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## Gps-tsc.upc.es

**NLOS MITIGATION BASED ON A TRELLIS SEARCH FOR WIRELESS LOCATION**
*Andreu Urruela, Helena Morata and Jaume Riba*
Signal Processing and Communications Group. Technical University of Catalonia (UPC)
Campus Nord, Ed. D5, Jordi Girona 1-3, D5, 08034, Barcelona (Spain)
Emails:

*{*andreu,alhelena,jriba

*}*@gps.tsc.upc.edu

**ABSTRACT**
estimating the variance of the incoming observation, so this ap-proach is not very suitable for high dynamic scenario.

Wireless location using Time of Arrival (TOA) and Time-Difference-
Other papers, like [4] are based on the idea that the NLOS
Of-Arrival (TDOA) measurements has received considerable at-
process could be intermittent. This means that the BSs can be in
tention over the last years to be the best selection for cell phone
LOS or in NLOS in an intermittent way. This model is intuitively
location. The major problem related with these types of mea-
correct when we are talking about in-car applications where the
surements is the Non-Line-Of-Sight problem (NLOS) that happens
mobile is supposed to be moving and seeing the BSs in LOS or
when the direct path between the Base Stations (BSs) and the mo-
NLOS intermittently. The idea proposed in [4] consists in exploit-
bile is blocked. This paper presents a new technique to mitigate
ing this intermittent effect of the NLOS to discard the NLOS mea-
the NLOS effect in dynamic scenarios based on a Trellis search
surements. The problem here is that in [4] it is assumed that the
of the NLOS state. Numerical simulation shows that the proposed
mobile is static during the observation period, which is contradic-
technique outperforms the previous contributions because it is able
tory with the hypothesis that the NLOS is intermittent.

to detect and reject the NLOS measurements.

Also based on the idea that the NLOS effect can be intermit-
tent, other papers like [5] and [6] try to pre-filter the timing mea-

**1. INTRODUCTION**
surements to, somehow, eliminate the effect of the NLOS from themeasurements. The techniques used are a biased Kalman filter in
Wireless location has received considerable attention in the last
[5] and a filter based on a certain scattering model in [6]. Obvi-
decades. This is partially motivated by the fact that location al-
ously, the authors in these last papers made assumptions about the
gorithms will be mandatory for public cell networks, and partially
nature of the NLOS errors, so the performance of such algorithms
motivated by the interesting market in position-based services as
is logically based on how well the real measurements errors fit this
car-routing or position-based publicity. Another interesting feature
model. On the other side, this type of approaches does not use the
is the possible optimization of the network planning based on the
relationship between the measurements captured simultaneously
from different BSs. This is, all the timing measurements coming
To estimate the position of the mobile, several techniques have
from a certain BS are pre-filtered independently from the rest of
been proposed during the last years. The most successful tech-
measurements, not exploiting the relationship between them.

nique seems to be based on timing measurements such as TOA
In this direction, [7] tries to exploit the relationships between
or TDOA. The advantages of using these observations are that the
three TOA measurements captured simultaneously from different
accuracy achieved is in the order of tens of meters, that is what
BSs, trying to mitigate the NLOS errors. However, it is difficult to
is required for emergency calls, and that the investments needed
extent the proposed algorithm to more TOA measurements or even
other types of measurements. Maybe the most significant contri-
The main obstacle between the use of TOA or TDOA obser-
butions in this direction are [8] and [9]. In both articles, the authors
vations towards an accurate location system is the Non-Line-Of-
formulate all the possible hypothesis of LOS/NLOS for a specific
Sight effect. This effect mentioned as the

*killing effect *in [1], is by
instant of time, i.e. each hypothesis is characterized by which BSs
far the most important cause of big errors in the positioning algo-
are under LOS and which BSs are under NLOS. Latter on, the ML
rithms based on timing measurements. Note that NLOS errors can
position estimates are obtained for each hypothesis. Finally the
be in the order of hundreds of meters.

solution presented in [8] consists in a linear weighted combination
Good algorithms to mitigate the NLOS effect have been pre-
of the partial position estimates associated to each hypothesis. The
sented in the previous literature. One of the first approaches pre-
algorithm presented in [9] tries to identify what is the most likely
sented was based on the observation that the timing measurements
hypothesis using the incoming timing measurements based on the
captured in a NLOS channel are normally corrupted with a stronger
application of the ML detection principle. The major problem with
noise, i.e. bigger variance. Using this concept, [2] and [3] pro-
these last approaches is that they exploit only the relationship be-
posed to discard the timing measurements with bigger variances
tween timing measurements captured at the same time (one snap-
using a certain detection criteria. The main problem here is that
shot), but they do not exploit the relationship between the timing
the variance of a timing observation in a LOS scenario depends
measurements along the time axis as in [5] and [6].

strongly on the type of scenario, i.e. urban, suburban or rural. An-
This paper tries to extend the work presented in [9] to dynamic
other problem is that the detection process could take some time
scenarios, exploiting also the relationship between the timing ob-
servations among different snapshots of measurements. In this
This work has been partially supported by the European Comission
under IST project EMILY IST-2000-26040, by the European Commis-
sense the new algorithm exploits the relationship between mea-
sion (FEDER) and by the Spanish/Catalan Government under projects
surements in both directions: inside the snapshot and along the
TIC2003-05482, TEC2004-04526 and 2001SGR-00268
time axis. The proposed algorithm assumes, like previous ap-
proaches [4], that the NLOS effect is intermittent, but it is coherent
A common assumption in wireless location is that the multi-
also with assuming that the mobile is in a dynamic scenario. No
path terms

*wk,n *are independent in time among sources, and Gaus-
specific assumption of the NLOS error is done, so the proposed al-
gorithm exploits only the coherence between the incoming timingmeasurements using certain assumptions about the mobile trajec-

*E *[

*wk,nwk ,n *] =

*σ*2

*kδk−k δn−n*
tory. As it will be seen, the natural extension of [9] for dynamic
scenarios leads to an unafordable complexity, so a Trellis-based

*p *is the Kronecker Delta. For a generalization with un-
known variance, we refer the reader to [10].

search algorithm is proposed to track the state (LOS or NLOS) of
previous one but three common assumptions are done in previous
A model for the intermittent effect of the NLOS is also pre-
sented to simulate the proposed algorithm. Simulation results willshow that the proposed algorithm outperforms the previous ap-

**(a) **The NLOS term for each source (subindex

*k*) is indepen-

proaches and it is near to the optimum that is ideally defined as
dent from that of the other sources.

the ML estimator that only uses the timing measurements under

**(b) **In NLOS conditions, a significant bias will be added to the

measurement and this bias will be positive for TOA mea-surements. The distribution for this bias is usually assumed

**2. SIGNAL MODEL**
as uniform but no standard models have been presented sothey will be assumed to be completely unknown.

Let us assume an scenario where

*L *BSs are under the visibility

**(c) **The NLOS effect in dynamic scenarios is intermittent, so

of the mobile. In this scenario, the mobile can periodically obtain
the NLOS term will be zero when the measurement is un-
snapshots of

*L *TOA measurements associated to each one of the
der LOS conditions and will be more or less constant in
BSs or

*L − *1 independent TDOA measurements associated to all
the independent pairs between the reference BSs and the neigh-bor BSs. To unify the notation of the development for the caseof TOA and TDOA observations, let us define

*K *as the number

**3. SINGLE SNAPSHOT ESTIMATION**
of measurements we have at each instant of time (each snapshot),i.e.

*K*=

*L *for TOA measurements and

*K*=

*L − *1 for TDOA mea-
This section presents the ML estimation of the position at the

*n*-th
surements. These

*K *measurements will be referred to as

*sources*
instant of time,

**x***n*, using only a single snapshot of

*K *measure-

(because they produce a new measurement at each instant of time)
ments as shown in (1). The goal here is to obtain the ML estima-
indicating single BSs for the TOA case and pairs of BSs for TDOA
tion of

**x***n *being robust against the NLOS effect. This represents

an introduction to the multiple snapshot case and a review of pre-
Let us define the

*K *observations (

*k*=[1,

*K*]) taken at the

*n*-th
If we have

*K *measurements that can be in LOS or in NLOS,
there are

*Q *= 2

*K *possible hypothesis regarding the NLOS state.

*tk,n *=

*gk *(

**x***n*) +

*wk,n *+

*lk,n*
For the

*q*-th hypothesis, we can define

*γLOS*
vectors containing the subindex of the sources that are supposed to
where

*wk,n *is the noise term caused by the additive noise and the
multipath effect in the TOA/TDOA estimation,

*lk,n *is the error
Under the

*q*-th hypothesis, we can divide the

*K *original sam-
caused by the NLOS effect and

*gk *(

**x***n*) is the non-linear function

ples

*tk,n *in two subvectors

**t***LOS*
and

**t***NLOS*
relating the observations and the parameter of interest, i.e.

**x***n *that

measurements that are in LOS and NLOS respectively as follows:
is the position of the mobile at the

*n*-th instant of time.

The non-linear function

*gk *(

**x***n*) in (1) does not depend on the

*tk,n ∈ ***t***LOS*
temporal subindex

*n *and it is defined specifically for the type of

*tk,n ∈ ***t***NLOS*
In a coherent way we can define also the non-linear functions as-
(

**x***n*)

**x***n − ***x***BS*
(

**x***n*)

**x***n − ***x***BS*
**x***n − ***x***BS*
*gk *(

**x***n*)

*∈ ***g***LOS*
*n,q *(

**x***n*)

where

**x***BS*
is the known position of the

*k*-th BS.

*gk *(

**x***n*)

*∈ ***g***NLOS*
(

**x***n*)

If the mobile captures

*N *snapshots of measurements at a con-
stant rate of

*r *snapshots per second, we would like to model the
In the same direction, we can divide the original

*K *variances

*σ*2

*k*
evolution of the mobile position at each instant of time as a func-
in two diagonal matrices containing the variance associated with
tion of a finite number of parameters as:
the measurements that are supposed to be in LOS and NLOS re-spectivelly under the

*q*-th hypothesis as follows:

**x***n *=

**h***n *(

**x***, ***u**)

*k ∈ *diag

**R**LOS

*n *(

**x***, ***u**) does not depend on the source subindex

*k *and

shows the mobile position at the

*n*-th instant of time based on the
position

**x **at the end of the observed window and the rest of move-

*k ∈ *diag

**R**NLOS

ment parameters included in

**u **such as speed, acceleration,etc. For

instance, for a constant speed linear trajectory, we have:
and finally, we can define the vector containing all the unknown

**h***n *(

**x***, ***u**) =

**x **+ (

*n − N*)

**s**
where

**u **=

**s **is the constant unknown speed vector of the mobile.

*lk,n ∈ ***l***n,q ⇔ k ∈ γNLOS*
Using these definitions, we can define the ML estimation of

**4. MULTIPLE SNAPSHOT ESTIMATION**
the position

**x***n *and the unknown NLOS biases

**l***n,q *under the

*q*-th

hypothesis (

**x**
The goal now is to extend the work presented in [9] to the multi-

*n,q *and

**l***n,q *respectively) as follows:

ple snapshot case using the movement model presented in (4). The

**x***n,q, ***l***n,q *= arg min Ψ

*LOS*
major drawback in [9] was that the decisions about the LOS/NLOS

*n,q *(

**x***n*) + Ψ

*NLOS*
(

**x***n, ***l***n,q*)

**x***n,***l***n,q*
hypothesis were taken instantaneously at each instant of time as
shown in (22), non exploiting the high correlation between the po-
sition in two consecutive instants of time shown in (4) or (5) northe high correlation in time of the NLOS phenomena.

*n,q *(

**x***n*) =

*φ ***t***LOS*
*− ***g***LOS*
*n,q *(

**x***n*)

*, ***R***LOS*
The goal now is not estimating the mobile position at each in-
stant of time in an independent way, but also estimating the mobile
(

**x***n, ***l***n,q*) =

*φ ***t***NLOS*
*− ***g***NLOS*
(

**x***n*)

*− ***l***n,q, ***R***NLOS*
position at the end of the observed window

**x **and the movement

*φ *(

**m***, ***R**) = ln

*|***R***| *+

**m***T ***R***−*1

**m**
parameters

**u **shown in (4).

Taking into account that we have

*Q *possible simple hypothe-
As shown in [11], under the assumtion that

**x***n *is known, the

sis regarding the NLOS state for each instant of time, now we have
ML estimation of

**l***n,q *can be obtained from (15) as:

*P *=

*QN *global hypothesis regarding the NLOS state for the entireobservation window of

*N *consecutive snapshots of measurements.

**l***n,q *(

**x***n*) =

**t***NLOS*
*− ***g***NLOS*
(

**x***n*)

Let us define

**q***p *as the vector containing the simple hypothesis for

and now using (17) in (15), the compressed ML estimation of

**x**
each instant of time under the

*p*-th global hypothesis. This is, its
-th element,

**q***p *(

*n*)

*∈ *[1

*, Q*], is the simple hypothesis selected

for the

*n*-th instant of time under the

*p*-th (

*p ∈ *[1

*, P *]) global hy-

**x***n,q *= arg min Ψ

*LOS*
*n,q *(

**x***n*) + Ψ

*NLOS*
**x***n, ***l***n,q *(

**x***n*) (18)

Under this

*p*-th global hypothesis we have as unknowns, the

**x***n,q *= arg min Ψ

*LOS*
position at the end of the observation window

**x**, the movement

*n,q *(

**x***n*)

parameters

**u **and the set of unknown NLOS biases for each instant

that can be solved using a closed form algorithm like in [12] or
[13]. Note that the second term in (18) becames constant with
respect to

**x**
*n *and can be removed from the minimization. This

*p *=

**l***T*
1

*,***q***p*(1)

*, . . . , ***l***T*
*N,***q***p*(

*N*)

last expression (19) basically demonstrates that, if we do not have
any knowledge of the NLOS biases, the ML position estimator
Analogously to the previous section, the ML formulation of
is obtained only using the measurements that are supposed to be
all the unknown parameters under the

*p*-th hypothesis can be for-
Another conclusion from (19) is that the covariance matrix of

*n,q*,

**C***n,q *=

*E *(

**x***n,q − ***x***n*) (

**x***n,q − ***x***n*)

*T*
*p, ***u***p, ***l***p *= arg min

*p *(

**x***, ***u***, ***l***p*)

**x***,***u***,***l***p*
*χp *(

**x***, ***u***, ***l***p*) =

**C***n,q ≈ ***G***Tn,q *(

**x***n,q*)

**R***LOS*
**G***n,q *(

**x***n,q*)

*n,***q***p*(

*n*) (

**h***n *(

**x***, ***u**)) + Ψ

*NLOS*
*n,***q***p*(

*n*)

**h***n *(

**x***, ***u**)

*, ***l***n,***q***p*(

*n*)

*∂***g***LOS*
*∂***g***LOS*
where matrix

**G***n,q *(

**x***n*)=

*n,q *(

**x***n*)

*n,q *(

**x***n*)

*∂*[

**x***n*]1

*∂*[

**x***n*]2

*n,q *(

**x***n*) and Ψ

*NLOS*
(

**x***n, ***l***n,q*) are defined in (16).

Until now, we have

*Q *possible position estimates

**x***n,q *,

*q *=

Again, as shown in [11] and assuming that all the unknown
1

*, · · · , Q *corresponding to the ML position estimates under the

*Q*
parameters included in

**l***p *are independent, we can obtain the ML

possible hypothesis regarding the NLOS state. In order to obtain a
estimation

**l***p *and compress the ML function

*χp *(

**x***, ***u***, ***l***p*). Con-

final position estimate considering the

*Q *possible hypothesis, two
cretely, the ML estimation of

**l**
approaches have been proposed to solve this last step. Pi-Chun

*p*,

**l***p *=

**l***T*
1

*,***q***p*(1)

*, . . . , ***l***T*
*N,***q***p*(

*N*) is

Chen proposed in [8] to obtain the estimations as a combination of
the

*Q *partial estimates as follows:

**l***n,***q***p*(

*n*) =

**t***NLOS*
*n,***q***p*(

*n*)

*− ***g***NLOS*
*n,***q***p*(

*n*)(

**h***n *(

**x***, ***u**))

and the compressed ML estimations of

**x **and

**u **are obtained as:

**x***n*=

*n,q *(

**x***n,q*)

*n,q *(

**x***n,q*)

**x***n,q*
**x***p, ***u***p *= arg min

*χp ***x***, ***u***, ***l***p*
**x***,***u**
In [9], the authors proposed just to extend the ML search into

**x***p, ***u***p *= arg min

*n,***q**
the hypothesis domain applying the ML detection principle as fol-

**x***,***u**
*p*(

*n*) (

**h***n *(

**x***, ***u**))

Here, we obtain a similar result: the mobile position

**x **and

**x***n , q *= arg min Ψ

*LOS*
the movement parameters

**u **are obtained as if we had only the

*n,q *(

**x***n*) + ln Γ

*−*1

available LOS measurements (under the

*p*-th hypothesis).

where Γ

*q *are some tuning parameters to modify the a-priori proba-
Finally, if we apply the same idea shown in (22), we can select
bility of each hypothesis and to compensate for the different num-
ber of unknown parameters in each hypothesis.

As exposed in [9], this second alternative obtains better per-

**x***, ***u***, p *= arg min

*n,***q***p*(

*n*) (

**h***n *(

**x***, ***u**)) + ln Γ

*−*1

formance, specially in severe NLOS scenarios since it is able to

**x***,***u***,p*
**5. PROPOSED ALGORITHM**
of simple hypothesis associated to all the instants of time until thecurrent iteration. These

*Q *partial hypothesis will be the most likely
It is clear that the complexity associated with the last expression
hypothesis that end with a particular NLOS state.

makes it useless for a real implementation since it is not possible
At the

*M *-th instant of time, these

*Q *partial hypothesis are de-
to test

*P *= 2

*K·N *global hypothesis to select the best one. This
fined by the vectors

**β**M
*q *,

*q *= 1

*, · · · , Q*. These vectors contain
section presents the proposed simplification for a real implemen-
the simple hypothesis for each instant of time until the current in-
tation. The proposed idea is inspired in the coding theory based on
stant of time indicated by

*M *, this is

**β**M
*q *(

*n*)

*∈ *[1

*, Q*]

*n ≤ M *, the
Trellis Coded Modulation (TCM), where it is not possible to eval-

*n*-th element of

**β**M
*q *, is the simple hypothesis considered for the
uate the likelihood of all possible paths, so only the most likely
paths survive at each iteration. While in TCM schemes we try to
Coherently with the definition, the last element inside the vec-
extract the codified source information, here we try to extract the
tor

**β**M
NLOS state at each iteration. While in TCM schemes there are

*q *has to be

*q*, this is

**β**M
*q *(

*M *) =

*q*. Note that

**β**M
length vector since it contains only the simple hypothesis selected
forbiden transitions, here we have low probability transitions (i.e.

for the first

*M *instants of time. Note also, that each partial hypoth-
all NLOS to all LOS). Finally, while in TCM schemes we extract
esis corresponds to a family of the

*P *global hypothesis since the
information from the received signal to, somehow, estimate the
partial hypothesis shows only the simple hypothesis selected until
probability of each state based on the previous probability of all
the states, here we do exactly the same.

Let us assume then that we are at the

*M *-th instant of time and
First of all, note that (28) shows the ML estimation of the posi-
that we have

*Q *partial hypothesis defined by the vectors

**β**M
tion

**x **and model parameters

**u **using only the LOS measurements

clear that initially, we can form

*Q ·Q *partial hypothesis as follows:
we have at different instants of time defined by the

*p*-th globalhypothesis. This estimation can be approximated in a two-stepsestimation process. In the first step, partial position estimates are

**β**M
*q,q *=

**β**M
*q *= 1

*, · · · , Q q *= 1

*, · · · , Q*
obtained for each instant of time using only the LOS we have. Inthe second step the partial position estimates are fused using the
Generically,

**β**M
*q,q *contains all the simple hypothesis considered
movement model shown in (4). This two steps estimation process
until the

*M *-th instant of time by

**β**M
can be mathematically expressed as follows:
about the NLOS state for the

*M *+ 1-th instant of time.

Since we have

*Q *new vectors that end with a certain hypoth-

**x***p, ***u***p *= arg min

*φ ***x***n,***q**
**x***,***u**
*p*(

*n*)

*− ***h***n *(

**x***, ***u**)

*, ***C***n,***q***p*(

*n*)

esis

*q *, these are

**β**M
*q,q ∀q*, we have to select the most likely one
using a simplified version of (32). Concretelly we implement this
where

**x***n,q *is the partial estimate under the

*q*-th hypothesis for the

*n*-th instant of time as defined in (19) and

**C***n,q *is its approximate

**β**M+1

**β**M
Under this approximation, we can redefine the hypothesis se-

*qsel *= arg min Ξ

**β**M
**x***, ***u***, p *=

Note that the search for each state (

*q *) is only over the

*Q *pos-
sibilities consisting in the transitions between all the previous state

*φ ***x***n,***q**
and the new one. It is clear that the unafordable complexity of (32)

**x***,***u***,p*
*p*(

*n*)

*− ***h***n *(

**x***, ***u**)

*, ***C***n,***q***p*(

*n*) +ln Γ

*−*1

**q***p*(

*n*) (31)

has been dramatically reduced since here we implement a searchof

*Q *candidates instead of the

*QM *candidates that we had in the
where it can be seen that all the

*P *hypothesis are based on the
partial position estimates and their covariance matrices. This
Finally, it can be seen that the minimization over the posi-
is, the

*P *=

*QN *global hypothesis are all the ways to combine the
tion

**x **and movement parameters

**u **in (33) can be implemented

partial position estimates that we have per each one of the

*N*
iteratively with a linear Kalman filter reducing the complexity of
evaluating the minimization at each iteration.

For clarity reasons, let us isolate the hypothesis selection in

*p *= arg min Ξ (

**q***p, N*)

**6. NUMERICAL SIMULATIONS**
Ξ (

**q***, M *) =

Numerical simulations have been performed in a scenario with fiveBSs uniformly distributed in a circle of radius 2 Km centered in
the origin of coordinates. The mobile is moving with a constant

*φ ***x***n,***q**(

*n*)

*− ***h***n *(

**x***, ***u**)

*, ***C***n,***q**(

*n*) + ln Γ

*−*1

**x***,***u**
**q***p*(

*n*)

speed of

**s **=

of coordinates in the middle of the time observed window. The
The cost function Ξ (

**q***, M *) shows the likelihood of a certain

measurement rate is

*r *= 1 snapshot of TOAs per second and the
hypothesis expressed by the vector

**q**. This includes the simple

length of the observed window is

*N *= 100 snapshots.

hypothesis for the

*M *instants of time.

Joining the assumptions

**(a)**,

**(b) **and

**(c) **shown in section 2

Note that (32) is just showing that we decide the hypothesis

*p*
about the NLOS effect, we have simulated the NLOS effect of
based on the likelihood of the possible

*P *global hypothesis using
each BS as an independent process. The NLOS phenomena for
all the

*N *available snapshots of measurements.

each BS is modelled with a Markov Chain with two states: LOS
The idea we propose is to limit the search over the entire

*P*
and NLOS. There is a transition at each instant of time with the
possible hypothesis, iteration by iteration as in TCM.

The proposed technique consists in only considering

*Q *partial
hypothesis at each iteration. A partial hypothesis is a collection
performs the previous contributions since it is able to exploit a
movement model over the intermittent effect of the NLOS.

**7. REFERENCES**
[1] M. I. Silventoinen and T. Rantalainen, “Mobile station emer-
gency locating in gsm,”

*IEEE International Conference on*
*Personal WirelessCommunications*, pp. Page(s): 232 –238,
[2] M. P. Wylie and J. Holtzman, “Non-line of sight problem in
mobile location estimation, the,”

*IEEE InternationalConfer-*
*ence on Universal Personal Communications*, vol. Volume:
Optimum per iteration Simple ML−Detection (max 2 NLOS)
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Simple ML−Detection (max 1 NLOS) Pi−Chun Chen approach (max 2 NLOS)
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**Fig. 1**. C.D.F. of the position error with

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[5] N. J. Thomas, D. G. M. Cruickshank, and D. I. Laurenson,
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*IEEE Vehicular Technology Conference VTC*, vol.

Spring 2002. IEEE 55th, Volume: 2, pp. Page(s): 861 –865,
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[7] S. Venkatraman, Caffery J., Jr., and H. R. You, “Location
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[8] Pi-Chun Chen, “A non-line-of-sight error mitigation algo-

**Fig. 2**. C.D.F. of the position error with

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*lk,n *using a uniform distribution in the
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*ence on Acoustics, Speech, and Signal Processing (ICASSP)*,
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vol. Volume: 2, pp. Pages:ii – 153–6 vol.2, May 2004.

Figures (1) and (2) show the sample cumulative density func-
[10] Andreu Urruela and Jaume Riba, “Efficient mobile loca-
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*meters*
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*σk *= 60

*meters *repectivelly.

In the same figure, two theoretical limits have been plotted for
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*Optimum per iteration *limit consisting
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*IEEE Inter-*
in selecting always the LOS measurements at each iteration and

*national Conference on Acoustics, Speech, and Signal Pro-*
compute the position

**x***n *iteration by iteration. On the other hand,

*cessing (ICASSP)*, vol. vol 3, pp. Pages:III–2473 – III–2476,
we can also see the

*Optimum *limit consisting in selecting always
the LOS measurements and compute the mobile position

**x **and the

[12] A. Urruela and J. Riba, “Novel closed-form ML position es-
movement parameters

**u **exploiting the movement model presented

timator for hyperbolic location,”

*IEEE International Confer-*
in (5). Logically, both limits can not be implemented since it is not

*ence on Acoustics, Speech, and Signal Processing (ICASSP)*,
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For comparison, we also present the performance of the al-
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*simple-ML-detection *and the
for hyperbolic location,”

*IEEE Transactions on Signal Pro-*
algorithm presented in (21) [8] as

*Pi-Chun Chen *algorithm. In

*cessing*, vol. Volume: 42 Issue: 8, pp. Page(s): 1905 –1915,
both cases, we have limited the NLOS hypothesis search to 1 or
2 NLOS. It can be seen how the proposed algorithm clearly out-

Source: http://gps-tsc.upc.es/comm/jriba/NLOS%20mitigation%20based%20on%20a%20Trellis%20Search%20for%20Wireless%20Location.pdf

8TH LASAIRE BIENNIAL 13 AND 14 JANUARY 2005 WORKSHOPS PROGRAMS Workshop 1 : How Europe can work efficiently with 25 member countries? Workshops leaders: Mario Dehove, Joël Maurice, Antonio Lettieri Thursday 13 January 2005 from 2:30 p.m to 4:15 p.m ROUND TABLE A: How should we assess Europe's economic performance? Speakers: Eva Belabed, European Economic

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