rao-blackwellized gaussian sum particle filtering for

Research ArticleRao-Blackwellized Gaussian Sum Particle Filtering forMultipath Assisted Positioning

Markus Ulmschneider Christian Gentner Thomas Jost and Armin Dammann

German Aerospace Center (DLR) Institute of Communications and Navigation Muenchner Str 20 82334 Wessling Germany

Correspondence should be addressed to Markus Ulmschneider MarkusUlmschneiderdlrde

Received 25 August 2017 Revised 27 November 2017 Accepted 18 January 2018 Published 23 April 2018

Academic Editor Huimin Chen

Copyright copy 2018 Markus Ulmschneider et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In multipath assisted positioning multipath components arriving at a receiver are regarded as being transmitted by a virtualtransmitter in a line-of-sight condition As the locations and clock offsets of the virtual and physical transmitters are in generalunknown simultaneous localization and mapping (SLAM) schemes can be applied to simultaneously localize a user and estimatethe states of physical and virtual transmitters as landmarks Hencemultipath assisted positioning enables localizing a user with onlyone physical transmitter depending on the scenario In this paper we present and derive a novel filtering approach for ourmultipathassisted positioning algorithm called Channel-SLAM Making use of Rao-Blackwellization the location of a user is tracked by aparticle filter and each landmark is represented by a sum of Gaussian probability density functions whose parameters are estimatedby unscented Kalman filters Since data association that is finding correspondences among landmarks is essential for robust long-term SLAM we also derive a data association scheme We evaluate our filtering approach for multipath assisted positioning bysimulations in an urban scenario and by outdoor measurements

1 Introduction

The amount of available and potential services requiringprecise localization of a user has steadily increased over therecent years Global navigation satellite systems (GNSSs) canoften satisfy the demands for localization in scenarios wherethe receiver has a clear view of the sky However if the viewof the sky is obstructed such as indoors in urban canyonsor in tunnels the positioning performance of GNSSs maybe drastically decreased or no positioning solution may beobtained at all [1] Reasons for this include a low receivedsignal power due to signal blocking or shadowing andmultipath propagation

In contrast to GNSS signals many kinds of terrestrialsignals are likely to have a good coverage in GNSS deniedplaces In particular cellular radio frequency (RF) signalsare designed to be reliably available at least in populatedareas and they may be used as signals of opportunity (SoOs)for positioning However also terrestrial signals experiencemultipath propagation Multipath propagation biases rangeestimates if standard correlator based methods are used

Various approaches to handle the multipath problem havebeen addressed in the literature for example in [2] Advancedmethods such as maximum likelihood (ML) mitigationalgorithms try to estimate the channel impulse response(CIR) and to mitigate the influence of multipath components(MPCs) on the line-of-sight (LoS) path [3]

The idea of multipath assisted positioning is contrarythough Instead of regarding multipath propagation as illthe spatial information of MPCs on the receiver positionis exploited In [4] the information of MPCs is used in afingerprinting scheme Going one step further each MPCcan be regarded as being transmitted by a virtual transmitterin a pure LoS condition and the virtual transmitters can beused to locate the user Such an approach is called multipathassisted positioning

The authors of [5 6] derived some theoretical boundsfor multipath assisted positioning Multipath assisted posi-tioning schemes have for example been applied in radarapplications [7] using ultrawideband (UWB) [8 9] or 5G [10]systems and in cooperative systems [11]

HindawiJournal of Electrical and Computer EngineeringVolume 2018 Article ID 4761601 15 pageshttpsdoiorg10115520184761601

2 Journal of Electrical and Computer Engineering

If the locations of physical transmitters and reflecting andscattering objects are known the locations of virtual trans-mitters can be calculated based on geometrical considera-tionsThe authors of [12] assume the room layout to be knownand focus on the association among virtual transmitters andreflecting walls In a general setting however the scenario isunknown to the user

The authors of [13 14] have presented amultipath assistedpositioning scheme namedChannel-SLAM that does not relyon prior information on the scenario Instead the locations ofthe physical and virtual transmitters are estimated simulta-neously with the user position in a simultaneous localizationand mapping (SLAM) [15 16] approach In general SLAMdescribes the simultaneous estimation of a user positionand the locations of landmarks In Channel-SLAM thelandmarks are the physical and virtual transmitters Previousextensions to Channel-SLAM include mapping of the userpositions [17] the consideration of vehicular applications[18] and data association methods [19 20] for example

Nonlinearities in the prediction and update equations ofthe Bayesian recursive estimation framework prohibit the useof optimal algorithms such as the Kalman filter since theintegrals involved in the estimation process cannot be solvedin closed form or become intractable A popular alternativeis the extended Kalman filter (EKF) [21] which linearizes thenonlinear terms using a first-order Taylor series expansionHowever such a linearization can introduce large errors inthe estimation process [22] The unscented Kalman filter(UKF) [23 24] uses a nonlinear transformation to deal withnonlinearities and outperforms the EKF in a wide range ofapplications [22 25]

UKF methods have found their way into localizationproblems for example in [27 28] The authors of [29]propose Gaussian sum cubature filters In [30 31] theauthors consider a Rao-Blackwellization scheme for SLAMwith a particle filter for the user state and UKFs for thelandmark states where the measurement model is based onlinearization though

The current Channel-SLAM algorithm uses a Rao-Blackwellized particle filter to estimate the user state andthe location of transmitters simultaneously Hence boththe user state probability density function (PDF) and thetransmitter state PDFs are represented by a large set ofparticles tending to result in a highmemory occupationThispaper is an extension of [32] where we proposed a novelestimation approach for Channel-SLAM scheme based onRao-Blackwellization and performed first simulations Werefer to this new estimation method as Rao-BlackwellizedGaussian sum particle filter (RBGSPF) In the RBGSPFthe user position is tracked by a sequential importanceresampling (SIR) particle filter while the physical and virtualtransmitter state PDFs are represented by Gaussian mixturemodels estimated byUKFsThis parametrized representationof the transmitter states is a key enabler for exchangingmaps of transmitters among users since the amount of datathat has to be communicated among users can be decreaseddrastically compared to the nonparametric representationwith particles Such an exchange of maps may be performeddirectly among users or via a central entity for example

in form of local dynamic maps (LDMs) in an intelligenttransportation system (ITS) context In this paper we providea full and detailed derivation of our novel algorithm Inparticular we derive the calculation of the particle weightsin the user particle filter given the representation of thetransmitters in the UKF framework Since data associationis an essential feature for the accuracy in long-term SLAMwe also derive a data association method based on [33]We evaluate our algorithm by both simulations in an urbanscenario and outdoor measurements

The remainder of this paper is structured as followsSection 2 describes the fundamental idea behind multipathassisted positioning and Channel-SLAM In Section 3 webriefly summarize some concepts of nonlinear Kalman filter-ing The derivation of the RBGSPF is presented in Section 4and a solution to data association is presented in Section 5After the experimental results in Section 6 Section 7 con-cludes the paper

Throughout the paper we use the following notation

(i) As indices 119894 stands for a user particle 119895 denotes atransmitter or a signal component ℓ is a componentin a Gaussian mixture model and 119898 stands for asigma point

(ii) (sdot)119879 denotes the transpose of a matrix or vector(iii) 1119899 denotes the identity matrix of dimension 119899 times 119899(iv) 0119899 and 0119898times119899 denote the zero matrices of dimensions119899 times 119899 and119898 times 119899 respectively(v) N(x120583C) denotes the PDF of a normal distribution

in x with mean 120583 and covariance C(vi) 1198880 denotes the speed of light(vii) sdot denotes the Euclidean norm of a vector

2 Multipath Assisted Positioning

21 Virtual Transmitters The idea of virtual transmitters isillustrated in Figure 1 The physical transmitter Tx transmitsan RF signal A mobile user equipped with an RF receiverreceives the transmitted signal via three different propagationpaths

In the first case the signal is reflected at the reflectingsurface The user treats the corresponding impinging MPCas being transmitted by the virtual transmitter vTx1 in a pureLoS condition The location of vTx1 is the location of thephysical transmitter Tx mirrored at the reflecting surfaceWhen the usermoves along the trajectory the reflection pointat the wall moves as well However the location of vTx1is static The two transmitters Tx and vTx1 are inherentlyperfectly synchronized

In the second case the signal from the physical trans-mitter is scattered by a point scatterer and then receivedby the user We define the effect of scattering such that theenergy of an electromagnetic wave impinging against thescatterer is distributed uniformly in all directions [34] Theuser regards the scattered signal as a LoS signal from thevirtual transmitter vTx2 which is located at the scattererlocation If the signal is scattered the physical and the virtual

Journal of Electrical and Computer Engineering 3

ReflectingSurface

UservTx2

vTx1vTx3

ScattererTx

Figure 1 Signals from the physical transmitter Tx are receivedat the two user positions via different propagation paths EachMPC arriving at the receiver is regarded as being transmitted by avirtual transmitter in a pure LoS condition The propagation pathscorrespond to a reflection at the wall (vTx1) a scattering at a pointscatterer (vTx2) and a scattering followed by a reflection at the wall(vTx3)

transmitter are not time synchronized the virtual transmitterhas an additional delay offset to the physical transmittercorresponding to the propagation time of the signal travelingfrom the physical to the virtual transmitter

In the third case the signal is first scattered at the scattererand then reflected at the surfaceThe user treats this signal asbeing sent from the virtual transmitter vTx3 The location ofvTx3 is the location of vTx2 that is the scatterer locationmirrored at the reflecting surface Accordingly the conceptof single reflections and scatterings can be generalized in astraightforwardmanner to the case ofmultiple reflections andscatterings by applying the first two cases iteratively In casethe signal undergoes only reflections the physical and thevirtual transmitters are inherently time synchronized If thesignal is scattered at least once the delay offset correspondsto the actual propagation time of the signal from the physicaltransmitter to the last scatterer the signal interacts withTherefore in Figure 1 the virtual transmitters vTx2 and vTx3have the same delay offset towards the physical transmitterNote that a delay offset can be interpreted as a clock offset

Throughout the paper we consider the physical trans-mitter and the environment to be static Hence the virtualtransmitters are static as well

22 Recursive Bayesian Estimation Recursive Bayesian esti-mation [35] is a method to recursively estimate the evolutionof a state vector x where the state evolution is modeled as

x119896 = f119896 (x119896minus1 k119896minus1) (1)

The index 119896 denotes the time instant the function f119896(sdot) isassumed to be known and k119896minus1 denotes a sample of theprocess noise with covariance matrix Q The state is relatedto the measurement z119896 by

z119896 = h119896 (x119896n119896) (2)

where h119896(sdot) is again a known function and n119896 is a sample ofthe measurement noise with covariance matrix R Recursive

PositionEstimation

ParameterEstimation

ReceivedSignal

IMU

PositionEstimate

Figure 2 Based on the received signal the parameters of the propa-gation paths are estimated in the first step by the KEST algorithm Inthe second step the estimates serve as measurements for estimatingthe positions of the user and the physical and virtual transmitters Inaddition user heading rates of change measurements from an IMUare incorporated in the second step

Bayesian estimation works in two steps the prediction andthe update step The corresponding PDFs can be calculatedrecursively by

p (x119896 | z1119896minus1) = int p (x119896 | x119896minus1) p (x119896minus1 | z1119896minus1) dx119896minus1 (3)

for the prediction step and by

p (x119896 | z1119896) = 1119888119896 p (z119896 | x119896) p (x119896 | z1119896minus1) (4)

for the update step where 119888119896 is a constant and z1119896 denotes themeasurements from time instant 1 to 119896 The state transitionprior p(x119896 | x119896minus1) and the measurement likelihood p(z119896 | x119896)are obtained from the movement model in (1) and themeasurement model in (2) respectively

23 Channel-SLAM In the following we will revise theChannel-SLAM algorithm from [13 17] Figure 2 gives anoverview of the two stages of Channel-SLAM In the firststage the parameters of the signal components received bythe user via different propagation paths are estimated Theresulting estimates are used as measurement input in thesecond stage where the states of the physical and virtualtransmitters and the user position are estimated simultane-ously in a SLAM scheme Further sensors such as an inertialmeasurement unit (IMU) may be included in the secondstageThe locations of both the physical and virtual transmit-ters are assumed to be unknown Thus Channel-SLAM doesnot differentiate between physical and virtual transmittersand the term transmitter comprises both physical and virtualtransmitters in the following Each signal component arrivingat the receiver corresponds to one transmitter

The RF propagation channel between the physical trans-mitter and the user equipped with a receiver is assumed to bea linear and time-variant multipath channel

The received signal ismodeled as a superposition of signalcomponents of the transmit signal 119904(119905) where the 119895th signalcomponent is defined by a complex amplitude 119886119895(119905119896) and adelay 119889119895(119905119896) at time 119905119896 The signal received by the user at timeinstant 119905119896 is

119910 (120591 119905119896) = sum119895

119886119895 (119905119896) 119904 (120591 minus 119889119895 (119905119896)) + 119899 (120591) (5)

where 119899(120591) is a sample from a colored noise sequence incor-porating both dense multipath components (DMCs) andadditive Gaussian noise The channel is assumed to be


constant during the short time interval when the receivedsignal is sampled at time instant 119896

The physical transmitter continuously broadcasts thesignal 119904(119905) that is known to the user At the user side theparameters of the signal components arriving at the receiverare estimated Such parameters can in general be the complexamplitude time of arrival (ToA) angle of arrival (AoA) orDoppler shift depending on the available hardware and thescenario For the signal parameter estimation we use theKEST algorithm [36] The KEST estimator works in twostages In an inner stage a ML parameter estimator suchas Space-Alternating Generalized Expectation-Maximization(SAGE) [37] estimates the parameters of the signal compo-nents jointly on a snapshot basis An outer stage tracks theseestimated parameters over time with a Kalman filter andkeeps track of the number of signal components The KESTestimator is in general able to handle the DMCs in the noiseterm in (5) However DMC handling is not implementedin our evaluations leading to a model mismatch in KESTand hence to a higher variance in the parameter estimationHowever we do not expect many DMCs in our evaluationscenarios In an indoor scenario for example DMCs need tobe considered [38]

In the literature there are alternatives to the KESTestimator For example the authors of [39] track signalcomponent parameters based on an EKF though the authorsof [40] showed that the KEST estimator is more robust inresolving signal components that are close to each other inthe state space In [41] an EKF is used as well for parameterestimation while the position estimation is based on thetime difference of arrival (TDoA) of virtual transmitters Theauthors of [42] consider the linearization of the observationmodel in the EKF a major drawback that might lead to atracking loss

In the second stage of Channel-SLAM we use onlythe delays that is ToAs and AoAs estimated by KESTas measurement inputs Hence after sampling the receivedsignal the KEST estimates at time instant 119896 are comprised inthe vector

z119896 = [d119879119896 120579119879119896 ]119879 (6)

where

d119896 = [1198891119896 sdot sdot sdot 119889119873TX 119896]119879 (7)

are the ToA estimates for the 119873TX signal components ortransmitters and

120579119896 = [1205791119896 sdot sdot sdot 120579119873TX 119896]119879 (8)

are the corresponding AoA estimates Note that the numberof signal components and thus transmitters may change overtime Nevertheless for notational convenience we omit thetime instant index 119896 in119873TX

In the second stage of Channel-SLAM the user state xu119896is estimated simultaneously with the state of the transmittersxTX119896 The entire state vector is hence

x119896 = [xu119896119879 xTX119896119879]119879= [xu119896119879 x⟨1⟩TX119896

119879 sdot sdot sdot x⟨119873TX⟩TX119896119879]119879 (9)

where x⟨119895⟩TX119896 is the state of the 119895th transmitter As we considera two-dimensional scenario the user state at time instant 119896 isdefined by

xu119896 = [119909119896 119910119896 V119909119896 V119910119896]119879 = [p119879u119896 k119879u119896]119879 (10)

where the user position is defined by pu119896 = [119909119896 119910119896]119879 andthe user velocity by ku119896 = [V119909119896 V119910119896]119879 Each transmitter isdefined by its location pTX119896 = [119909TX119896 119910TX119896]119879 and a clockoffset 1205910119896 at time instant 119896 The state vector of the 119895thtransmitter is hence defined by

x⟨119895⟩TX119896 = [119909⟨119895⟩TX119896 119910⟨119895⟩TX119896 120591⟨119895⟩0119896

]119879 = [p⟨119895⟩TX119896119879 120591⟨119895⟩0119896

]119879 (11)

Our goal is to find the minimum mean square error(MMSE) estimator for x119896 which is defined as

x119896 = int x119896p (x119896 | z1119896) dx119896 (12)

where z1119896 denotes all measurements up to time instant 119896We use a recursive Bayesian estimation scheme as in

Section 22 to estimate the posterior PDF p(x119896 | z1119896) Thisposterior can be factorized as

p (x119896 | z1119896) = p (xTX119896 xu119896 | z1119896)= p (xTX119896 | xu119896 z1119896) p (xu119896 | z1119896) (13)

The signal components arriving at the receiver areassumed to be independent of each other that is we assumethey interact with distinct objects Assuming independenceamong the measurements for distinct transmitters (on theone hand the parameters of the signal components areestimated jointly by the KEST algorithm and hence theseestimates might be correlated between signal componentsand between the parameters on the other hand the correla-tion is likely to have effect only on a short term basis as KESTestimates are unbiasedwhenobserved over a longer time) thefirst factor in (13) can be factorized further as

p (xTX0119896 | xu0119896 z1119896) =119873TXprod119895=1

p (x⟨119895⟩TX0119896 | xu0119896 z⟨119895⟩1119896) (14)

With the above factorization the transmitter states areestimated independently from each other

As we consider a static scenario the virtual transmittersare static as well and the transition prior for the 119895thtransmitter is calculated as

p (x⟨119895⟩TX119896 | x⟨119895⟩TX119896minus1) = 120575 (x⟨119895⟩TX119896 minus x⟨119895⟩TX119896minus1) (15)

where 120575(sdot) denotes the Dirac distributionFor the prediction of the user additional sensors such

as an IMU carried by the user may be integrated intothe movement model Within this paper we assume onlyheading change rate measurements from a gyroscope to beavailable though and no knowledge on the user speed


Txj

djk

jk

vuk

Δ k

Figure 3 The user moves in the direction ku119896 at time instant 119896The heading change rate from the IMU is Δ 120573119896 The ToA and AoAmeasurements for the signal from the 119895th transmitter are 119889119895119896 and120579119895119896 respectively where 120579119895119896 describes the angle between the headingdirection ku119896 of the user and the arriving signal

With the gyroscope heading change rate Δ 120573119896 we predict themovement of the user by

xu119896 = [12 1198791198961202 R (Δ 120573119896)] xu119896minus1 = Fu119896xu119896minus1 (16)

where 119879119896 denotes the time between instants 119896 minus 1 and 119896 Thetwo-dimensional rotation matrix R(Δ 120573119896) is defined as

R (Δ 120573119896) = [[cos (Δ 120573119896 + 119908119896) minus sin (Δ 120573119896 + 119908119896)sin (Δ 120573119896 + 119908119896) cos (Δ 120573119896 + 119908119896) ]

] (17)

where 119908119896 is the heading noise which is distributed followinga von Mises distribution Hence the function f119896 in (1) can beexpressed in our case as

f119896 (x119896minus1 k119896minus1) = [ Fu119896 04times3119873TX

03119873TXtimes4 13119873TX

] x119896minus1 + k119896minus1 (18)

where the process noise covariance matrixQ is diagonalAs depicted in Figure 3 an AoA measurement for a

transmitter 119895 describes the angle 120579119895119896 between the userheading direction ku119896 and the incoming signal from thetransmitter The measurement noise for the ToA and AoAmeasurements is assumed to be zero-mean Gaussian dis-tributed with variances 1205902119889119895 and 1205902120579119895 respectively for the 119895thtransmitter Also we assume no cross-correlation betweenthe single ToA andAoAmeasurementsThe likelihood for themeasurement vector z119896 conditioned on the state vector x119896 istherefore the product

p (z119896 | x119896) =119873TXprod119895=1

N (119889119895119896 119889119895119896 1205902119889119895)N (120579119895119896 120579119895119896 1205902120579119895) (19)

where the predicted ToA between the user and the 119895thtransmitter is

119889119895119896 = 11198880100381710038171003817100381710038171003817pu119896 minus p⟨119895⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119895⟩0119896

(20)

and the predicted AoA is calculated as

120579119895119896 = atan2 (119910119896 minus 119910⟨119895⟩TX119896 119909119896 minus 119909⟨119895⟩TX119896)minus atan2 (V119910119896 V119909119896) (21)

The function atan2(119910 119909) calculates the four-quadrant inversetangent function It returns the counterclockwise anglebetween the positive 119909-axis and the point given by thecoordinates (119909 119910)3 Nonlinear Kalman Filtering

31 Unscented Transform If a random variable x is trans-formed by a function g(sdot) such that y = g(x) the statisticsof y cannot always be calculated in closed form Monte Carlo(MC) methods try to estimate the statistics of y from a setof randomly chosen sample points of x that undergo thetransformation g(sdot) For the unscented transform a set of theso-called sigma points is propagated through the functiong(sdot) to obtain transformed sigma points yielding the statisticsof y However the sigma points are not chosen randomlybut in a deterministic manner which is the fundamentaldifference to MC methods

Based on the unscented transform numerical approxi-mations of integrals can be derived In particular for the casewhen the integrand is a product of an arbitrary function g(x)of the integration variable x and aGaussian PDFN(x120583xCx)an integration rule of the form

int g (x)N (x120583xCx) dx asymp 119873sigsum119898=1

120596119898g (X119898) (22)

can be applied where X119898 is the 119898th of the 119873sig sigmapoints with its associated weight 120596119898 The idea of the UKFis to approximate the posterior PDF in recursive Bayesianestimation by a Gaussian PDF Hence the integral in theprediction step is approximated by the integration rulein (22) The authors of [43] provide further insight intosigma point methods and their relation to Gaussian processquadrature

32 Choice of Sigma Points In the literature different sets ofsigmapoints have been proposed for the unscented transform[44] Let X119898 be the 119898th sigma point and 120596119898 its associatedweight The dimension mean and covariance of the randomvariable x are denoted by119873 120583x and Cx respectively In [23]the sigma points and their weights are defined for some 120581 isin R

as

X0 = 120583x 1205960 = 120581120581 + 119873X119898 = 120583x + (radic(119873 + 120581)Cx)

119898 120596119898 = 12 (120581 + 119873)

X119898+119873 = 120583x minus (radic(119873 + 120581)Cx)119898120596119898+119873 = 12 (120581 + 119873)

(23)

where119898 = 1 119873 (A)119898 denotes the119898th row or column ofthe matrix A and (119873 + 120581)Cx is factorized into

(119873 + 120581)Cx = radic(119873 + 120581)Cxradic(119873 + 120581)Cx119879 (24)


1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩

Figure 4 Structure of the RBGSPF representation the user state xuis represented by a number of particles Each particle estimates thetransmittersrsquo states on its own Each of the 119873TX transmitter statesis represented by a sum of 119873UKF Gaussian distributions Nℓ withassociated weights 119908⟨ℓ⟩

This definition leads to 119873sig = 2119873 + 1 sigma points Theauthors of [45] presented the cubature Kalman filter (CKF)with an intuitive derivation of the choice of sigma pointsand their weights The CKF differs from the UKF only inthe choice of the sigma points Its derivation is based on thefact that the approximation of an integral using the unscentedtransform as in (22) is exact for g(x) being a monomial of anorder not greater than some integer 119889 The resulting sigmapoints are the points in (23) for 120581 = 0 Since the weight of thefirst sigma point is zero there are only 2119873 effective pointsAlthough the derivation in [45] gives useful insight into theUKF the same choice of sigma points had been proposed in[44] before

The Appendix summarizes the equations for the pre-diction and update steps of the UKF If the state transitionmodel in (1) or the measurement model in (2) are linearor if Gaussian noise is assumed in the state transition ormeasurement model methods from [46] can be applied todecrease the computational complexity of the UKF

4 Derivation of the Gaussian SumParticle Filter

41 The Rao-Blackwellized Gaussian Sum Particle Filter Thefactorization in (13) allows for estimating the user stateindependently from the transmitter states For the estimationof the user state in the RBGSPF we use a SIR particle filter[26 47]The single transmitter states x⟨119895⟩TX119896 are estimated inde-pendently from each other following (14) Each transmitterstate is represented by a Gaussian mixture model or Gaussiansum model [48] The posterior PDF of each of the 119873UKFGaussian components in a Gaussian mixture is estimated bya UKFThe structure of the resulting RBGSPF representationis shown in Figure 4

A particle filter is a MC based method where theposterior PDF is represented by a number of random samplescalled particles with associated weights The user posteriorPDF is approximated as

p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)

where x⟨119894⟩u119896 is the 119894th user particle 119908⟨119894⟩119896

its associated weightand 119873119901 the number of particles in the particle filter Fromthe structure of (14) we see that the transmitter states areestimated for each user particle independently

The posterior distribution of the state of each transmitteris approximated by a Gaussian mixture model In a Gaussianmixture model a PDF is described as a sum of weightedGaussian PDFs each described by a mean and a covarianceHence the posterior PDF of the state of the 119895th transmitter ofthe 119894th user particle is represented as [48]

p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (x⟨119894119895⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (26)

where z⟨119895⟩1119896

are the measurements for the 119895th transmitterand x⟨119894119895ℓ⟩TX119896|119896 P

⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩

119896are the mean the covariance

matrix and the weight respectively of the ℓth Gaussiancomponent of the Gaussian mixture for the 119895th transmitterBoth x⟨119894119895ℓ⟩TX119896|119896 and P⟨119894119895ℓ⟩

119896|119896are obtained from the update step

of the corresponding UKF Similarly the likelihood for themeasurement of the 119895th transmitter of the 119894th user particle is

p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896

is the measurement noise covariance matrix forthe 119895th transmitter The number 119873UKF of Gaussian compo-nents might differ between transmitters user particles andtime instants However for notational convenience we dropthe particle transmitter and time instant indices of 119873UKFThe predicted measurement of the ℓth Gaussian componentfor the 119895th transmitter of the 119894th user particle

z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)

consists of the predicted ToA measurement

119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)

and the predicted AoA measurement

120579⟨119894119895ℓ⟩119896

= atan2 (119910⟨119894⟩119896 minus 119910⟨119894119895ℓ⟩TX119896 119909⟨119894⟩119896 minus 119909⟨119894119895ℓ⟩TX119896 )minus atan2 (V⟨119894⟩119910119896 V⟨119894⟩119909119896)

(30)

where the 119894th user particle is

x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)


and the mean of the corresponding ℓth Gaussian componentof the 119895th transmitter is

x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)

In the prediction step of the user particle filter newparticles are sampled based on the transition prior p(xu119896 |xu119896minus1) Hence the particle x⟨119894⟩u119896 is drawn as

x⟨119894⟩u119896 = fu119896 (x⟨119894⟩u119896minus1 ku119896minus1) (33)

where the function fu119896(sdot) describes the user movementmodel and ku119896minus1 is a noise sample drawn from the userprocess noise PDF For the prediction and the update step ofa Gaussian component of a transmitterrsquos state the equationsof the UKF are summarized in the Appendix

42 Derivation of the Particle Weight Calculation In thefollowing we will derive the calculation of the particleweights in the user particle filter and of the weights forthe Gaussian components in the Gaussian mixture modelsused to describe the PDFs of the transmitter states As theimportance density of the SIR particle filter is the statetransition prior and resampling of the particles is performedat every time instant the weight for the 119894th particle at timeinstant 119896 is given by [26]

119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)

This expression can be written as

119908⟨119894⟩119896 prop int p (z119896 | x⟨119894⟩u0119896 x⟨119894⟩TX119896 z1119896minus1)times p (x⟨119894⟩TX119896 | x⟨119894⟩u0119896 z1119896minus1) dx⟨119894⟩TX119896prop int p (z119896 | x⟨119894⟩u119896 x⟨119894⟩TX119896)times p (x⟨119894⟩TX119896 | x⟨119894⟩u119896 z119896minus1) dx⟨119894⟩TX119896

(35)

where we use the assumption of a first-order hidden Markovmodel With the assumption that the measurements areindependent for different transmitters (35) can be expressedas

119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896)times p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩119896minus1) dx⟨119894119895⟩TX119896

(36)

Furthermore using the Gaussian mixture model rep-resentation from (26) and (27) and assuming Gaussianmeasurement noise the integrand can be expressed as a sumof weighted Gaussian PDFs namely

p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩119896minus1)= 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)

In (37) we defined for notational brevity

p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896

is the measurement noise covariance matrix forthe 119895th transmitter and x⟨119894119895ℓ⟩TX119896 denotes the state of the ℓthGaussian component of the 119895th transmitter of the 119894th userparticle As we assume no correlation among the ToA andAoA measurements we have

R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)

Inserting (37) into (36) leads to

119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)

Thepredictedmeasurement z⟨119894119895ℓ⟩119896

in (38) defined in (28)is a nonlinear function of x⟨119894119895ℓ⟩TX119896 We express themeasurementlikelihood in (38) explicitly as a function g(sdot) of x⟨119894119895ℓ⟩TX119896 resulting in

g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)

Due to the nonlinearity in (42) stemming from (29) and (30)the integral in (41) cannot be solved analytically Instead weuse the approximation from (22) to calculate the integral as

int g (x⟨119894119895ℓ⟩TX119896 )N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) dx⟨119894119895ℓ⟩TX119896

asymp 119873sigsum119898=1

120596119898g (X119898) (43)

where the sigma points X119898 and their weights 120596119898 can becalculated by (23) where

120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)

and119873 is the dimension of a transmitterrsquos state that is119873 = 3Finally the weight of the 119894th particle is calculated as

119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)

It follows directly from (45) that the weights of the Gaussiancomponents are updated by

119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩

TX119896minus1 and 119908⟨119894119895ℓ⟩119896minus1

for 119894 = 1 119873119901 119895 = 1 119873TX ℓ = 1 119873UKF z119896Output x⟨119894⟩u119896 x

⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896

for 119894 = 1 119873119901 119895 = 1 119873TX ℓ = 1 119873UKF(1) for 119894 = 1 119873119901 do(2) draw new user particle x⟨119894⟩u119896 using (33)(3) if any new signal components detected then(4) initialize the new transmitter(s) based on z119896(5) if track of any signal components lost then(6) delete the corresponding transmitter(s)(7) for 119895 = 1 119873TX do(8) for ℓ = 1 119873UKF do(9) perform UKF prediction and update to calculate x⟨119894119895ℓ⟩TX119896 using the UKF equations in the Appendix(10) calculate the weight 119908⟨119894119895ℓ⟩

119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896

lt 120588 then(12) prune this Gaussian component set 119908⟨119894119895ℓ⟩

119896= 0

(13) calculate the weight 119908⟨119894⟩119896

with (45)(14) for 119894 = 1 119873119901 do(15) for 119895 = 1 119873TX do(16) normalize the weights 119908⟨119894119895ℓ⟩

119896for ℓ = 1 119873UKF

(17) normalize the weights 119908⟨119894⟩119896

for 119894 = 1 119873119901(18) resample the user particles x⟨119894⟩u119896 [26]

Algorithm 1 RBGSPF for time instant 119896 gt 0

Note that the weights 119908⟨119894⟩119896

and 119908⟨119894119895ℓ⟩119896

in (45) and (46) ofthe user particles and the Gaussian components respectivelyare not yet normalized Since resampling is performed atevery time instant in the SIR particle filter the user particleweights 119908⟨119894⟩

119896do not depend on the weights 119908⟨119894⟩

119896minus1from the

previous time instant [26]

43 Merging and Pruning of Gaussian Components Whenthe KEST estimator detects a new signal component a newtransmitter is initialized for each user particle based onthe ToA and AoA measurement for that new transmitterat the current time instant The posterior PDF of the newtransmitter is represented by a number of Gaussian PDFswhose means are initially placed on a grid dependent on themeasurementThe number of Gaussian components dependson the measurement as well

As the user travels through a scenario the meanscovariances and weights of the Gaussian components of atransmitterrsquos state posterior PDF change over time dependingon the available measurements The mean and covariance ofa Gaussian component may be regarded as a hypothesis anda corresponding uncertainty respectively for the state of atransmitter If the weight of a Gaussian component becomessmaller the hypothesis for that state of the transmitterbecomes less likely Hence if the weight of a Gaussian com-ponent falls below a threshold 120588 the Gaussian component ispruned that is its weight 119908⟨119894119895ℓ⟩

119896minus1is set to zero If the means

of two Gaussian components get very close to each otherthey may be merged in order to reduce the computationalcomplexity

The final algorithm for one time instant 119896 gt 0 of theRBGSPF is summarized in Algorithm 1 For a particle filter

(III)(II)

(I)

Tx

Figure 5 A user moves along the trajectory The LoS signal to thetransmitter in Region (I) is lost in Region (II) temporarily due toblocking by an obstacle and received again in Region (III)

resampling algorithm we refer to [26] Note again that wehave dropped the particle and transmitter indices in119873UKF

5 Data Association

Data association is of crucial importance for robust long-termSLAM It describes the correspondences among landmarkswhich are transmitters in multipath assisted positioning InFigure 5 a user travels along its trajectory In Region (I)the LoS signal from the transmitter Tx can be tracked Thissignal is lost in Region (II) and regained in Region (III)However KEST is not able to retrack a former path Hencewhen the user enters Region (III) KEST detects a new signalcomponent and consequently a new transmitter is initializedHowever the transmitter is the same as that which had beenobserved in Region (I)

We define the set of transmitters that had been observedpreviously but are not detected any more as old transmittersWhen a new signal component is detected by KEST a new


transmitter has to be initialized Consequently two casesmayarise

(1) the new signal component corresponds indeed to anew transmitter or

(2) the new signal component corresponds to an oldtransmitter that had been observed before already

Data association is the decision on the above two cases whena new signal component is detected In the first case anew transmitter is initialized for the newly detected signalcomponent In the second case the newly detected signalcomponent is associated with a previously observable that isold transmitter

In [33] amultiple hypothesis tracking (MHT) associationmethodwas introduced and derived for FastSLAMwhere theuser state is represented by a particle filter and each landmarkstate by an EKF In [19] the samemethod has been derived fora Rao-Blackwellized particle filter In the following we willderive the method for the RBGSPF derived in Section 4

Each user particle decides for associations individu-ally and thus carries a hypothesis for associations Henceassociation decisions are hard decisions for each particleRegarding the ensemble of user particles though there aremany different hypotheses on associations in the user stateestimate and the associationmethod can be regarded as a softdecision method Consequently the state vector of the user isincreased by data association

In the following we describe how to make an associationdecision for a single user particle where we omit the particleindex 119894 in the association variables for notational brevity Thevalue of the association variable 119899119896 denotes an association ofthe new transmitter with the old transmitter 119899119896 We denotethe marginalized likelihood of the measurement of the newtransmitter that is to be initialized at time instant 119896 by 119901119899119896assuming that the new transmitter is associated with the oldtransmitter 119899119896 The set of association decisions up to timeinstant 119896 minus 1 is denoted by119873119896minus1 From [33] we have

119901119899119896 = p (z119896 | 119899119896 119873119896minus1 x⟨119894⟩u119896 z1119896minus1)= int p (z119896 | x⟨119894119899119896⟩TX119896 119899119896 119873119896minus1 x⟨119894⟩u119896 z1119896minus1)times p (x⟨119894119899119896⟩TX119896 | 119899119896 119873119896minus1 x⟨119894⟩u119896 z1119896minus1) dx⟨119894119899119896⟩TX119896

(47)

where x⟨119894119899119896⟩TX119896 denotes the state vector of the 119899119896th transmitterfor the 119894th user particle

Assuming a first-order hidden Markov model the firstintegrand in (47) can be simplified to

p (z119896 | x⟨119894119899119896⟩TX119896 119899119896 119873119896minus1 x⟨119894⟩u119896 z1119896minus1)= p (z119896 | x⟨119894119899119896⟩TX119896 119899119896 119873119896minus1 x⟨119894⟩u119896)

(48)

Since we use a Gaussian mixture model to represent thesingle transmitter states the second integrand in (47) can berewritten as

p (x⟨119894119899119896⟩TX119896 | 119899119896 119873119896minus1 x⟨119894⟩u119896 z1119896minus1)

= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

N (x⟨119894119899119896⟩TX119896 x⟨119894119899119896ℓ⟩TX119896|119896minus1P⟨119894119899119896ℓ⟩119896|119896minus1)

(49)

Inserting (48) and (49) into (47) yields

119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)

where p⟨119894119899119896ℓ⟩z119896 is defined as in (38) and

p⟨119894119899119896ℓ⟩

TX119896|119896minus1 = N (x⟨119894119899119896ℓ⟩TX119896 x⟨119894119899119896ℓ⟩TX119896|119896minus1P⟨119894119899119896ℓ⟩119896|119896minus1) (51)

Similar to (42) we define

g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)

which is nonlinear in x⟨119894119899119896ℓ⟩TX119896 and rewrite the integral in (50)as

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩

TX119896 = int g (x⟨119894119899119896ℓ⟩TX119896 )timesN (x⟨119894119899119896ℓ⟩TX119896 x⟨119894119899119896ℓ⟩TX119896|119896minus1P⟨119894119899119896ℓ⟩119896|119896minus1

) dx⟨119894119899119896ℓ⟩TX119896 (53)

Approximating the integral using (22) and inserting itinto (50) finally yield

119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)

The sigma points X119898 and their weights 120596119898 can again becalculated by (23) with

120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)

The authors of [33] propose two ways to come to anassociation decision a ML method and data associationsampling (DAS) The probability for making no associationis defined and denoted by 1199010 The set of indices of oldtransmitters that have not yet been and hence might beassociated is denoted by Γ119896

For the ML association method the association of thenew transmitter with the old transmitter 119899119896 is chosen to be

119899ML119896 = arg max119899119896isinΓ119896cup0

119901119899119896 (56)

In DAS an association is sampled based on the likelihoods119888119901119899119896 for 119899119896 isin Γ119896 cup 0 where 119888 is a normalization constantIf a detected transmitter is associated with an old

transmitter the new transmitter can be initialized with the


Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)

Figure 6 The simulation scenario with thick black lines represent-ing reflecting walls and black circles representing objects that scatterthe RF signalsThe physical transmitter is marked by the red upwardtriangle labeled TxThe user travels along the blue line from STARTto END with one loop around the central building

posterior PDF of the associated old transmitter Thus dataassociation has to be incorporated in Line (4) of Algorithm 1

The above method describes how to take associationdecisions if no more than one new transmitter is initializedat one time instant that is if no more than one new signalcomponent is detected by KEST at a time instant In caseof multiple transmitters being initialized at the same timeinstant a greedy algorithm [20] may be applied

6 Evaluations

In the following we evaluate the RBGSPF derived inSection 4 by means of simulations and actual outdoor mea-surements For the evaluations we implemented a square-root version of the cubature Kalman filter as in [45] fornumerical stability The sigma points are the ones in (23) for120581 = 0 Since the movement model of the transmitters is linearand we assume Gaussian noise the prediction step can becalculated analytically For the description of a prediction stepof a square-root version of the conventional Kalman filter werefer to [48]

61 Simulations in an Urban Scenario A top view of theurban simulation scenario is depicted in Figure 6 The thickblack lines represent walls for example from buildings thatreflect RF signals and the black circles are objects such astraffic light poles acting as scatterers There is one physicaltransmitter in the scenariomarked by the red upward trianglelabeled Tx The user travels with a constant speed of 10msalong the blue line with a loop around the central buildingThe initial and final user positions are labeled START and

100 200 300 400 5000User Traveled Distance (m)

0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)

Figure 7 The results of the KEST estimator for the simulationsshowing the propagation distances of signal components versusthe user traveled distance The propagation distances are the ToAmultiplied by the speed of light Only signal components that areobservable for a traveled distance of at least 35m are shown Thecolor indicates the normalized amplitude in linear domain

END respectively The traveled distances of the user aremarked for every 50m

The transmitter continuously broadcasts a signal that isknown to the user and has a rectangular shape in frequencydomain with a center frequency of 15 GHz and a bandwidthof 100MHz As we know the environment a CIR and thereceived signal can be modeled for every user position witha simple ray-tracing approach We incorporate first- andsecond-order reflections and scattering that is single anddouble reflections andor scattering The power loss for thesignal being reflected is 3 dB and 6 dB when the signal isscattered at a point scattererThe average signal-to-noise ratio(SNR) at the user is 7 dB

The user is equipped with an RF receiver and a two-dimensional rectangular antenna array consisting of nineelements Hence both the ToA and the AoA estimates fromKEST are incorporated in the estimation of the user andthe transmittersrsquo states Based on the received signal KESTestimates the ToAs and AoAs every 50ms

The results of the KEST estimator are plotted in Figure 7It shows the propagation distance which is the ToA mul-tiplied by the speed of light of the signal componentsversus the traveled distance of the user Each continuousline represents one signal component and its evolution asthe user travels through the scenario The color of each lineindicates the normalized absolute value of the amplitude ofthe corresponding signal component in linear domain Sincesignal components that are observable for a long time cancontribute much better to Channel-SLAM than componentswhich are observable only for a short time only signalcomponents that are observable for a user traveled distanceof at least 35m are plotted and used Using all detected signalcomponents would dramatically increase the computationalcomplexity and hardly increase the positioning performance


no associationsML associationDAS

100 200 300 400 500 6000User Traveled Distance (m)

0

5

10

15

20

25

30

Use

r RM

SE (m

)

Figure 8 The RMSE of the user position versus the user traveleddistance for the simulations The red curve shows the RMSE if noassociations among transmitters are made the blue curve if the MLmethod for associations is applied and the green curve for usingDAS

In Channel-SLAM the user position is estimated relativeto the physical and virtual transmitters in the scenario Thusto create a local coordinate system the initial state of the useris assumed to be known However no prior knowledge onany transmitter is assumed In the Rao-Blackwellized particlefilter the number of user particles in the particle filter is4000 while the number of Gaussian components for eachtransmitter depends on the first ToA measurement for thattransmitter

The root mean square error (RMSE) of the user positionversus its traveled distance is plotted in Figure 8 The redcurve shows the RMSE if no associations among transmittersare made that is every signal component that is detectedby the KEST algorithm is assumed to be a new transmitterThe RMSEs with the ML association method and DASfrom Section 5 being applied are plotted in blue and greenrespectively Since the particle filter is a MC based methodall RMSE curves are averaged over 100 simulations

As we assume the starting position of the user to beknown all three curves start with a low RMSE that increaseslinearly during the first 200m as expected The increaseof the RMSE is less due to a bias in the position estimatebut more due to an increasing uncertainty that is varianceabout the user position After approximately 200m theRMSE tends to decrease for all three curves As more andmore transmitters are observed the weight for some userparticles becomes small and these particles are unlikely tobe resampled Towards the end of the track the geometricaldelusion of precision (GDOP) causes an increase in theRMSE since most of the transmitters are observed from thesame direction After a traveled distance of around 370mseveral transmitters that had been observed in the beginningare observed again and correspondences among them canbe found If data association methods are used the RMSEdecreases particularly in that region The ML method andDAS show a similar performance Note that there are severalreasons for which associations among transmitters can befound Examples are signal blocking or the geometry of the

Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)

Figure 9 Top viewof themeasurement scenario in front of a hangarThe physical transmitter location is marked by the red trianglelabeled Tx The user travels along the blue track from START toEND The expected virtual transmitter locations are marked by themagenta downward triangles

environment causing virtual transmitters to be observableonly from certain regions In addition when KEST loses andregains track of a signal component if its received powerfluctuates or if another signal component arrives at thereceiver with a very small difference in delay transmittersmay be discarded and initialized again at a later point andassociations among them may be found This explains theincreasing positioning performance gain after approx 50musing data association

62 OutdoorMeasurements In addition to the simulations asdescribed above we performed outdoormeasurements on anairfield A top view of the measurement scenario is depictedin Figure 9 The grey area is an airplane hangar with solidmetallic doors The user track with a total length of 1125m isplotted in blueThe user walked along the track starting fromthe light blue cross labeled START to the black cross labeledEND The traveled distance of the user is marked after 25m75m and 100mThere is one physical transmitter marked bythe red upward triangle labeled Tx The user is in LoS to thephysical transmitter throughout the entire track

In the scenario we have three fences labeled Fence 1Fence 2 and Fence 3 We expect these fences and thehangar door to reflect the RF signal emitted by the physicaltransmitter Hence we expect a virtual transmitter for eachof the fences and for the hangar door following Section 21The virtual transmitter corresponding to the reflection ofthe signal at Fence 1 is the magenta downward trianglelabeled vTx2 It is located at the physical transmitter positionmirrored at Fence 1 Likewise the location of the virtualtransmitter corresponding to Fence 2 is labeled as vTx3 Forthe reflection of the signal at the hangar doors we expectthe virtual transmitter located at the magenta triangle labeledas vTx1 The expected virtual transmitter corresponding toFence 3 vTx4 is outside of the boundaries of Figure 9


0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)


GLoS TxGLoS vTx1GLoS vTx2

GLoS vTx3GLoS vTx4

Figure 10 The results of the KEST estimator for the outdoormeasurements showing the propagation distances that is the ToAsmultiplied by the speed of light of signal components versus theuser traveled distance Only signal components that are observablefor long traveled distance are shown The color indicates theestimated received power for the signal components in dB

The Medav RUSK broadband channel sounder [49] wasused to perform the measurements The transmit signal is amultitone signal with a center frequency of 151 GHz and abandwidth of 100 MHz The signal has 1281 subcarriers withequal gains and a total transmit power of 10mW

The user was equipped with an RF receiver recordinga snapshot of the received signal every 1024ms For laterevaluation the user carried a prism mounted next to thereceiver antenna that was tracked by a tachymeter (LeicaGeosystems TCRP1200) to obtain the ground truth of theuser location in centimeter accuracy In addition the usercarried an XsensMTI-G-700 IMU Only heading change ratemeasurements were used from the IMU

On both transmitter and receiver side single antennaswere used Hence no AoA information about the imping-ing signal components can be used for Channel-SLAMInstead only ToA estimates from KEST are incorporatedThe likelihood function in (19) is adapted accordingly for theevaluation

The results of the KEST estimator for the outdoor mea-surements are plotted in Figure 10 The colors indicate thepower estimated by KEST in dBm As for the simulationsonly signal components that are observable for a long usertraveled distance are plotted and used In addition theground truth geometrical line-of-sight (GLoS) propagationdistances from the physical and the expected virtual trans-mitters as in Figure 9 to the user are plotted by black linesTheymatch theKEST estimates verywell justifying the signalmodel in (5) without considering DMCs in KEST for themeasurement scenario


0

1

2

3

4

5

6

Use

r RM

SE (m

)


Figure 11 The RMSE of the user position versus the user traveleddistance for the outdoor measurements The red curve shows theRMSE if no associations among transmitters are made the bluecurve if the ML method for associations is applied and the greencurve for using DAS

The RMSE of the user position versus its traveled distancefor the outdoor measurements is plotted in Figure 11 wherethe RMSE is averaged over 50 particle filter simulationsAs for Figure 8 the red curve denotes the RMSE with noassociation method applied the blue and green curves showthe RMSE if the ML method and DAS respectively areincorporated for data association

Theuser is always in a LoS condition to the physical trans-mitter vTx and the corresponding LoS signal component istracked by the user throughout the track as becomes evidentin Figure 10 Likewise the signal component correspondingto the virtual transmitter vTx2 can be tracked after a traveleddistance of approximately 22m until the end

The almost continuous presence of the signals fromthese two transmitters is reflected in the user RMSE inFigure 11 The RMSE without data association methodsapplied increases in the beginning but then stays constantin the order of 3-4m with some fluctuations This is due tothe fact that once the variance on the states of transmittersvTx and vTx2 has decreased far enough they serve asreliable anchors throughout the track Hence they preventthe uncertainty about the user state from increasing furtheralthoughwemeasure only the ToA for each signal componentin the outdoor measurement scenario

For the same reason the data association methodscannot really improve the user positioning performance inthe outdoor measurements From another point of view acorrect data association is inherently made for vTx and vTx2throughout the track since once these transmitters havebeen initialized they stay observable throughout the trackHowever if a user was to go through the same scenario asecond time with prior information of the transmitter statesas estimated during the first run data association wouldimprove the positioning performance as correspondencesamong transmitters estimated during the first and second runcould be found and exploited


In contrast the user RMSE in Figure 8 without dataassociation in the simulations keeps increasing since thereare constantly new transmitters showing up and currenttransmitters disappear As mentioned above the uncertaintyabout transmitters is high upon initialization since themeasurements obtained from KEST are of fewer dimensionsthan the transmitter states In addition the current useruncertainty adds up to the transmitter uncertainty Notransmitter can be tracked throughout the scenario andthe overall uncertainty keeps increasing In the simulationsdata association relates new transmitters with previouslyobservable transmitters and decreases the uncertainty aboutthe states of transmitters drastically Consequently also theuncertainty about and hence the RMSE of the user statedecrease

7 Conclusion

Within this paper we derived a novel filtering approachfor Channel-SLAM Using Rao-Blackwellization the userstate is represented by a number of particles and estimatedby a particle filter The states of the landmarks which arethe physical and virtual transmitters in Channel-SLAM arerepresented by a sum of Gaussian PDFs where eachGaussiancomponent PDF is filtered by a UKF The approach can beapplied to SLAM problems in general

We evaluated our approach in simulations in an urbanscenario as well as with outdoormeasurement data where wecould track a userrsquos position with only one physical transmit-ter whose location was unknown For the simulations in theurban scenario the user RMSE was always below 21m Withthe presented data associationmethods applied it was alwaysbelow 165m For the measurements on an airfield the userRMSE was in the order of 3-4m

Appendix

A UKF Prediction and Update Equations

A1 Prediction Step

(1) Given the Gaussian state PDF p(x119896minus1 | z119896minus1) =N(x119896minus1|119896minus1P119896minus1|119896minus1) calculate the set of 119873sig sigmapoints X119898119896minus1|119896minus1 and their weights 120596119898 for 119898 = 1 119873sig for example using (23)

(2) Propagate the sigma points through the movementmodel

Xlowast119898119896|119896minus1 = f119896 (X119898119896minus1|119896minus1) (A1)

(3) Calculate the predicted state

x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)

(4) Calculate the predicted error covariance

P119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1Xlowast119898119896|119896minus1119879 minus x119896|119896minus1x119879119896|119896minus1

+Q119896minus1(A3)

A2 Update Step

(1) Given the Gaussian state PDF p(x119896 | z119896minus1) =N(x119896|119896minus1P119896|119896minus1) calculate the set of 119873sig predictedsigma points X119898119896|119896minus1 and their weights 120596119898 as for theprediction

(2) Propagate the predicted sigma points through themeasurement function

Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)

(3) Calculate the predicted measurement

z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)

(4) Calculate the estimated innovation covariancematrix

P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1Z119898119896|119896minus1119879 minus z119896|119896minus1z119879119896|119896minus1

+ R119896minus1(A6)

(5) Calculate the cross-covariance matrix

P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1

120596119898X119898119896|119896minus1Z119898119896|119896minus1119879 minus x119896|119896minus1z119879119896|119896minus1 (A7)

(6) Calculate the Kalman gain

W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)

(7) Calculate the updated state estimate

x119896|119896 = x119896|119896minus1 +W119896 (z119896 minus z119896|119896minus1) (A9)

(8) Calculate the updated error covariance

P119896|119896 = P119896|119896minus1 minusW119896P⟨119911119911⟩119896|119896minus1W

119879119896 (A10)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was partially supported by the EuropeanUnionrsquos Project HIGHTS (High Precision Positioningfor Cooperative-ITS Applications) MG-35a-2014-636537and the DLR project Navigation 40

References

[1] J Li Y Zhao J Zhang R Jiang C Tao and Z Tan ldquoRadiochannel measurements and analysis at 245GHz in subwaytunnelsrdquo China Communications vol 12 no 1 pp 36ndash45 2015


[2] M Lentmaier and B Krach ldquoMaximum likelihood multipathestimation in comparison with conventional delay lock loopsrdquoin Proceedings of the Institute of Navigation - 19th InternationalTechnical Meeting of the Satellite Division ION GNSS 2006 pp1741ndash1751 usa September 2006

[3] F Antreich J A Nossek and W Utschick ldquoMaximum likeli-hood delay estimation in a navigation receiver for aeronauticalapplicationsrdquo Aerospace Science and Technology vol 12 no 3pp 256ndash267 2008

[4] M Triki D T M Slock V Rigal and P Francois ldquoMobileterminal positioning via power delay profile fingerprintingReproducible validation simulationsrdquo in Proceedings of the 2006IEEE 64th Vehicular Technology Conference VTC-2006 Fall pp2840ndash2844 September 2006

[5] Y Shen and M Z Win ldquoOn the use of multipath geometry forwideband cooperative localizationrdquo in Proceedings of the 2009IEEE Global Telecommunications Conference GLOBECOM2009 December 2009

[6] P Setlur and N Devroye ldquoMultipath exploited Bayesianand Cramer-Rao bounds for single-sensor target localizationrdquoEURASIP Journal onAdvances in Signal Processing vol 2013 no1 article no 53 2013

[7] P Setlur G E Smith F Ahmad and M G Amin ldquoTargetlocalization with a single sensor via multipath exploitationrdquoIEEE Transactions on Aerospace and Electronic Systems vol 48no 3 pp 1996ndash2014 2012

[8] P Meissner E Leitinger M Frohle and K Witrisal ldquoAccurateand robust indoor localization systems using ultra-widebandsignalsrdquo in Proceedings of the in European Conference onNavigation (ENC) 2013

[9] E Leitinger P Meissner M Lafer and K Witrisal ldquoSimulta-neous localization and mapping using multipath channel infor-mationrdquo in Proceedings of the IEEE International Conference onCommunicationWorkshop ICCW 2015 pp 754ndash760 June 2015

[10] K Witrisal P Meissner E Leitinger et al ldquoHigh-accuracylocalization for assisted living 5G systems will turn multipathchannels from foe to friendrdquo IEEE Signal Processing Magazinevol 33 no 2 pp 59ndash70 2016

[11] M Froehle E Leitinger P Meissner and K Witrisal ldquoCoop-erative multipath-assisted indoor navigation and tracking (Co-MINT) using UWB signalsrdquo in Proceedings of the 2013 IEEEInternational Conference on Communications Workshops ICC2013 pp 16ndash21 Hungary June 2013

[12] A OrsquoConnor P Setlur and N Devroye ldquoSingle-sensor RFemitter localization based on multipath exploitationrdquo IEEETransactions on Aerospace and Electronic Systems vol 51 no 3pp 1635ndash1651 2015

[13] C Gentner and T Jost ldquoIndoor positioning using time differ-ence of arrival between multipath componentsrdquo in Proceedingsof the 2013 International Conference on Indoor Positioning andIndoor Navigation IPIN 2013 Montbeliard France October2013

[14] C Gentner R Pohlmann M Ulmschneider T Jost and ADammann ldquoMultipath Assisted Positioning for Pedestriansrdquo inProceedings of the in Proceedings of the ION GNSS+ Tampa FLUSA 2015

[15] H Durrant-Whyte and T Bailey ldquoSimultaneous localizationand mapping Part Irdquo IEEE Robotics and Automation Magazinevol 13 no 2 pp 99ndash108 2006

[16] T Bailey and H Durrant-Whyte ldquoSimultaneous localizationand mapping (SLAM) Part IIrdquo IEEE Robotics and AutomationMagazine vol 13 no 3 pp 108ndash117 2006

[17] C Gentner B MaM Ulmschneider T Jost and A DammannldquoSimultaneous localization and mapping in multipath envi-ronmentsrdquo in Proceedings of the IEEEION Position Locationand Navigation Symposium PLANS 2016 pp 807ndash815 GeorgiaApril 2016

[18] M Ulmschneider R Raulefs C Gentner and M WalterldquoMultipath assisted positioning in vehicular applicationsrdquo inProceedings of the 13thWorkshop on Positioning Navigation andCommunication WPNC 2016 Germany October 2016

[19] M Ulmschneider C Gentner T Jost and A Dammann ldquoAsso-ciation of Transmitters in Multipath-Assisted Positioningrdquoin Proceedings of the GLOBECOM 2017 - 2017 IEEE GlobalCommunications Conference pp 1ndash7 Singapore December 2017

[20] M Ulmschneider C Gentner T Jost and A DammannldquoMultiple hypothesis data association for multipath-assistedpositioningrdquo in Proceedings of the 2017 14th Workshop onPositioning Navigation and Communications (WPNC) pp 1ndash6Bremen October 2017

[21] Y Bar-Shalom X Li and T Kirubarajan Estimation withApplications to Tracking and Navigation John Wiley amp SonsNew York NY USA 2001

[22] E A Wan and R van der Merwe ldquoThe unscented Kalman filterfor nonlinear estimationrdquo in Proceedings of the IEEE AdaptiveSystems for Signal Processing Communications and ControlSymposium (AS-SPCC rsquo00) pp 153ndash158 Lake Louise CanadaOctober 2000

[23] S J Julier J K Uhlmann and H F Durrant-Whyte ldquoA newapproach for filtering nonlinear systemsrdquo in Proceedings of theAmerican Control Conference pp 1628ndash1632 June 1995

[24] S J Julier and J K Uhlmann ldquoA new extension of theKalman filter to nonlinear systemsrdquo in Proceedings of the SignalProcessing Sensor Fusion andTarget RecognitionVI vol 3068 ofProceedings of SPIE pp 182ndash193 Orlando Fla USA 1997

[25] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[26] M S Arulampalam S Maskell N Gordon and T Clapp ldquoAtutorial on particle filters for online nonlinearnon-GaussianBayesian trackingrdquo IEEE Transactions on Signal Processing vol50 no 2 pp 174ndash188 2002

[27] R Martinez-Cantin and J A Castellanos ldquoUnscented SLAMfor large-scale outdoor environmentsrdquo in Proceedings of theIEEE IRSRSJ International Conference on Intelligent Robots andSystems (IROS rsquo05) pp 3427ndash3432 Edmonton Canada August2005

[28] S Thrun W Burgard and D Fox Probabilistic Robotics serIntelligent robotics and autonomous agents MIT Press 2005

[29] P H Leong S Arulampalam T A Lamahewa and T DAbhayapala ldquoA Gaussian-sum based cubature Kalman filter forbearings-only trackingrdquo IEEE Transactions on Aerospace andElectronic Systems vol 49 no 2 pp 1161ndash1176 2013

[30] C Kim R Sakthivel and W K Chung ldquoUnscented Fast-SLAM a robust algorithm for the simultaneous localization andmapping problemrdquo in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA rsquo07) pp 2439ndash2445 Roma Italy April 2007

[31] C Kim R Sakthivel andW K Chung ldquoUnscented FastSLAMa robust and efficient solution to the SLAM problemrdquo IEEETransactions on Robotics vol 24 no 4 pp 808ndash820 2008

[32] M Ulmschneider C Gentner R Faragher and T Jost ldquoGaus-sian mixture filter for multipath assisted positioningrdquo in Pro-ceedings of the 29th International Technical Meeting of the


Satellite Division of the Institute of Navigation ION GNSS 2016pp 1263ndash1269 usa September 2016

[33] SThrunMMontemerlo D Koller BWegbreit J Nieto and ENebot ldquoFastSLAM An Efficient Solution to the SimultaneousLocalization AndMapping Problemwith UnknownData Asso-ciationrdquo Journal of Machine Learning Research vol 4 no 3 pp380ndash407 2004

[34] T S RappaportWireless Communications Principles and Prac-tice ser Prentice Hall communications engineering and emergingtechnologies Prentice Hall PTR 2002

[35] S SarkkaBayesian Filtering and Smoothing CambridgeUniver-sity Press Cambridge UK 2013

[36] T Jost W Wang U-C Fiebig and F Perez-Fontan ldquoDetectionand tracking of mobile propagation channel pathsrdquo IEEETransactions on Antennas and Propagation vol 60 no 10 pp4875ndash4883 2012

[37] B H Fleury M Tschudin R Heddergott D Dahlhaus and KI Pedersen ldquoChannel parameter estimation in mobile radioenvironments using the SAGE algorithmrdquo IEEE Journal onSelected Areas in Communications vol 17 no 3 pp 434ndash4501999

[38] J Poutanen enGeometry-based radio channel modeling Prop-agation analysis and concept development [PhD thesis] AaltoUniversity 2011

[39] J Salmi A Richter and V Koivunen ldquoDetection and track-ing of MIMO propagation path parameters using state-spaceapproachrdquo IEEE Transactions on Signal Processing vol 57 no4 pp 1538ndash1550 2009

[40] WWang T Jost and A Dammann ldquoEstimation andmodellingof NLoS time-variant multipath for localization channel modelin mobile radiosrdquo in Proceedings of the 53rd IEEE Global Com-munications Conference GLOBECOM 2010 USA December2010

[41] M Zhu J Vieira Y Kuang K Astrom A F Molisch and FTufvesson ldquoTracking and positioning using phase informationfrom estimated multi-path componentsrdquo in Proceedings of theIEEE International Conference on Communication WorkshopICCW 2015 pp 712ndash717 June 2015

[42] G E Kirkelund G Steinbock X Yin and B Fleury ldquoTrackingof the temporal behaviour of path components in the radiochannel - A comparison betweenmethodsrdquo inProceedings of the2008 Annual IEEE Student Paper Conference AISPC DenmarkFebruary 2008

[43] S Sarkka J Hartikainen L Svensson and F Sandblom ldquoOnthe relation between Gaussian process quadratures and sigma-point methodsrdquo Journal of Advances in Information Fusion vol11 no 1 pp 31ndash46 2016

[44] D SimonOptimal State Estimation Kalman Hinfin and Nonlin-ear Approaches Wiley-Interscience 2006

[45] I Arasaratnam and S Haykin ldquoCubature Kalman filtersrdquoInstitute of Electrical and Electronics Engineers Transactions onAutomatic Control vol 54 no 6 pp 1254ndash1269 2009

[46] M Briers S R Maskell and R Wright ldquoA rao-blackwellisedunscented Kalman filterrdquo in Proceedings of the 6th InternationalConference on Information Fusion FUSION 2003 pp 55ndash61Australia July 2003

[47] N J Gordon D J Salmond and A F M Smith ldquoNovelapproach to nonlinearnon-gaussian Bayesian state estimationrdquoIEE proceedings F vol 140 no 2 pp 107ndash113 1993

[48] B Anderson and JMooreOptimal Filtering ser Dover Books onElectrical Engineering Dover Publications 2012

[49] R S Thoma M Landmann G Sommerkorn and A RichterldquoMultidimensional high-resolution channel sounding inmobileradiordquo in Proceedings of the 21st IEEE Instrumentation andMeasurement Technology Conference IMTC04 pp 257ndash262ita May 2004

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If the locations of physical transmitters and reflecting andscattering objects are known the locations of virtual trans-mitters can be calculated based on geometrical considera-tionsThe authors of [12] assume the room layout to be knownand focus on the association among virtual transmitters andreflecting walls In a general setting however the scenario isunknown to the user

The authors of [13 14] have presented amultipath assistedpositioning scheme namedChannel-SLAM that does not relyon prior information on the scenario Instead the locations ofthe physical and virtual transmitters are estimated simulta-neously with the user position in a simultaneous localizationand mapping (SLAM) [15 16] approach In general SLAMdescribes the simultaneous estimation of a user positionand the locations of landmarks In Channel-SLAM thelandmarks are the physical and virtual transmitters Previousextensions to Channel-SLAM include mapping of the userpositions [17] the consideration of vehicular applications[18] and data association methods [19 20] for example

Nonlinearities in the prediction and update equations ofthe Bayesian recursive estimation framework prohibit the useof optimal algorithms such as the Kalman filter since theintegrals involved in the estimation process cannot be solvedin closed form or become intractable A popular alternativeis the extended Kalman filter (EKF) [21] which linearizes thenonlinear terms using a first-order Taylor series expansionHowever such a linearization can introduce large errors inthe estimation process [22] The unscented Kalman filter(UKF) [23 24] uses a nonlinear transformation to deal withnonlinearities and outperforms the EKF in a wide range ofapplications [22 25]

UKF methods have found their way into localizationproblems for example in [27 28] The authors of [29]propose Gaussian sum cubature filters In [30 31] theauthors consider a Rao-Blackwellization scheme for SLAMwith a particle filter for the user state and UKFs for thelandmark states where the measurement model is based onlinearization though

The current Channel-SLAM algorithm uses a Rao-Blackwellized particle filter to estimate the user state andthe location of transmitters simultaneously Hence boththe user state probability density function (PDF) and thetransmitter state PDFs are represented by a large set ofparticles tending to result in a highmemory occupationThispaper is an extension of [32] where we proposed a novelestimation approach for Channel-SLAM scheme based onRao-Blackwellization and performed first simulations Werefer to this new estimation method as Rao-BlackwellizedGaussian sum particle filter (RBGSPF) In the RBGSPFthe user position is tracked by a sequential importanceresampling (SIR) particle filter while the physical and virtualtransmitter state PDFs are represented by Gaussian mixturemodels estimated byUKFsThis parametrized representationof the transmitter states is a key enabler for exchangingmaps of transmitters among users since the amount of datathat has to be communicated among users can be decreaseddrastically compared to the nonparametric representationwith particles Such an exchange of maps may be performeddirectly among users or via a central entity for example

in form of local dynamic maps (LDMs) in an intelligenttransportation system (ITS) context In this paper we providea full and detailed derivation of our novel algorithm Inparticular we derive the calculation of the particle weightsin the user particle filter given the representation of thetransmitters in the UKF framework Since data associationis an essential feature for the accuracy in long-term SLAMwe also derive a data association method based on [33]We evaluate our algorithm by both simulations in an urbanscenario and outdoor measurements

The remainder of this paper is structured as followsSection 2 describes the fundamental idea behind multipathassisted positioning and Channel-SLAM In Section 3 webriefly summarize some concepts of nonlinear Kalman filter-ing The derivation of the RBGSPF is presented in Section 4and a solution to data association is presented in Section 5After the experimental results in Section 6 Section 7 con-cludes the paper

Throughout the paper we use the following notation

(i) As indices 119894 stands for a user particle 119895 denotes atransmitter or a signal component ℓ is a componentin a Gaussian mixture model and 119898 stands for asigma point

(ii) (sdot)119879 denotes the transpose of a matrix or vector(iii) 1119899 denotes the identity matrix of dimension 119899 times 119899(iv) 0119899 and 0119898times119899 denote the zero matrices of dimensions119899 times 119899 and119898 times 119899 respectively(v) N(x120583C) denotes the PDF of a normal distribution

in x with mean 120583 and covariance C(vi) 1198880 denotes the speed of light(vii) sdot denotes the Euclidean norm of a vector

2 Multipath Assisted Positioning

21 Virtual Transmitters The idea of virtual transmitters isillustrated in Figure 1 The physical transmitter Tx transmitsan RF signal A mobile user equipped with an RF receiverreceives the transmitted signal via three different propagationpaths

In the first case the signal is reflected at the reflectingsurface The user treats the corresponding impinging MPCas being transmitted by the virtual transmitter vTx1 in a pureLoS condition The location of vTx1 is the location of thephysical transmitter Tx mirrored at the reflecting surfaceWhen the usermoves along the trajectory the reflection pointat the wall moves as well However the location of vTx1is static The two transmitters Tx and vTx1 are inherentlyperfectly synchronized

In the second case the signal from the physical trans-mitter is scattered by a point scatterer and then receivedby the user We define the effect of scattering such that theenergy of an electromagnetic wave impinging against thescatterer is distributed uniformly in all directions [34] Theuser regards the scattered signal as a LoS signal from thevirtual transmitter vTx2 which is located at the scattererlocation If the signal is scattered the physical and the virtual


ReflectingSurface

UservTx2

vTx1vTx3

ScattererTx






x119896 = f119896 (x119896minus1 k119896minus1) (1)


z119896 = h119896 (x119896n119896) (2)


PositionEstimation

ParameterEstimation

ReceivedSignal

IMU

PositionEstimate





p (x119896 | z1119896) = 1119888119896 p (z119896 | x119896) p (x119896 | z1119896minus1) (4)





119910 (120591 119905119896) = sum119895

119886119895 (119905119896) 119904 (120591 minus 119889119895 (119905119896)) + 119899 (120591) (5)







z119896 = [d119879119896 120579119879119896 ]119879 (6)

where



120579119896 = [1205791119896 sdot sdot sdot 120579119873TX 119896]119879 (8)



x119896 = [xu119896119879 xTX119896119879]119879= [xu119896119879 x⟨1⟩TX119896



xu119896 = [119909119896 119910119896 V119909119896 V119910119896]119879 = [p119879u119896 k119879u119896]119879 (10)


x⟨119895⟩TX119896 = [119909⟨119895⟩TX119896 119910⟨119895⟩TX119896 120591⟨119895⟩0119896

]119879 = [p⟨119895⟩TX119896119879 120591⟨119895⟩0119896

]119879 (11)


x119896 = int x119896p (x119896 | z1119896) dx119896 (12)





p (xTX0119896 | xu0119896 z1119896) =119873TXprod119895=1

p (x⟨119895⟩TX0119896 | xu0119896 z⟨119895⟩1119896) (14)







Txj

djk

jk

vuk

Δ k






] (17)



03119873TXtimes4 13119873TX

] x119896minus1 + k119896minus1 (18)



p (z119896 | x119896) =119873TXprod119895=1

N (119889119895119896 119889119895119896 1205902119889119895)N (120579119895119896 120579119895119896 1205902120579119895) (19)


119889119895119896 = 11198880100381710038171003817100381710038171003817pu119896 minus p⟨119895⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119895⟩0119896

(20)







120596119898g (X119898) (22)



as

X0 = 120583x 1205960 = 120581120581 + 119873X119898 = 120583x + (radic(119873 + 120581)Cx)

119898 120596119898 = 12 (120581 + 119873)

X119898+119873 = 120583x minus (radic(119873 + 120581)Cx)119898120596119898+119873 = 12 (120581 + 119873)

(23)




1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩







p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)




p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896


where z⟨119895⟩1119896


⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩





p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896


z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)


119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)


120579⟨119894119895ℓ⟩119896


(30)


x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)



x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



ReflectingSurface

UservTx2

vTx1vTx3

ScattererTx






x119896 = f119896 (x119896minus1 k119896minus1) (1)


z119896 = h119896 (x119896n119896) (2)


PositionEstimation

ParameterEstimation

ReceivedSignal

IMU

PositionEstimate





p (x119896 | z1119896) = 1119888119896 p (z119896 | x119896) p (x119896 | z1119896minus1) (4)





119910 (120591 119905119896) = sum119895

119886119895 (119905119896) 119904 (120591 minus 119889119895 (119905119896)) + 119899 (120591) (5)







z119896 = [d119879119896 120579119879119896 ]119879 (6)

where



120579119896 = [1205791119896 sdot sdot sdot 120579119873TX 119896]119879 (8)



x119896 = [xu119896119879 xTX119896119879]119879= [xu119896119879 x⟨1⟩TX119896



xu119896 = [119909119896 119910119896 V119909119896 V119910119896]119879 = [p119879u119896 k119879u119896]119879 (10)


x⟨119895⟩TX119896 = [119909⟨119895⟩TX119896 119910⟨119895⟩TX119896 120591⟨119895⟩0119896

]119879 = [p⟨119895⟩TX119896119879 120591⟨119895⟩0119896

]119879 (11)


x119896 = int x119896p (x119896 | z1119896) dx119896 (12)





p (xTX0119896 | xu0119896 z1119896) =119873TXprod119895=1

p (x⟨119895⟩TX0119896 | xu0119896 z⟨119895⟩1119896) (14)







Txj

djk

jk

vuk

Δ k






] (17)



03119873TXtimes4 13119873TX

] x119896minus1 + k119896minus1 (18)



p (z119896 | x119896) =119873TXprod119895=1

N (119889119895119896 119889119895119896 1205902119889119895)N (120579119895119896 120579119895119896 1205902120579119895) (19)


119889119895119896 = 11198880100381710038171003817100381710038171003817pu119896 minus p⟨119895⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119895⟩0119896

(20)







120596119898g (X119898) (22)



as

X0 = 120583x 1205960 = 120581120581 + 119873X119898 = 120583x + (radic(119873 + 120581)Cx)

119898 120596119898 = 12 (120581 + 119873)

X119898+119873 = 120583x minus (radic(119873 + 120581)Cx)119898120596119898+119873 = 12 (120581 + 119873)

(23)




1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩







p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)




p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896


where z⟨119895⟩1119896


⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩





p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896


z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)


119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)


120579⟨119894119895ℓ⟩119896


(30)


x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)



x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







z119896 = [d119879119896 120579119879119896 ]119879 (6)

where



120579119896 = [1205791119896 sdot sdot sdot 120579119873TX 119896]119879 (8)



x119896 = [xu119896119879 xTX119896119879]119879= [xu119896119879 x⟨1⟩TX119896



xu119896 = [119909119896 119910119896 V119909119896 V119910119896]119879 = [p119879u119896 k119879u119896]119879 (10)


x⟨119895⟩TX119896 = [119909⟨119895⟩TX119896 119910⟨119895⟩TX119896 120591⟨119895⟩0119896

]119879 = [p⟨119895⟩TX119896119879 120591⟨119895⟩0119896

]119879 (11)


x119896 = int x119896p (x119896 | z1119896) dx119896 (12)





p (xTX0119896 | xu0119896 z1119896) =119873TXprod119895=1

p (x⟨119895⟩TX0119896 | xu0119896 z⟨119895⟩1119896) (14)







Txj

djk

jk

vuk

Δ k






] (17)



03119873TXtimes4 13119873TX

] x119896minus1 + k119896minus1 (18)



p (z119896 | x119896) =119873TXprod119895=1

N (119889119895119896 119889119895119896 1205902119889119895)N (120579119895119896 120579119895119896 1205902120579119895) (19)


119889119895119896 = 11198880100381710038171003817100381710038171003817pu119896 minus p⟨119895⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119895⟩0119896

(20)







120596119898g (X119898) (22)



as

X0 = 120583x 1205960 = 120581120581 + 119873X119898 = 120583x + (radic(119873 + 120581)Cx)

119898 120596119898 = 12 (120581 + 119873)

X119898+119873 = 120583x minus (radic(119873 + 120581)Cx)119898120596119898+119873 = 12 (120581 + 119873)

(23)




1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩







p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)




p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896


where z⟨119895⟩1119896


⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩





p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896


z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)


119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)


120579⟨119894119895ℓ⟩119896


(30)


x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)



x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Txj

djk

jk

vuk

Δ k






] (17)



03119873TXtimes4 13119873TX

] x119896minus1 + k119896minus1 (18)



p (z119896 | x119896) =119873TXprod119895=1

N (119889119895119896 119889119895119896 1205902119889119895)N (120579119895119896 120579119895119896 1205902120579119895) (19)


119889119895119896 = 11198880100381710038171003817100381710038171003817pu119896 minus p⟨119895⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119895⟩0119896

(20)







120596119898g (X119898) (22)



as

X0 = 120583x 1205960 = 120581120581 + 119873X119898 = 120583x + (radic(119873 + 120581)Cx)

119898 120596119898 = 12 (120581 + 119873)

X119898+119873 = 120583x minus (radic(119873 + 120581)Cx)119898120596119898+119873 = 12 (120581 + 119873)

(23)




1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩







p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)




p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896


where z⟨119895⟩1119896


⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩





p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896


z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)


119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)


120579⟨119894119895ℓ⟩119896


(30)


x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)



x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



1

xu x４８

N４８

1

1 w⟨1⟩

N５＋

wN５＋

w⟨1⟩

w⟨NUKF⟩

⟨N５＋⟩







p (xu119896 | z1119896) =119873119901sum119894=1

119908⟨119894⟩119896 120575 (xu119896 minus x⟨119894⟩u119896) (25)




p (x⟨119894119895⟩TX119896 | x⟨119894⟩u119896 z⟨119895⟩1119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896


where z⟨119895⟩1119896


⟨119894119895ℓ⟩

119896|119896 and 119908⟨119894119895ℓ⟩





p (z⟨119895⟩119896

| x⟨119894⟩u119896 x⟨119894119895⟩TX119896) = 119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896

timesN (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (27)

where R⟨119895⟩119896


z⟨119894119895ℓ⟩119896

= [119889⟨119894119895ℓ⟩119896

120579⟨119894119895ℓ⟩119896

]119879 (28)


119889⟨119894119895ℓ⟩119896

= 11198880100381710038171003817100381710038171003817p⟨119894⟩u119896 minus p⟨119894119895ℓ⟩TX119896

100381710038171003817100381710038171003817 + 120591⟨119894119895ℓ⟩0119896 (29)


120579⟨119894119895ℓ⟩119896


(30)


x⟨119894⟩u119896 = [119909⟨119894⟩119896 119910⟨119894⟩119896

V⟨119894⟩119909119896

V⟨119894⟩119910119896]119879 = [p⟨119894⟩u119896119879 k⟨119894⟩u119896

119879]119879 (31)



x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




x⟨119894119895ℓ⟩TX119896|119896 = [119909⟨119894119895ℓ⟩TX119896 119910⟨119894119895ℓ⟩TX119896 120591⟨119894119895ℓ⟩0119896

]119879= [p⟨119894119895ℓ⟩TX119896

119879 120591⟨119894119895ℓ⟩0119896

]119879 (32)





119908⟨119894⟩119896 prop p (z119896 | x⟨119894⟩u0119896 z1119896minus1) (34)



(35)


119908⟨119894⟩119896 prop 119873TXprod119895=1

int p (z⟨119895⟩119896


(36)


p (z⟨119895⟩119896


119908⟨119894119895ℓ⟩119896minus1

p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 (37)


p⟨119894119895ℓ⟩

z119896 = N (z⟨119895⟩119896

z⟨119894119895ℓ⟩119896

R⟨119895⟩119896

) (38)

p⟨119894119895ℓ⟩

TX119896 = N (x⟨119894119895ℓ⟩TX119896 x⟨119894119895ℓ⟩TX119896|119896P⟨119894119895ℓ⟩119896|119896 ) (39)

where R⟨119895⟩119896


R⟨119895⟩119896

= [1205902119889119895 00 1205902120579119895] (40)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

int p⟨119894119895ℓ⟩

z119896 p⟨119894119895ℓ⟩

TX119896 dx⟨119894119895ℓ⟩

TX119896 (41)



g (x⟨119894119895ℓ⟩TX119896 ) = p⟨119894119895ℓ⟩

z119896

= N (119889119895119896 119889⟨119894119895ℓ⟩119896 1205902119889119895)timesN (120579119895119896 120579⟨119894119895ℓ⟩119896 1205902120579119895)

(42)




120596119898g (X119898) (43)


120583x = x⟨119894119895ℓ⟩TX119896|119896Cx = P⟨119894119895ℓ⟩

119896|119896 (44)


119908⟨119894⟩119896 prop 119873TXprod119895=1

119873UKFsumℓ=1

119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (45)


119908⟨119894119895ℓ⟩119896

prop 119908⟨119894119895ℓ⟩119896minus1

119873sigsum119898=1

120596119898g (X119898) (46)


Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Input x⟨119894⟩u119896minus1 x⟨119894119895ℓ⟩



⟨119894119895ℓ⟩

TX119896 and 119908⟨119894119895ℓ⟩119896


119896with (46)

(11) if 119908⟨119894119895ℓ⟩119896


119896= 0



119896for ℓ = 1 119873UKF





and 119908⟨119894119895ℓ⟩119896










(III)(II)

(I)

Tx



5 Data Association












(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











(47)




(48)



= 119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896


(49)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896 dx⟨119894119899119896ℓ⟩TX119896 (50)


p⟨119894119899119896ℓ⟩



g (x⟨119894119899119896ℓ⟩TX119896 ) = p⟨119894119899119896ℓ⟩

z119896

= N (119889119899119896 119896 119889⟨119894119899119896ℓ⟩119896 1205902119889119899119896)

timesN (120579119899119896 119896 120579⟨119894119899119896ℓ⟩119896 1205902120579119899119896)

(52)


int p⟨119894119899119896ℓ⟩

z119896 p⟨119894119899119896ℓ⟩

TX119896|119896minus1dx⟨119894119899119896ℓ⟩


) dx⟨119894119899119896ℓ⟩TX119896 (53)


119901119899119896 =119873UKFsumℓ=1

119908⟨119894119899119896ℓ⟩119896

119873sigsum119898=1

120596119898g (X119898) (54)


120583x = x⟨119894119899119896ℓ⟩TX119896|119896minus1Cx = P⟨119894119899119896ℓ⟩

119896|119896minus1 (55)




119901119899119896 (56)




Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Tx

START

END

50m

450m

100m

150m 200 m

250 m

350 m

300 m

400 m

500m

550m

1008020 40 140 1600 12060minus20minus40

x (m)

minus80

minus60

minus40

minus20

0

20

40

60

80

100

120

140y

(m)




6 Evaluations




0

01

02

03

04

05

06

07

08

09

1

Nor

mal

ized

Abs

olut

e Am

plitu

de

0

50

100

150

200

250

Prop

agat

ion

Dist

ance

(m)









0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





0

5

10

15

20

25

30

Use

r RM

SE (m

)





Tx

vTx1

vTx2

vTx3

y (m

) Fence 1

Fence 2

Fence 3

Hangar

START

END

Hangar Doors100 m

25m

75m

0

20

40

60

80

20 40 60 80 1000x (m)






0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0

50

100

150

Prop

agat

ion

Dist

ance

(m)

minus90

minus74

minus58

minus42

minus26

Estim

ated

Pow

er (d

Bm)



GLoS vTx3GLoS vTx4







0

1

2

3

4

5

6

Use

r RM

SE (m

)









7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




7 Conclusion



Appendix


A1 Prediction Step





x119896|119896minus1 =119873sigsum119898=1

120596119898Xlowast119898119896|119896minus1 (A2)


P119896|119896minus1 =119873sigsum119898=1


+Q119896minus1(A3)

A2 Update Step



Z119898119896|119896minus1 = h119896 (X119898119896|119896minus1) (A4)


z119896|119896minus1 =119873sigsum119898=1

120596119898Z119898119896|119896minus1 (A5)


P⟨119911119911⟩119896|119896minus1 =119873sigsum119898=1


+ R119896minus1(A6)


P⟨119909119911⟩119896|119896minus1 =119873sigsum119898=1



W119896 = P⟨119909119911⟩119896|119896minus1P⟨119911119911⟩119896|119896minus1

minus1 (A8)





119879119896 (A10)



Acknowledgments


References























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


rao-blackwellized gaussian sum particle filtering for

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