Tracking of Multiple Wideband Targets using Passive Sensor Arrays and Particle Filters

1. Tracking of Multiple Wideband Targets using Passive Sensor Arrays and Particle Filters Volkan Cevher cevher@ieee.org

2. Georgia Institute of Technology Center for Signal and Image Processing 2 Outline Simulations Problem Definition Data Model Probability Density Construction and Optimal Importance Function Conclusions

3. Georgia Institute of Technology Center for Signal and Image Processing 3 Simulations

4. Georgia Institute of Technology Center for Signal and Image Processing 4 Simulations

5. Georgia Institute of Technology Center for Signal and Image Processing 5 Simulations

6. Georgia Institute of Technology Center for Signal and Image Processing 6 Simulations

7. Georgia Institute of Technology Center for Signal and Image Processing 7 Simulations

8. Georgia Institute of Technology Center for Signal and Image Processing 8 Simulations

9. Georgia Institute of Technology Center for Signal and Image Processing 9 Simulations

10. Georgia Institute of Technology Center for Signal and Image Processing 10 Simulations

11. Georgia Institute of Technology Center for Signal and Image Processing 11 Problem Definition Consider K far-field targets coplanar with a sensor node consisting of P acoustic sensors. The derivations do not assume any special structure on the array node. The results are demonstrated with a 15-element circular array. Use the ambient radiation emanating from the targets (i.e. acoustic signals). Track the direction-of-arrivals of multiple targets Achieve higher resolution than snapshot techniques (i.e. MUSIC)

12. Georgia Institute of Technology Center for Signal and Image Processing 12 Data Model State/Observation model type of formulation is used State model: State vector: Consists of partitions for each target (taste the rainbow)

13. Georgia Institute of Technology Center for Signal and Image Processing 13 Data Model State Model (Cont.)

14. Georgia Institute of Technology Center for Signal and Image Processing 14 Data Model State Model (Cont.): The state update equations can be derived by relating the DOA's of the target at times t and t+T using the geometrical relation of position 1 to position 2 Other assumptions: Constant velocity Constant heading direction Known frequency rate of change for each available frequency- a(t)

15. Georgia Institute of Technology Center for Signal and Image Processing 15 Data Model State Model (Cont.): State update equation: where State model is updated every T seconds.

16. Georgia Institute of Technology Center for Signal and Image Processing 16 Data Model Observation Model: The acoustic array outputs are used: Observations are updated every ?= T/M seconds. (M is defined as the batch size.) The steering matrix A is

17. Georgia Institute of Technology Center for Signal and Image Processing 17 Data Model Observation Model (Cont.): A steering vector associated with the array defines the complex array response for a source at DOA ?, and has the following form

18. Georgia Institute of Technology Center for Signal and Image Processing 18 Data Model Observation Model (Cont.): For notational convenience and tractability, the data collected at each time is stacked to form the following data vector Yt: The signal vector St and the noise vector Wt are formed in the same manner.

19. Georgia Institute of Technology Center for Signal and Image Processing 19 Data Model Observation Model (Cont.): The array data (or observation) model for the batch period can be compactly written as the following: where implicitly incorporates the DOA information of the targets

20. Georgia Institute of Technology Center for Signal and Image Processing 20 Probability Density Construction The particle filter is a convenient way of recursively updating a target posterior of interest While formulating these update equations in our problem, one encounters two nuisance parameters: the signal vector St the noise variance for the additive Gaussian noise vector Wt

21. Georgia Institute of Technology Center for Signal and Image Processing 21 Probability Density Construction

22. Georgia Institute of Technology Center for Signal and Image Processing 22 Conclusions The new filter is implemented and tested for synthetic data Need real data to work on Need a time-frequency filter to jointly estimate ak(t) The algorithm is flexible The state vector can be extended to other parameters of interest New pdf�s can be derived for different set of signals of interest (i.e. signals having white noise characteristics)

23. Georgia Institute of Technology Center for Signal and Image Processing 23 Probability Density Construction The data likelihood given the signal and noise vectors can be written as follows: If the priors are known for the signals and noise variance given the state vector at time t, they can be integrated out from the Gaussian pdf

24. Georgia Institute of Technology Center for Signal and Image Processing 24 Probability Density Construction If one desires to assume the least about these parameters and let the observed data speak for itself, then the use of reference priors come into play The reference priors can be derived using an estimation model based on communication channel with a source and data. The reference prior maximizes the mutual information between the source and the data.

25. Georgia Institute of Technology Center for Signal and Image Processing 25 Probability Density Construction Two nuisance parameters leads to two different reference priors Unknown Signal Vector and Known Noise Variance: Unknown Signal Vector and Noise Variance:

26. Georgia Institute of Technology Center for Signal and Image Processing 26 Probability Density Construction Concentrate on the former case (Unknown Signal Vector and Known Noise Variance) Also need a pdf for state transition Define

27. Georgia Institute of Technology Center for Signal and Image Processing 27 Probability Density Construction PDF for state transition: where

28. Georgia Institute of Technology Center for Signal and Image Processing 28 Optimal Importance Function An appropriate choice of the importance function may reduce the variance of the simulation errors For our choice of the importance function, the objective is to minimize the variance of the importance weights Local linearization of the importance function is used to draw samples Technical details were given in the previous presentation

29. Georgia Institute of Technology Center for Signal and Image Processing 29 Optimal Importance Function

30. Georgia Institute of Technology Center for Signal and Image Processing 30 Optimal Importance Function

31. Georgia Institute of Technology Center for Signal and Image Processing 31 Optimal Importance Function The gradient and Hessians in the previous slide need to be evaluated They are analytically evaluated. Approximations are made to guarantee positive definiteness of the Hessian matrices Hessians approximate covariance matrices and hence need to be positive definite. Crucial assumption: Importance function can be factored out for each partition

32. Georgia Institute of Technology Center for Signal and Image Processing 32 The Algorithm The algorithm in M. Orton and W. Fitzgerald, �A Bayesian approach to tracking multiple targets using sensor arrays and particle filters,� IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 50, no. 2, February 2002, pp.216-223 is used with minor modifications.