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Folie 1 > Positioning Based on Factor Graphs > Christian Mensing2005-12-15

Positioning Based on Factor GraphsChristian MensingNEWCOM, DPT 1, SWP 1, Cergy-Pontoise, France, 2005-12-15


German Aerospace Center (DLR)

Institute of Communications and NavigationDepartment of Communications Systems

Topics of the Mobile Radio Transmission Group:Multi-carrier transmission systems

Multiple access (MC-CDMA)Detection for MC-CDMA and related schemesChannel estimationMulti-cellular environmentsMultiple antenna techniques (Cyclic Delay/Doppler Diversity, …)

Combination of communications and navigationPerformance improvements of 4G mobile radio systems using GALILEONavigation by means of 4G systems

Involvement in EU-projects NEWCOM, WINNER and 4MORE


Outline

IntroductionIterative Algorithms for Time Difference of Arrival (TDoA)Factor Graphs and the Sum-Product AlgorithmPositioning Based on Factor GraphsConclusions and Outlook


Introduction


Introduction, Time of Arrival (ToA)

Measures the absolute time for a signal travelling from BS to MSExact time knowledge is necessary at the MS (synchronization)At least three BSs have to be visible for triangularizationPropagation time proportional to distance:


Introduction, Time Difference of Arrival (TDoA)

Measures the time difference of signals received from various BSsNo synchronization between MS and BS network necessaryMS lies on hyperbolas with foci at the two related BSsTDoAs are defined w.r.t. BS 1:


Time Difference of Arrival (TDoA)

TDoAs are defined w.r.t. BS 1:

System model:

Find position estimate by minimization of a cost function, i.e.

Weighted Nonlinear Least Squares (WNLS):


Iterative Algorithms for TDoA

Gauss-Newton (GN) MethodLinearization and linear least squares solution

Steepest Descent (SD) MethodGradient method

Levenberg-Marquardt (LM) AlgorithmDamped GN procedure


Performance Criteria

Mean Square Error (MSE):

Root Mean Square Error (RMSE):

Cramer Rao Lower Bound (CRLB):

Bound for minimum variance of the estimation error GDOP Valid for Gaussian noise only


Simulation Results (TDoA)


Factor Graphs

A function that can be factorized in local functions

e.g.

can be represented by a factor graph

Often we are interested in computing the marginal functionse.g.


Sum-Product Algorithm

We are interested in all (redundancy)Use of a message passing algorithm in the FG Sum-Product Algorithm (SPA)

Variable to local function:

Local function to variable:

Termination step:


Applications of Factor Graphs

Decoding of codes (Low-Density Parity Check, Hamming, Turbo, …)Maximum A Posteriori (MAP) algorithm (forward/backward, BCJR)Viterbi algorithm (change of SPA metric)Kalman filterFFT algorithms…


Positioning Based on Factor Graphs (TDoA)



Idea: Process x and y coordinates independently Principal axis transformation, shift and mapping operation

TDoA equations can be rewritten as

1. Principal axis transformation / rotation (R) by EVD

2. Shift operation (S)

3. Mapping operation (M)

with



Local functions:

……

…

…

…

…



……

…

…

…

…

with



……

…

…

…

…


Conclusions and Outlook

Conclusions:Positioning in cellular networks using TDoALM algorithm with good trade-off between accuracy and complexityFactor graphs and SPA as tool for several applicationsPositioning based on factor graphs for TDoA with promising results

Outlook:Synchronization aspectsHybrid approachesTracking (e.g. by Kalman filters)Cycles in factor graphsScheduling methodsMultipath mitigation, NLoS problem, MAI


German Aerospace Center (DLR)

Institute of Communications and NavigationDepartment of Communications SystemsMobile Radio Transmission Group

Possibilities of cooperation:Joint publicationsMeetings, seminars, colloquia, talksExchange of researchers

UndergraduateDiploma/Master theses at DLRInternships at DLR

Graduate, Postgraduate, …Visit as guest scientists at DLR

Visit of DLR research staff at NEWCOM partner institutions


Simulation Results


Further Positioning Methods

Angle of Arrival (AoA):Measures the angle of the incident waveMultiple antennas at BS necessaryOnly two BSs for positioning

Received Signal Strength (RSS):Measures the received power from the BSRange is calculated according to path loss modelsGeometric similar to ToA

Fingerprinting Solutions:Position characterized by channel impulse responseOnly one BS for positioning

(Cell ID)


Gauss-Newton (GN) Method

Linearizes the signal model about some initial value and applies linear LSLinearization Step:

with the Jacobian matrix

and

Finally, the iterated solution becomes


Steepest Descent (SD) Method

Gradient method, starting from an initial positionGradient vector of search direction

is weighted with the optimum stepsize (optimum Line Search)

and the iterated solution becomes


Levenberg-Marquardt (LM) Method

Gauss-Newton:Good estimates for accurate initial valuesDivergent for inaccurate initial values and bad geometric conditions

Steepest Descent:Always finds (sometimes local) minimumSlow convergence in the final iteration stages

Levenberg-Marquardt:Combination of GN and SD methodFast convergence for arbitrary initial values

Idea: Damped GN method

Parameter depends on quality of the recent estimates


Levenberg-Marquardt (LM) Method


Chan-Ho (CH) Method

Non-iterative methodExtension of the Spherical Intersection (SI) method

Idea:1. TDoA equations yield linear relation

for the three unknowns where independence is assumed2. The relation is used to improve the estimate3. Ambiguity due to squaring operation is resolved

Very good performance (CRLB) for low noise powerVery bad performance for high noise power


Simulation Results


Factor Graphs

Tree representation of a factor graph



Calculate



Termination step:



Messages can be reused for calculating all


Decoding of Hamming Codes

Characteristic function for a Hamming Code C is defined as

with

Factor graph represents factorization of the code‘s characteristic function

e.g. (7, 4, 3) Hamming Code with


Positioning Based on Factor Graphs (ToA)



with:

Local functions:

… …

…



… …

…

with



… …

…



… …

…

with



… …

…

Termination step


Simulation Results (ToA)


31 Research Institutes andScientific/TechnicalFacilities at

8 Sites4 BranchesGerman-DutchWind Tunnels (DNW)European Transonic Wind Tunnel (ETW)

Total Staff: 5000

German Aerospace Center (DLR)The national aerospace research center and space agency

Institute of Communications and NavigationActivities and Research in Communications

Research Activities in Satellite Communications• Signal design of the satellite navigation system

Galileo• Optical free space communications

(inter satellite links)• Broadband satellite services to aircraft for wireless

access technologies

Research Activities in Terrestrial Communications• DoCoMo Euro-Labs research collaboration• Combination of communications and navigation• EU projects:

• 4MORE (4G MC-CDMA Multiple Antenna Systemon Chip for Radio Enhancements)

• WINNER (Wireless World Initiative New Radio)• NEWCOM (Network of Excellence in Wireless Communications)

DLR Research Center Oberpfaffenhofen• Institute of Radio Frequency Technology and

Radar Systems• Institute of Communications and Navigation• Institute of Methods of Remote Sensing• Institute of Optoelectronics• Institute of Physics of the Atmosphere• Institute of Robotics and Mechatronics • German Space Operation Center• German Remote Sensing Data Center

Total Staff in Oberpfaffenhofen: 1000

german aerospace center (dlr)

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