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German Aerospace Center (DLR)

Positioning Based on Factor Graphs Christian Mensing NEWCOM, DPT 1, SWP 1, Cergy-Pontoise, France, 2005-12-15. German Aerospace Center (DLR). Institute of Communications and Navigation Department of Communications Systems Topics of the Mobile Radio Transmission Group:

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German Aerospace Center (DLR)

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

  2. German Aerospace Center (DLR) Institute of Communications and Navigation Department of Communications Systems Topics of the Mobile Radio Transmission Group: • Multi-carrier transmission systems • Multiple access (MC-CDMA) • Detection for MC-CDMA and related schemes • Channel estimation • Multi-cellular environments • Multiple antenna techniques (Cyclic Delay/Doppler Diversity, …) • Combination of communications and navigation • Performance improvements of 4G mobile radio systems using GALILEO • Navigation by means of 4G systems • Involvement in EU-projects NEWCOM, WINNER and 4MORE

  3. Outline • Introduction • Iterative Algorithms for Time Difference of Arrival (TDoA) • Factor Graphs and the Sum-Product Algorithm • Positioning Based on Factor Graphs • Conclusions and Outlook

  4. Introduction

  5. Introduction, Time of Arrival (ToA) • Measures the absolute time for a signal travelling from BS to MS • Exact time knowledge is necessary at the MS (synchronization) • At least three BSs have to be visible for triangularization • Propagation time proportional to distance:

  6. Introduction, Time Difference of Arrival (TDoA) • Measures the time difference of signals received from various BSs • No synchronization between MS and BS network necessary • MS lies on hyperbolas with foci at the two related BSs • TDoAs are defined w.r.t. BS 1:

  7. 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):

  8. Iterative Algorithms for TDoA • Gauss-Newton (GN) Method Linearization and linear least squares solution • Steepest Descent (SD) Method Gradient method • Levenberg-Marquardt (LM) Algorithm Damped GN procedure

  9. 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

  10. Simulation Results (TDoA)

  11. 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 functions e.g.

  12. 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:

  13. 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 filter • FFT algorithms • …

  14. Positioning Based on Factor Graphs (TDoA)

  15. 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 • Principal axis transformation / rotation (R) by EVD • Shift operation (S) • Mapping operation (M) with 

  16. … … … … … … Positioning Based on Factor Graphs (TDoA) Local functions:

  17. … … … … … … Positioning Based on Factor Graphs (TDoA) with

  18. … … … … … … Positioning Based on Factor Graphs (TDoA)

  19. Simulation Results (TDoA)

  20. Conclusions and Outlook Conclusions: • Positioning in cellular networks using TDoA • LM algorithm with good trade-off between accuracy and complexity • Factor graphs and SPA as tool for several applications • Positioning based on factor graphs for TDoA with promising results Outlook: • Synchronization aspects • Hybrid approaches • Tracking (e.g. by Kalman filters) • Cycles in factor graphs • Scheduling methods • Multipath mitigation, NLoS problem, MAI

  21. German Aerospace Center (DLR) Institute of Communications and Navigation Department of Communications Systems Mobile Radio Transmission Group Possibilities of cooperation: • Joint publications • Meetings, seminars, colloquia, talks • Exchange of researchers • Undergraduate • Diploma/Master theses at DLR • Internships at DLR • Graduate, Postgraduate, … • Visit as guest scientists at DLR • Visit of DLR research staff at NEWCOM partner institutions

  22. Simulation Results

  23. Further Positioning Methods Angle of Arrival (AoA): • Measures the angle of the incident wave • Multiple antennas at BS necessary • Only two BSs for positioning Received Signal Strength (RSS): • Measures the received power from the BS • Range is calculated according to path loss models • Geometric similar to ToA Fingerprinting Solutions: • Position characterized by channel impulse response • Only one BS for positioning (Cell ID)

  24. Gauss-Newton (GN) Method Linearizes the signal model about some initial value and applies linear LS Linearization Step: with the Jacobian matrix and Finally, the iterated solution becomes

  25. Steepest Descent (SD) Method Gradient method, starting from an initial position Gradient vector of search direction is weighted with the optimum stepsize (optimum Line Search) and the iterated solution becomes

  26. Levenberg-Marquardt (LM) Method • Gauss-Newton: • Good estimates for accurate initial values • Divergent for inaccurate initial values and bad geometric conditions • Steepest Descent: • Always finds (sometimes local) minimum • Slow convergence in the final iteration stages • Levenberg-Marquardt: • Combination of GN and SD method • Fast convergence for arbitrary initial values Idea: Damped GN method Parameter depends on quality of the recent estimates

  27. Levenberg-Marquardt (LM) Method

  28. Chan-Ho (CH) Method • Non-iterative method • Extension of the Spherical Intersection (SI) method Idea: • TDoA equations yield linear relation for the three unknowns where independence is assumed 2. The relation is used to improve the estimate • Ambiguity due to squaring operation is resolved • Very good performance (CRLB) for low noise power • Very bad performance for high noise power

  29. Simulation Results (TDoA)

  30. Simulation Results (TDoA)

  31. Simulation Results

  32. Simulation Results

  33. Factor Graphs Tree representation of a factor graph

  34. Sum-Product Algorithm  Calculate

  35. Sum-Product Algorithm  Calculate

  36. Sum-Product Algorithm  Calculate

  37. Sum-Product Algorithm  Calculate

  38. Sum-Product Algorithm Termination step:

  39. Sum-Product Algorithm  Messages can be reused for calculating all

  40. 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

  41. Positioning Based on Factor Graphs (ToA)

  42. Positioning Based on Factor Graphs (ToA) with: Local functions: … … …

  43. Positioning Based on Factor Graphs (ToA) … … … with

  44. Positioning Based on Factor Graphs (ToA) … … … with

  45. Positioning Based on Factor Graphs (ToA) … … …

  46. Positioning Based on Factor Graphs (ToA) … … … with

  47. Positioning Based on Factor Graphs (ToA) … … … Termination step

  48. Simulation Results (ToA)

  49. German Aerospace Center (DLR)The national aerospace research center and space agency • 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 31 Research Institutes and Scientific/Technical Facilities at 8 Sites 4 Branches German-DutchWind Tunnels (DNW) European Transonic Wind Tunnel (ETW) Total Staff:  5000 Institute of Communications and Navigation Activities and Research in Communications • 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) • 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

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