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Posterior Cramer-Rao Lower Bound for Mobile Tracking
in Mixed Line-of-Sight/Non Line-of-Sight Conditions
Chen Liang1,2, Wu Lenan2, Robert Piché1
1 Tampere University of Technology, Finland 2 Southeast University, China
Eusipco 2009, Aug 25, Glasgow, Scotland
www.math.tut.fi/posgroup
Chen Liang, et al. CR bound for (N)LOS tracking 2
Outline
IntroductionSystem modelPosterior Cramer-Rao Lower BoundNumerical resultsConclusions
Chen Liang, et al. CR bound for (N)LOS tracking 3
Introduction
Non Line-of-Sight (NLOS) condition
Chen Liang, et al. CR bound for (N)LOS tracking 4
Introduction
Non Line-of-Sight (NLOS) condition
State of the art mitigation methods
methods for static position system
－Y. –T. Chan et al (IEEE T. VT 2006)
methods for mobile tracking system
－Two-step Kalman Filter; －Interactive Multiple Model (IMM); －Modified EKF banks + data fusion; －Rao Blackwellized Particle Filtering
(RBPF)
Chen Liang, et al. CR bound for (N)LOS tracking 5
Introduction
Non Line-of-Sight (NLOS) condition
State of the art mitigation methods
The motivation
assess different algorithms
predict the performance
Note:
Compute P-CRLB assuming sight condition sequence is known.
Chen Liang, et al. CR bound for (N)LOS tracking 6
System Model
1. Mobile State Model
Constant velocity model:
1. Mobile State Model2 . Measurement Model 3. Problem Formulation
Constant velocity matrix
Chen Liang, et al. CR bound for (N)LOS tracking 7
System Model 1. Mobile State Model2 . Measurement Model 3. Problem Formulation
2. Measurement Model
True distance from BSi at time k
NLOSNLOSNLOSNLOSNLOSNLOS
LOS
Chen Liang, et al. CR bound for (N)LOS tracking 8
System Model 1. Mobile State Model2 . Measurement Model 3. Problem Formulation
2. Measurement Model
True distance from BSi at time k
NLOSNLOSNLOSNLOSNLOSNLOS
LOS
LOS
NLOS
Chen Liang, et al. CR bound for (N)LOS tracking 9
System Model
3. Problem Formulation
Overall mobility tracking model
1. Mobile State Model2 . Measurement Model 3. Problem Formulation
Markov chain for LOS/NLOS state
Chen Liang, et al. CR bound for (N)LOS tracking 10
System Model
3. Problem Formulation
Overall mobility tracking model
Mobile tracking in mixed LOS/NLOS conditions
Measurement Sequence:
Sight condition sequence:
Mobile state Inference
1. Mobile State Model2 . Measurement Model 3. Problem Formulation
Chen Liang, et al. CR bound for (N)LOS tracking 11
System Model
3. Problem Formulation
Overall mobility tracking model
Mobile tracking in mixed LOS/NLOS conditions
Measurement Sequence:
Sight condition sequence:
Mobile state Inference
1. Mobile State Model2 . Measurement Model 3. Problem Formulation
Optimal Bayesian Solution cannot be analytically computed!
Chen Liang, et al. CR bound for (N)LOS tracking 12
Posterior CRLB (1)
-- an estimate of
The estimate covariance is bounded by the P-CRLB (Van Trees 1968)
where
posterior Fisher information matrix
Chen Liang, et al. CR bound for (N)LOS tracking 13
Posterior CRLB (1)
-- an estimate of
The estimate covariance is bounded by the P-CRLB (Van Trees 1968)
where
Recursive formula (P Tichavsky et al 1998)
where
posterior Fisher information matrix
Chen Liang, et al. CR bound for (N)LOS tracking 14
Posterior CRLB (1)
-- an estimate of
The estimate covariance is bounded by the P-CRLB (Van Trees 1968)
where
Recursive formula (P Tichavsky et al 1998)
where
posterior Fisher information matrix
Linear motion model
Chen Liang, et al. CR bound for (N)LOS tracking 15
Posterior CRLB (1)
-- an estimate of
The estimate covariance is bounded by the P-CRLB (Van Trees 1968)
where
Recursive formula (P Tichavsky et al 1998)
where
posterior Fisher information matrix
Linear motion model
Chen Liang, et al. CR bound for (N)LOS tracking 16
Posterior CRLB (2)
Using linearization approximation
where，
--- the measurement covariance matrix
Chen Liang, et al. CR bound for (N)LOS tracking 17
Posterior CRLB (2)
Using linearization approximation
where，
--- the measurement covariance matrix
Chen Liang, et al. CR bound for (N)LOS tracking 18
Posterior CRLB (2)
Using linearization approximation
where，
--- the measurement covariance matrix
1. Analyze the distribution
2. Sample from the distribution
3. Approximate
Chen Liang, et al. CR bound for (N)LOS tracking 19
Posterior CRLB (3)
1. Use decentralized EKF to approximately compute
where
，
1. Analyze the posterior distribution 2 . Sample from the distribution 3. Approximate
Chen Liang, et al. CR bound for (N)LOS tracking 20
Posterior CRLB (3)
，
2. Deterministically sampling from , using sigma points
1. Analyze the posterior distribution 2 . Sample from the distribution 3. Approximate
Chen Liang, et al. CR bound for (N)LOS tracking 21
Posterior CRLB (3)
，
3.
1. Analyze the posterior distribution 2 . Sample from the distribution 3. Approximate
Chen Liang, et al. CR bound for (N)LOS tracking 22
Posterior CRLB (4)
Recursively compute the Posterior CRLB
，
I. Initialization: set
；
II. Recursive estimation: for1) Predict the mean and covariance of mobile state:2) Update the using decentralized EKF method
3) Deterministically choose a set of sigma points
4) Calculate
5) Update
6) The position Mean Square Error (MSE) bound is:
where are the bounds on the MSE corresponding to
Chen Liang, et al. CR bound for (N)LOS tracking 23
Numerical Results
Simulation parameters
，
-- Actual sight condition. ( for LOS and for NLOS)
-- 3 base stations (M=3)-- compare 3 algorithms with the Posterior CRLBInteractive Multiple Model (IMM);Modified EKF banks + data fusion (Modi-EKF)Improved Rao-Blackwellized Particle Filtering (I-RBPF)
Time(k) 1-200 201-600 601-1600s1,k 1 0 0s2,k 0 0 0s3,k 0 1 0
-- LOS: NLOS:
Chen Liang, et al. CR bound for (N)LOS tracking 24
Numerical Results
，
(MC = 50 realization)
Chen Liang, et al. CR bound for (N)LOS tracking 25
Numerical Results
，
Chen Liang, et al. CR bound for (N)LOS tracking 26
Conclusions & Future work
，
We presented the derivation of a posterior Cramer-Rao lower bound of the mobile tracking problem in mixed LOS/NLOS conditions. (Modified EKF + Sigma point sampling + Unscented Transformation)
Conclusions:
Chen Liang, et al. CR bound for (N)LOS tracking 27
Conclusions & Future work
，
We presented the derivation of a posterior Cramer-Rao lower bound of the mobile tracking problem in mixed LOS/NLOS conditions. (Modified EKF + Sigma point sampling + Unscented Transformation)
In simulations, the error performance of all three algorithms showed agreement with the theoretical bounds.
Conclusions:
Chen Liang, et al. CR bound for (N)LOS tracking 28
Conclusions & Future work
，
We presented the derivation of a posterior Cramer-Rao lower bound of the mobile tracking problem in mixed LOS/NLOS conditions. (Modified EKF + Sigma point sampling + Unscented Transformation)
In simulations, the error performance of all three algorithms showed agreement with the theoretical bounds.
calculate the theoretical bound without assuming sight conditions to be known.
tests using field data.
Conclusions:
Future work: