works better with a small
BS set size, while using the current reference location gives
smaller false detection rate for a large BS set size. For NLOS
errors with a known distribution, the ML estimator reduces the
NLOS location error. Simulation results demonstrate that the
proposed ML estimator with NLOS correction performs consistently better than that using only LOS measurements, especially when the number of available BSs is small and/or the
LOS BSs have an undesirable geometric layout.
1
Fig. 5. Comparison of 4 BSs location error, with BS 3 having an NLOS error.
0
N =4
0.1
0.5
0.1
0.5
VI. C ONCLUSIONS
[1]
0
NLOS
1
Fig. 6. Comparison of 4 BSs location error, with BS 2 having an NLOS error
(NLOS = 0:5km).
684
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