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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,0st−τ(j)

n,0+

K(j)

n

X

k=1

α(j)

n,kst−τ(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 = (c√8π 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|xn−1),

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=(m∈M(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

1−p(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) = kpn−p(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 fL(ˆu(j)

n,m|u(j)

n)∝

fRice(ˆu(j)

n,m; 1, u(j)

n)ι(ˆu(j)

n,m −γ)and the NLOS amplitude LHF

as fNL(ˆu(j)

n,m)∝fRayl (ˆu(j)

n,m; 1) ι(ˆu(j)

n,m −γ), where ι(ˆu(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

f(ˆu(j)

n,m|u(j)

n, a(j)

n) = (fL(ˆu(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)

n−1). 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)

n−1=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 f(ˆu(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)f(ˆu(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 (ˆu(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 ˜

fxn−1(·),˜

f(j)

un−1(·)and

˜p(j)

qn−1(·)are messages of the previous time n−1, as

φn(xn) = ZΥ(xn|xn−1)˜

fxn−1(xn−1) dxn−1,(17)

ψ(j)

n(u(j)

n) = ZΦ(u(j)

n|u(j)

n−1)˜

f(j)

un−1(u(j)

n−1) du(j)

n−1,(18)

η(j)

n(q(j)

n) =

Nq

X

q(j)

n−1=1

Ψ(q(j)

n|q(j)

n−1) ˜p(j)

qn−1(q(j)

n−1).(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)/ξ(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|xn−1), is modelled by a linear, constant

velocity and stochastic acceleration model [27, p. 273], i.e.

xn=A xn−1+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 A∈R4x4 and B∈R4x2

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)

n−1)

=N(u(j)

n;u(j)

n−1, 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 2≤k≤9,

[Q]k,k = 0.85,[Q]k−1,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.

0 5 10 15 20

0

20

40

ˆ

d(j)

LOSnin m

0 5 10 15 20

0

20

40 measurements

estimate

0 5 10 15 20

0

20

40

0 5 10 15 20

0

5

10

15

20

t0in s

ˆu(j)MMSE

n

0 5 10 15 20

0

10

20

30

t0in s

0 5 10 15 20

0

10

20

t0in s

0 5 10 15 20

0.1

0.5

1

ˆq(j)MMSE

n

j= 1

0 5 10 15 20

0.1

0.5

1

j= 2

0 5 10 15 20

0.1

0.5

1

j= 3

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

fL(ˆu(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

n−pnk2],

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

10−2

10−1

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.

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