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A Message Passing based Adaptive PDA Algorithm for Robust Radio-based Localization and Tracking

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Abstract

We present a message passing algorithm for localization and tracking in multipath-prone environments that implicitly considers obstructed line-of-sight situations. The proposed adaptive probabilistic data association algorithm infers the position of a mobile agent using multiple anchors by utilizing delay and amplitude of the multipath components (MPCs) as well as their respective uncertainties. By employing a nonuniform clutter model, we enable the algorithm to facilitate the position information contained in the MPCs to support the estimation of the agent position without exact knowledge about the environment geometry. Our algorithm adapts in an online manner to both, the time-varying signal-to-noise-ratio and line-of-sight (LOS) existence probability of each anchor. In a numerical analysis we show that the algorithm is able to operate reliably in environments characterized by strong multipath propagation, even if a temporary obstruction of all anchors occurs simultaneously.
A Message Passing based Adaptive PDA Algorithm
for Robust Radio-based Localization and Tracking
Alexander Venus1,2, Erik Leitinger1, Stefan Tertinek3, and Klaus Witrisal1,2
1Graz University of Technology, Austria, 3NXP Semiconductors, Austria,
2Christian Doppler Laboratory for Location-aware Electronic Systems
Abstract—We present a message passing algorithm for lo-
calization and tracking in multipath-prone environments that
implicitly considers obstructed line-of-sight situations. The pro-
posed adaptive probabilistic data association algorithm infers the
position of a mobile agent using multiple anchors by utilizing
delay and amplitude of the multipath components (MPCs) as
well as their respective uncertainties. By employing a non-
uniform clutter model, we enable the algorithm to facilitate
the position information contained in the MPCs to support
the estimation of the agent position without exact knowledge
about the environment geometry. Our algorithm adapts in an
online manner to both, the time-varying signal-to-noise-ratio
and line-of-sight (LOS) existence probability of each anchor.
In a numerical analysis we show that the algorithm is able
to operate reliably in environments characterized by strong
multipath propagation, even if a temporary obstruction of all
anchors occurs simultaneously.
Index Terms—Obstructed Line-Of-Sight, Multipath, Message
Passing, Probabilistic Data Association, Belief Propagation
I. INTRODUCTION
Radio-based localization in environments such as indoor
or urban territories is still a challenging task [1], [2]. These
environments are characterized by strong multipath propaga-
tion and frequent obstructed line-of-sight (OLOS) situations,
which can prevent the correct extraction of the line-of-sight
(LOS) component (see Fig. 1). For safety and security critical
applications, such as keyless entry systems [3] or autonomous
driving [4], robustness, i.e., a low probability of localization
outage, is of critical importance.
Therefore, new systems take advantage of multipath chan-
nels by estimating multipath components (MPCs) for local-
ization [5], [6], exploiting cooperation among agents [7], or
signal processing against multipath propagation and clutter
measurements, i.e., outliers, in general [8]–[10].
The probabilistic data association filter [11] is a Gaussian
variant of probabilistic data association (PDA), which is able
to incorporate multiple anchors (sensors) [11] and amplitude
information (AI-PDAF [12]), but suffers from being compu-
tationally intractable or its dependence on mode-matching.
This paper proposes a low-complexity message passing
based multi-sensor PDA algorithm that estimates and tracks
the state of a mobile agent by utilizing delay and amplitude
of multipath components (MPCs) as well as their respective
The financial support by the Christian Doppler Research Association,
the Austrian Federal Ministry for Digital and Economic Affairs and the
National Foundation for Research, Technology and Development is gratefully
acknowledged.
LOS
partial OLOS
full OLOS
multipath
Fig. 1. The mobile agent is walking alongside the anchors on an
example trajectory. Due to an obstacle, the LOS to all anchors is not
always available. There occur partial , as well as full OLOS situations.
uncertainties. The proposed algorithm adapts in an online man-
ner to both, the time-varying signal-to-noise-ratio (SNR) [5]
and LOS existence probability of each anchor [13]. Further-
more, we use a non-uniform non-LOS (NLOS) model, which
comprises measurements originating from MPCs, as well as
false alarms (FAs), which do not have a physical explanation
(similar to [14]). This model enables the algorithm to utilize
the position information contained in the MPCs in order to
support the estimation of the agent state without specific map
information and, hence, to operate reliably in environments
with strong multipath propagation and temporary obstructed
LOS situations. In that sense, the proposed algorithm allows
to indirectly exploit MPCs. The key contributions of this paper
are as follows.
We present a multi-sensor message passing algorithm with
combined SNR and LOS existence probability tracking.
We employ a non-uniform NLOS probability density
function (PDF) using a double-exponential model for the
multipath likelihood function (LHF).
We show the applicability of our algorithm in the context
of OLOS mitigation and evaluate the influence of the
features of our algorithm in a numerical analysis.
Note that for this work it is assumed that the parameters of the
NLOS object are known constants. This shortcoming shall be
addressed in an extended version of this work, and is further
discussed in Section VII.
II. SI GNA L MOD EL
At each discrete time n∈ {1, ... , N }, the mobile agent
at position pntransmits a signal s(t)and each anchor j
arXiv:2103.13188v2 [cs.IT] 25 Mar 2021
{1, ... , J }at anchor position p(j)
A= [p(j)
Ax p(j)
Ay ]Tacts as a
receiver. The complex baseband signal received at the jth
anchor is modeled as
r(j)
n(t) = α(j)
n,0stτ(j)
n,0+
K(j)
n
X
k=1
α(j)
n,kstτ(j)
n,k+w(j)
n(t)(1)
The first and second term describe the LOS component and the
sum of K(j)
nspecular MPCs with their corresponding complex
amplitudes α(j)
n,k and delays τ(j)
n,k, respectively. The third term
w(j)
n(t)is additive white Gaussian noise with double-sided
power spectral density N0/2. The MPCs arise from reflection
or scattering by unkown objects, since we assume that no map
information is available as indicated in Fig. 1 by green lines.
III. CHA NN EL ESTIMATION
The received signal (1) is sampled and, by applying a
suitable snapshot based channel estimation and detection al-
gorithm [15], [16], one obtains at each time nand anchor
j, a number of M(j)
nmeasurements denoted by z(j)
n,m, with
measurement indices m∈ M(j)
n={1, ... , M (j)
n}. Each
z(j)
n,m = [ ˆ
d(j)
n,m ˆu(j)
n,m ˆσ(j)
dn,m]Tcontains a distance measurement
ˆ
d(j)
n,m =cˆτ(j)
n,m, a normalized signal amplitude measurement
ˆu(j)
n,m =|ˆα(j)
n,m|/ˆσ(j)
αn,m corresponding to the square root
of the SNR, and a distance standard deviation measurement
ˆσ(j)
dn,m, where ˆτ(j)
n,m and ˆα(j)
n,m represent the corresponding
delay and complex amplitude and cis the speed-of-light. If
not implicitly provided by the channel estimator, we can obtain
ˆσ(j)
dn,m by means ˆu(j)
n,m using ˆσ(j)
dn,m = (c8π β ˆu(j)
n,m)1, with
βbeing the effective bandwidth. This equation corresponds to
the Cram´
er-Rao lower bound (CRLB) for a single-distance
measurement [1] and is used for the simulations in Sec. VI.
Consider that assuming the statistical model to be correct,
the distance variance will attain the CRLB for a single-
distance measurement, as the NLOS measurements are taken
into account by the data association algorithm.
We define the nested vectors z(j)
n= [z(j)T
n,1... z(j)T
n,M(j)
n
]T
and zn= [z(1) T
n... z(J)T
n]T, where the latter denotes the joint
measurement vector per time n. All of its components are used
as noisy “measurements” by the proposed algorithm.
IV. SYS TE M MOD EL
We consider a mobile agent to be moving along an unknown
trajectory as depicted in Fig. 1. The current state of the agent
is described by the state vector xn= [pT
nvT
n]T, which is
composed of the mobile agent’s position pn= [pxnpyn]T
and velocity vn= [vxnvyn]T. The state evolves over time n
according to a predefined state transition PDF Υ(xn|xn1),
where the usual first-order Markov assumption is applied [17].
A. Data Association Model
At each time nand for each anchor j, the measurements,
i.e., the components of z(j)
nare subject to data association
uncertainty. Thus, it is not known which measurement z(j)
n,m
originated from the LOS, or which one is due to an MPC. It
is also possible that a measurement z(j)
n,m did not originate
from any physical component, but from FAs of the prior
channel estimation and detection algorithm. Our model only
distinguishes between “LOS measurements” originating from
the LOS and “NLOS measurements”, i.e., measurements due
to MPCs or FAs. Based on the concept of PDA [11], we define
an association variable
a(j)
n=(mM(j)
n,z(j)
n,m is the LOS measurement in z(j)
n
0,there is no LOS measurement in z(j)
n
(2)
Assuming the number of NLOS measurements to follow a
uniform distribution (so called “non-parametric model”), the
joint probability mass function (PMF) of a(j)
nand M(j)
ncan
be shown to be proportional to the function [11]
h(a(j)
n, M (j)
n;u(j)
n, q(j)
n) =
p(j)
En(u(j)
n,q(j)
n)
M(j)
n
, a(j)
n∈ M(j)
n
1p(j)
En(u(j)
n, q(j)
n), a(j)
n= 0
(3)
where p(j)
En(u(j)
n, q(j)
n)is the probability that there is a LOS
measurement for the current set of measurements defined
in Sec. IV-D and u(j)
nand q(j)
nare defined in Sections
IV-C and IV-D, respectively. We also define the joint vectors
an= [a(1)
n... a(J)
n]Tand Mn= [M(1)
n... M (J)
n]T.
B. Delay/Distance Model
To incorporate the data association procedure into the mea-
surement process we define two LHFs, one for the LOS event
and one for the NLOS event.
Let the range-only measurement vector be ˜
z(j)
n,m =
[ˆ
d(j)
n,m ˆσ(j)
dn,m]T. First we define the LOS LHF as
fL(˜
z(j)
n,m|pn) = N(ˆ
d(j)
n,m;d(j)
LOSn(pn),ˆσ(j)
dn,m)(4)
where N(·)denotes a Gaussian PDF of the random variable
(RV) ˆ
d(j)
n,m with mean d(j)
LOSn(pn)and standard deviation
ˆσ(j)
dn,m. The LOS distance is geometrically related to the agent
position via
d(j)
LOSn(pn) = kpnp(j)
Ak.(5)
Next, we define the NLOS LHF as [14]
fNL(ˆ
d(j)
n,m|pn) = PMP fMP (ˆ
d(j)
n,m|pn) + (1 PMP)fFA(ˆ
d(j)
n,m),
(6)
which represents a weighted sum of two LHFs, with PMP
acting as a weighting coefficient: The FA LHF fFA(ˆ
d(j)
n,m) =
U(0, dmax), which is a uniform distribution with the maximum
distance dmax and the multipath LHF
fMP(ˆ
d(j)
n,m|pn) =
γf+γr
γ2
f
(1 e(j)
n,m
γr)e(j)
n,m
γf,(j)
n,m >0
0,(j)
n,m 0
(7)
which is a double exponential function [18] with the distance
difference (j)
n,m =ˆ
d(j)
n,m d(j)
LOSn(pn)B.γris the rise
distance, γfthe fall distance, and Bis a bias value.
Finally, we define the overall-distance LHF as
f(˜
z(j)
n,m|pn, a(j)
n) = (fL(˜
z(j)
n,m|pn), a(j)
n=m
fNL(ˆ
d(j)
n,m|pn), a(j)
n6=m.(8)
The shape of (8) is depicted in Fig. 2a.
C. Amplitude Model
As for the delay model in Sec. IV-B, we start by
defining the LOS amplitude LHF as fLu(j)
n,m|u(j)
n)
fRice(ˆu(j)
n,m; 1, u(j)
n)ιu(j)
n,m γ)and the NLOS amplitude LHF
as fNL(ˆu(j)
n,m)fRayl u(j)
n,m; 1) ι(ˆu(j)
n,m γ), where ιu(j)
n,m γ)
is the unit step function with threshold γthat truncates the
distributions. The PDF fRice is a Rician distribution and PDF
fRayl is a Rayleigh distribution with non-centrality parameter
u(j)
nand respective spread parameters equal to 1. Note that
due to truncation the functions need to be scaled to represent
proper PDFs [12]. The overall-amplitude LHF is given by
fu(j)
n,m|u(j)
n, a(j)
n) = (fLu(j)
n,m|u(j)
n), a(j)
n=m
fNL(ˆu(j)
n,m), a(j)
n6=m(9)
which is shown in Fig. 2b. This model represents the distri-
bution of amplitude estimates of a single complex baseband
signal in additive Gaussian noise obtained using maximum
likelihood estimation and generalized likelihood ratio test
detection [19]–[21]. Consider that for the model to be true, the
MPCs in (1), i.e., all components except for the LOS, have
to be represented by a stochastic process with zero mean (i.e.
dense multipath component model [1]). The above definition is
similar to [12]. However, we use the normalized amplitude u(j)
n
[22], i.e., the spreading parameters of the Rayleigh and Rician
distribution, which constitute the overall-amplitude LHF (9),
are equal to 1, avoiding the necessity to track these parameters.
We model the temporal evolution of u(j)
nas a first order
Markov process, which is defined by a state transition PDF
Φ(u(j)
n|u(j)
n1). The amplitudes of all anchors jare assumed
to be independent stochastic processes, ignoring possible ge-
ometric information available, as for channels with strong
multipath propagation received signal strength measurements
tend to be error prone. We also define the joint amplitude
vector un= [u(1)
n... u(J)
n]T.
D. LOS Existence Probability Model
We model the LOS existence probability given in (3) as
p(j)
En(u(j)
n, q(j)
n) = p(j)
Dn(u(j)
n)q(j)
n. The so-called probability
of detection p(j)
Dn(u(j)
n)is modelled according to Sec. IV-C
by assuming that the proposed algorithm is applied after a
generalized likelihood ratio test detector. That is, p(j)
Dn(u(j)
n)is
completely determined by the normalized amplitude u(j)
nand
γ, which represents the detection threshold and is a constant
to be chosen. q(j)
nis the probability of the event that the LOS
is not obstructed, which is referred to as LOS probability in
the following, and acts as a prior probability to the detection
event. According to [13], [23], we model q(j)
nas discrete RV
that takes its values from a finite set Q={ω1, ... , ωQ}, where
ωi(0,1]. The temporal evolution of q(j)
nis modelled by
a first-order Markov process, which results in a conventional
Markov chain, with [Q(j)]i,k = Ψ(q(j)
n=ωi|q(j)
n1=ωk) being
the elements of the transition matrix. The LOS probabilities
0d(j)
LOS n(pn)dmax
ˆσ(j)
dn,m
B
˜γr˜γf
distance measurement ˆ
d(j)
n,m
LHF
a(j)
n=m a(j)
n6=m
(a) overall-distance LHF f(˜
z(j)
n,m|pn, a(j)
n).
0γu(j)
n
normalized amplitude measurement ˆu(j)
n,m
LHF
a(j)
n=m a(j)
n6=m
(b) overall-amplitude LHF fu(j)
n,m|u(j)
n, a(j)
n).
Fig. 2. Graphical representation of the stochastic models constituting the
overall LHF for a single measurement.
for different sensors jare assumed to be independent. We also
define the joint LOS probability vector qn= [q(1)
n... q(J)
n]T.
E. Joint Measurement Likelihood Function
Under commonly used assumptions about the statistics of
the measurements [24], the joint LHF for all measurements
per anchor jand time ncan be written as
f(z(j)
n|pn,u(j)
n,a(j)
n) =
M(j)
n
Y
m=1
f(˜
z(j)
n,m|pn, a(j)
n)fu(j)
n,m|u(j)
n,a(j)
n)
(10)
By neglecting all constant terms, we define the pseudo LHF
g(z(j)
n;pn, u(j)
n, a(j)
n)
=
M(j)
n
Y
m=1
fNL(ˆ
d(j)
n,m|pn)×(1, a(j)
n= 0
Λ(z(j)
n,a(j)
n|pn, u(j)
n), a(j)
n∈ M(j)
n
(11)
where
Λ(z(j)
n,m|pn, u(j)
n) = fL(˜
z(j)
n,m|pn)fL(ˆu(j)
n,m|u(j)
n)
fNL(˜
z(j)
n,m|pn)fNL u(j)
n,m)(12)
is the likelihood ratio. Note that the product of all NLOS
events in (11) is not a constant and thus cannot be neglected.
F. Joint Posterior and Factor Graph
Let z= [zT
1... zT
n]T,x= [xT
1... xT
n]T,a= [aT
1... aT
n]T,
u= [uT
1... uT
n]T,q= [qT
1... qT
n]T, and M= [MT
1... MT
n]T.
Applying Bayes’ rule as well as some commonly used inde-
pendence assumptions [9], [24] the joint posterior for all states
up to time nand all Janchors, can be derived up to a constant
factor as
f(x,a,u,q,M|z)
f(z|x,a,u,q)f(x,a,u,q)
=f(z|x,a,u,q)f(a|u,q)f(x)p(q)f(u)
f(x0)
J
Y
j=1
p(q(j)
0)f(u(j)
0)
n
Y
n0=1
Υ(xn0|xn0
1) Φ(u(j)
n0|u(j)
n0
1)
×Ψ(q(j)
n0|q(j)
n0
1) ˜g(z(j)
n0;pn0, u(j)
n0, a(j)
n0, q(j)
n0),(13)
j=J
q(J)
0
u(J)
0
Ψq(j)
n
˜g(j)
zn
a(j)
n
˜p(j)
qn
Φu(j)
n˜
f(j)
un
η(j)
n
β(j)
n
η(j)
n
1
ψ(j)
n
ψ(j)
n
ν(j)
n
ξ(j)
n
j= 1
q(1)
0
u(1)
0
Ψq(j)
n
˜g(j)
zn
a(j)
n
˜p(j)
qn
Φu(j)
n˜
f(j)
un
η(j)
n
β(j)
n
η(j)
n
1
ψ(j)
n
ψ(j)
n
ν(j)
n
x0Υxn
φn˜
fxn
χ(1)
nχ(J)
n
ξ(j)
n
˜
fx0
˜p(J)
q0
˜
f(J)
u0
˜p(1)
q0
˜
f(1)
u0
Fig. 3. Factor graph representing the factorization of the joint posterior PDF
in (13) and the messages according to the SPA (see Sec. V-B).
with ˜g(z(j)
n;pn,u(j)
n,a(j)
n,q(j)
n) = h(a(j)
n;u(j)
n, q(j)
n)g(z(j)
n;pn,
u(j)
n,a(j)
n). For the sake of brevity, we refer to this expression
as ˜g(j)
zn(·)in the rest of the work. Note that Mis fixed and
thus constant, as it is defined implicitly by the measurements
z. This factorization of the joint posterior PDF can be visually
represented by the factor graph shown in Fig. 3. Further note
that (13) is a mixture of discrete PMFs and continuous PDFs.
V. ALGORITHM
A. Problem Statement
Our goal is to estimate the agent state xn. This can be done
by calculating the minimum mean-square error (MMSE) [20]
ˆ
xMMSE
n,Zxnf(xn|z) dxn.(14)
with ˆ
xMMSE
n= [ ˆ
pMMSE T
nˆ
vMMSE T
n]T. Furthermore, we also
calculate
ˆu(j)MMSE
n,Zu(j)
nf(u(j)
n|z) du(j)
n,(15)
ˆq(j)MMSE
n,X
ωi∈Q
ωip(q(j)
n=ωi|z).(16)
In order to obtain (14), (15), and (16), marginalization of the
joint posterior has to be performed. In general this is compu-
tationally infeasible [24]. To counteract this problem, we use
a sum-product algorithm (SPA) based algorithm introduced in
the next section.
B. Marginal Posterior and Sum-Product Algorithm (SPA)
The marginal posterior can be calculated efficiently by pass-
ing messages on the factor graph according to the SPA [25].
The presented algorithm is an adaptation of the algorithms
presented in [24], [26] to the factor graph shown in Fig. 3. As
the filter shall be executable online, we only pass messages
forward in time. This makes the factor graph in Fig. 3 an
acyclic graph. For acyclic graphs the SPA yields exact results
for the marginal posterior [25]. At time n, the following
calculations are performed for all Janchors; We start by
defining the prediction messages, where ˜
fxn1(·),˜
f(j)
un1(·)and
˜p(j)
qn1(·)are messages of the previous time n1, as
φn(xn) = ZΥ(xn|xn1)˜
fxn1(xn1) dxn1,(17)
ψ(j)
n(u(j)
n) = ZΦ(u(j)
n|u(j)
n1)˜
f(j)
un1(u(j)
n1) du(j)
n1,(18)
η(j)
n(q(j)
n) =
Nq
X
q(j)
n1=1
Ψ(q(j)
n|q(j)
n1) ˜p(j)
qn1(q(j)
n1).(19)
Next, we define the measurement update messages as
ξ(j)
n(xn) =Zψ(j)
n(u(j)
n)
Nq
X
q(j)
n=1
η(j)
n(q(j)
n)
M(j)
n
X
a(j)
n=1
˜g(j)
zn(·) du(j)
n,(20)
χ(j)
n(xn) = φn(xn)
J
Y
j0=1
ξ(j0)
n(xn)(j)
n(xn),(21)
ν(j)
n(u(j)
n) =
Nq
X
q(j)
n=1
η(j)
n(q(j)
n)Zχ(j)
n(xn)
M(j)
n
X
a(j)
n=1
˜g(j)
zn(·) dxn,(22)
β(j)
n(q(j)
n) =ZZψ(j)
n(u(j)
n)χ(j)
n(xn)
M(j)
n
X
a(j)
n=1
˜g(j)
zn(·) dxndu(j)
n.
(23)
Finally, we calculate the posterior distributions as f(xn|z)
˜
fxn(xn) = φn(xn)QJ
j=1 ξ(j)
n(xn),f(u(j)
n|z)˜
f(j)
un(u(j)
n) =
ψ(j)
n(u(j)
n)ν(j)
n(u(j)
n)and p(q(j)
n|z)˜p(j)
qn(q(j)
n) = η(j)
n(q(j)
n)
×β(j)
n(q(j)
n). Since a direct calculation of the integrals in equa-
tions (17)-(23) is intractable, a particle-based approximation
[17] is used. See [26] and [24] for details.
C. Initialization
We propose to initialize the normalized amplitude PDFs as
˜
f(j)
u0(u(j)
0) = U(0, umax), where umax is a constant to be chosen
according to hardware specific limitations. The LOS PMFs
are initialized at q(j)
0= 1. Regarding the agent state x0, we
assume the velocity to be initialized at v0=0, as we do
not know in which direction we are moving. Since we cannot
make any assumptions about the angle that the agent takes
with respect to any of the anchors, it is reasonable to draw
the positions p0uniformly on two-dimensional discs around
each anchor j, which are bounded by the maximum possible
distance dmax and a sample is drawn from each of the Jdiscs
with equal probability.
VI. CO MP UTATIONA L RES ULTS
We evaluate the proposed algorithm using numerical simu-
lation. To investigate the performance independently of the
channel estimation and detection algorithm implementation
and possible resulting artifacts, we directly generate the mea-
surement vector zaccording to the system model in Sec. IV.
A. Simulation Model
In the example scenario investigated, the agent moves along
a curvy trajectory from a distant point to the centre of an
object where three anchors are mounted (e.g. a car or a
door). The trajectory is illustrated in Fig. 4. It is observed
0 10 20
0
10
20
A0
A1
A2
pxin m
pyin m
anchors
reference
partial NLOS
total NLOS
estimate
0 10 20
0
10
20
A0
A1
A2
pxin m
pyin m
anchors
reference
partial NLOS
total NLOS
estimate
0 10 20
0
10
20
A0
A1
A2
pxin m
pyin m
anchors
reference
partial NLOS
total NLOS
estimate
Fig. 4. Simulated trajectory and anchor setup together with a single position
estimate, corresponding to the measurement shown in Fig. 5
over a continuous measurement time t0[0,20] s, with a
constant sampling rate of T= 50 ms, resulting in N= 400
discrete times steps n∈ {1... N }. It comprises two OLOS
situations, a partial one, where only the LOS to one anchor
is blocked, and a full one, where the LOS to all anchors is
blocked. kvnkis set to vary around a velocity of 1.4 m/s.
The normalized amplitudes are set to 30 dB at d(j)
LOSn= 1 m,
with an exponential path-loss factor as low as 0.4to consider
multipath propagation. We used an average rate of 10 NLOS
measurements per time n. The parameters of the NLOS LHF,
were set to PMP = 0.9,γr= 1.5 m,γf= 6 m,B= 0.2 m
and dmax = 50 m. Fig. 5 shows simulated measurements
corresponding to a single realization of the trajectory.
B. Inference Model
In the estimation algorithm, the agent motion, i.e. the state
transition PDF Υ(xn|xn1), is modelled by a linear, constant
velocity and stochastic acceleration model [27, p. 273], i.e.
xn=A xn1+B wn, with the acceleration process wnbeing
i.i.d. across n, zero mean, and Gaussian with covariance matrix
σaI, where Iis a 2x2 identity matrix, σa= 0.3 m/s2is the
acceleration standard deviation, and AR4x4 and BR4x2
are defined according to [27, p. 273], with Tas defined in
Sec. VI-A. The state transition PDF of the normalized am-
plitudes is modelled as as Gaussian distribution Φ(u(j)
n|u(j)
n1)
=N(u(j)
n;u(j)
n1, σu= 0.2) , which is independent across n
and j. Thus, unlike the simulation model in Sec. VI-A, the
amplitudes of all sensors jare assumed to be independent
(see Sec. IV-C). We use γ= 0, which is equivalent to using
no detection threshold at all. Thus p(j)
Dn(u(j)
n)=1, which leads
to p(j)
En(u(j)
n, q(j)
n)q(j)
n. The set of possible LOS probabilities
is chosen as Q={0.1,0.2, ... , 1}. The state transition matrix
Q(j)=Qis set as follows: [Q]1,1= 0.9,[Q]10,10 = 0.95,
[Q]2,1= 0.1and [Q]9,10 = 0.05. For 2k9,
[Q]k,k = 0.85,[Q]k1,k = 0.05 and [Q]k+1,k = 0.1. For
all other tuples {i, k},[Q]i,k = 0. We used 104particles for
initialization and 103particles for inference during the track.
C. Performance Results
We analyze the influence of the individual features of
our algorithm with respect to the scenario described in Sec.
Fig. 5. A single measurement realization and the respective estimates using
the proposed algorithm (AL5). ˆ
d(j)
LOSnis calculated using (5) and (14).
VI-A. Fig. 6 shows the algorithm variants implemented and
the corresponding features that are enabled for an algorithm
(x) or not ( ). When “q(j)
ntracking” is deactivated, we set
q(j)
n= 0.999 for all n,j. When we do not use “CRLB
based ˆσ(j)
dn,m” measurements (see Sec. III), it is set constant to
ˆσ(j)
dn,m = 0.1 m. Not applying the “non-uniform fNL” means
PMP = 0, and deactivating “amplitude information” means
fLu(j)
n,m|u(j)
n)/fNL(ˆu(j)
n,m),1in (12). All simulation results
are shown in terms of the root mean squared error (RMSE) of
the estimated agent position eRMSE
n=pE[kˆ
pMMSE
npnk2],
evaluated using a numerical simulation with 500 realizations.
The RMSE is shown in two ways. First, as a function of the
continuous measurement time t0and, second, as the cumulative
frequency of the RMSE evaluated over the whole time span
(t0[0,20] s), as well as over the time span before the total
NLOS situation (t0[0,14.2] s). As a performance benchmark
we provide the CRLB for a single position measurement with-
out tracking (SP-CRLB) [28]. Comparing the curves of Fig. 6,
one can conclude that the RMSE is significantly lowered when
additional features are activated. The RMSE of AL1, which
represents a conventional multi-sensor PDA, is constantly
above 2m. This is due to the large percentage of outliers, i.e.,
realizations where the algorithm completely loses the track.
This is slightly improved by tracking q(j)
n(AL2), which leads
to a reduced number of lost tracks. For AL3, we activate
the amplitude information feature, which, in case of sufficient
component SNR, significantly improves the performance as
NLOS and LOS measurements can be separated better. AL4
can additionally support the state estimation using NLOS
measurements, as (6) depends on the agent position pndue
to the non-uniform NLOS LHF. This is especially beneficial
in the full OLOS situation as it significantly reduces the
probability of a lost track. Finally we use CRLB based ˆσ(j)
dn,m
measurements for AL5. This additionally reduces the error, as
the variance of the inference model is correctly adjusted to the
variance of the channel estimation and detection algorithm.
AL1 AL2 AL3 AL4 AL5
q(j)
ntracking x x x x
amplitude information x x x
non-uniform fNL x x
CRLB based ˆσ(j)
dn,m x
0 2 4 6 8 10 12 14 16 18 20
102
101
100
101
t0in s
eRMSE
nin m
SP-CRLB
partial OLOS
full OLOS
0 0.5 1 1.522.5 3
0%
68%
90%
97%
99%
99.7%
eRMSE
nin m
cumulative frequency
before full OLOS
whole track
SP-CRLB
Fig. 6. Performance in terms of the RMSE of the estimated agent position
determined using numerical simulation. The upper plot shows the estimated
eRMSE
nas a function of the measurement time t0. The lower plot shows the
cumulative frequency of the eRMSE
nin inverse logarithmic scale.
VII. CONCLUSION
We have presented a message passing based algorithm that
is able to robustly estimate and track the agent’s position in
multipath channels based on range and amplitude information
of multiple sensors as well as their respective uncertainties
in both partial and total OLOS situations. We analyzed the
performance of the algorithm using numerical simulation and
showed that the additional information provided by amplitude
information as well as by the NLOS object can support the
estimation of the agent state and, thus, reduce the number of
lost tracks. In partial OLOS situations the performance of the
proposed algorithm attained the CRLB (i.e., no lost tracks).
For this work, we assumed the parameters of the NLOS LHF
to be known constants. To overcome this issue, the parameters
of the multipath LHF (7) need to be jointly inferred with the
agent state. This non-trivial extension requires extended object
PDA [29] and shall be addressed in future work.
REFERENCES
[1] K. Witrisal, P. Meissner et al., “High-accuracy localization for assisted
living: 5G systems will turn multipath channels from foe to friend,”
IEEE Signal Process. Mag., vol. 33, no. 2, pp. 59–70, Mar. 2016.
[2] A. Shahmansoori, G. E. Garcia et al., “Position and orientation esti-
mation through millimeter-wave MIMO in 5G systems,IEEE Trans.
Wireless Commun., vol. 17, no. 3, pp. 1822–1835, Mar. 2018.
[3] A. Kalyanaraman, Y. Zeng, S. Rakshit, and V. Jain, “CaraoKey : Car
states sensing via the ultra-wideband keyless infrastructure,” in Proc.
IEEE SECON-20, 2020, pp. 1–9.
[4] R. Karlsson and F. Gustafsson, “The future of automotive localization
algorithms: Available, reliable, and scalable localization: Anywhere and
anytime,” IEEE Signal Process. Mag., vol. 34, no. 2, pp. 60–69, 2017.
[5] E. Leitinger, F. Meyer, F. Hlawatsch, K. Witrisal, F. Tufvesson, and
M. Z. Win, “A belief propagation algorithm for multipath-based SLAM,”
IEEE Trans. Wireless Commun., vol. 18, no. 12, pp. 5613–5629, 2019.
[6] C. Gentner, T. Jost et al., “Multipath assisted positioning with simultane-
ous localization and mapping,” IEEE Trans. Wireless Commun., vol. 15,
no. 9, pp. 6104–6117, Sep. 2016.
[7] H. Wymeersch, J. Lien, and M. Z. Win, “Cooperative localization in
wireless networks,” Proc. IEEE, vol. 97, no. 2, pp. 427 –450, Feb. 2009.
[8] H. Wymeersch, S. Maran`
o, W. M. Gifford, and M. Z. Win, “A machine
learning approach to ranging error mitigation for UWB localization,”
IEEE Trans. Wireless Commun., vol. 60, no. 6, pp. 1719–1728, 2012.
[9] E. Leitinger, F. Meyer, P. Meissner, K. Witrisal, and F. Hlawatsch,
“Belief propagation based joint probabilistic data association for
multipath-assisted indoor navigation and tracking,” in Proc. ICL-GNSS-
16, Barcelona, Spain, June 2016, pp. 1–6.
[10] F. Meyer, Z. Liu, and M. Z. Win, “Network localization and navigation
using measurements with uncertain origin,” in Proc. FUSION-18, July
2018, pp. 1–7.
[11] Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Princi-
ples and Techniques. Storrs, CT, USA: Yaakov Bar-Shalom, 1995.
[12] D. Lerro and Y. Bar-Shalom, “Automated tracking with target amplitude
information,” in 1990 American Control Conference, May 1990, pp.
2875–2880.
[13] G. Soldi, F. Meyer, P. Braca, and F. Hlawatsch, “Self-tuning algorithms
for multisensor-multitarget tracking using belief propagation,IEEE
Trans. Signal Process., vol. 67, no. 15, pp. 3922–3937, Aug. 2019.
[14] Z. Yu, Z. Liu, F. Meyer, A. Conti, and M. Z. Win, “Localization based
on channel impulse response estimates,” in Proc. IEEE/ION PLANS-20,
2020, pp. 1014–1021.
[15] A. Richter, “Estimation of Radio Channel Parameters: Models and
Algorithms,” Ph.D. dissertation, Ilmenau University of Technology,
2005.
[16] M. A. Badiu, T. L. Hansen, and B. H. Fleury, “Variational Bayesian
inference of line spectra,” IEEE Trans. Signal Process., vol. 65, no. 9,
pp. 2247–2261, May 2017.
[17] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial
on particle filters for online nonlinear/non-Gaussian Bayesian tracking,”
IEEE Trans. Signal Process., vol. 50, no. 2, pp. 174–188, Feb. 2002.
[18] J. Karedal, S. Wyne, P. Almers, F. Tufvesson, and A. Molisch, “A
measurement-based statistical model for industrial ultra-wideband chan-
nels,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 3028–3037,
Aug. 2007.
[19] A. Venus, E. Leitinger, S. Tertinek, and K. Witrisal, “Reliability and
threshold-region performance of TOA estimators in dense multipath
channels,” in Proc. IEEE ICC-WS-20, 2020, pp. 1–7.
[20] S. Kay, Fundamentals of Statistical Signal Processing: Estimation
Theory. Upper Saddle River, NJ, USA: Prentice Hall, 1993.
[21] ——, Fundamentals of Statistical Signal Processing: Detection Theory.
Upper Saddle River, NJ, USA: Prentice Hall, 1998.
[22] E. Leitinger, S. Grebien, and K. Witrisal, “Multipath-based SLAM
exploiting AoA and amplitude information,” in Proc. IEEE ICCW-19,
Shanghai, China, May 2019, pp. 1–7.
[23] G. Papa, P. Braca, S. Horn, S. Marano, V. Matta, and P. Willett,
“Adaptive Bayesian tracking with unknown time-varying sensor network
performance,” in Proc. IEEE ICASSP-15, 2015, pp. 2534–2538.
[24] F. Meyer, T. Kropfreiter, J. L. Williams, R. Lau, F. Hlawatsch, P. Braca,
and M. Z. Win, “Message passing algorithms for scalable multitarget
tracking,” Proc. IEEE, vol. 106, no. 2, pp. 221–259, Feb. 2018.
[25] F. Kschischang, B. Frey, and H.-A. Loeliger, “Factor graphs and the
sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp.
498–519, Feb. 2001.
[26] E. Leitinger, F. Meyer, F. Tufvesson, and K. Witrisal, “Factor graph
based simultaneous localization and mapping using multipath channel
information,” in Proc. IEEE ICCW-17, Paris, France, May 2017, pp.
652–658.
[27] Y. Bar-Shalom, T. Kirubarajan, and X.-R. Li, Estimation with Applica-
tions to Tracking and Navigation. New York, NY, USA: John Wiley
& Sons, Inc., 2002.
[28] D. B. Jourdan, D. Dardari, and M. Z. Win, “Position error bound for
UWB localization in dense cluttered environments,IEEE Trans. Aerosp.
Electron. Syst., vol. 44, no. 2, pp. 613–628, 2008.
[29] F. Meyer and J. L. Williams, “Scalable detection and tracking of
extended objects,” in Proc. IEEE ICASSP-20, Barcelona, Spain, May
2020, pp. 8916–8920.
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