Summary of lecture 8. FIR Wiener filter: computed by solving a finite number of Wiener-Hopf equations, h(i)r yy (k i) = R sy (k); k = 0; : : : ; m
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1 Summar of lecture 8 FIR Wiener filter: computed b solving a finite number of Wiener-Hopf equations, mx i= h(i)r (k i) = R s (k); k = ; : : : ; m Whitening filter: A filter that removes the correlation from a signal, i.e. it has a white noise output. Causal Wiener filters: Computed b truncating the non-causal Wiener filter after first whitening the signal, H c (z) = z m T (z) z m Φ s (z) 2 e T (1=z) where we have m = for filtering, m > for prediction and m < for smoothing. + F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
2 Outline Lecture 9 Kalman filter derivation 1 Optimal estimation 2 Signal model 3 Markov propert 4 Time update 5 Measurement update 6 Eamples F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
3 Kalman filter: signal model Signal model: (t + 1) =A(t) + w(t) (t) =C(t) + v(t) (t) state vector, w(t) process noise, v(t) measurement noise Both w(t) and v(t) are (possibl vector valued) white noise, with E [w(t)] = E [v(t)] = E h w(t)v(t) Ti = S E E h w(t)w(t) Ti = Cov(w(t)) = Q h v(t)v(t) Ti = Cov(v(t)) = R The initial state () is assumed to be a stochastic variable with E [()] = ; Cov(()) = E h (() ) (() ) Ti = P Tpicall, all stochastic variables are assumed to be Gaussian distributed. F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
4 The optimal estimation problems Optimal estimation: Compute an estimate ˆ(tjs) of the state (t) using the information in the measurements (); (1); : : : ; (s) such that the covariance of the estimation error (t) ˆ(tjs) is minimized. There are three variants of the problem: Filtering: t = s We will look at filtering toda Prediction: t > s Smoothing: t < s F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
5 Markov propert of the state space model The state sequence (); (1); : : : ; (t) is a Markov chain with Markov propert. This means that given the current state (t), the future state (t + 1) is independent of the past states (); (1); : : : ; (t 1). In terms of probabilit densities, the following holds Pr((t + 1)j(); (1); : : : ; (t)) = Pr((t + 1)j(t)) In the contet of Kalman filtering, this means that if we have a state estimate, i.e. either ˆ(t + 1jt) or ˆ(tjt), there is no further information available in the previous measurements (); (1); : : : ; (t) or the previous state estimates. F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
6 Notation Previous measurements up to and including time s, Y s = f(); (1); (2); : : : ; (s)g State estimate at time t given Y s, ˆ(tjs) = E [(t)jy s ] Estimation error at time t given Y s, (tjs) = (t) ˆ(tjs) Covariance matri for the estimation error, h P(tjs) = E (tjs) T (tjs) Innovation at time t given Y t 1, e(tjt 1) = (t) ŷ(tjt 1) = (t) C ˆ(tjt 1) i F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
7 The Kalman filter Input: Models A, C, Q, R, measurements (); : : : ; (N), and initial state estimate and covariance P. Recursion: For each time step t, iterate Prediction or time update: ˆ(t + 1jt) =Aˆ(tjt) P(t + 1jt) =AP(tjt)A T + Q Use model to propagate Increase uncertaint Correction or measurement update: e(tjt 1) =(t) C ˆ(tjt 1) Innovation S(t) =CP(tjt 1)C T + R Innovation covariance K(t) =P(tjt 1)C T S 1 (t) Kalman gain ˆ(tjt) =ˆ(tjt 1) + K(t)e(tjt 1) Correct estimate P(tjt) =P(tjt 1) K(t)CP(tjt 1) Decrease uncertaint F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
8 Target tracking eample (t) = A = Q = C = h (t) (t) (t) (t) 1 T 1 T i T T 4 =4 T 3 =2 T 4 =4 T 3 =2 T 3 =2 T 2 T 3 =2 T " # " 2 # 1 4 ; R = (t) P Initial estimate 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
9 4 Measurement update: (t t) =( ) (t) () P 1 ˆ 1 P ˆ () 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
10 4 (t) P ˆ P 1 ˆ 1 Time update: (t t 1) =(1 ) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
11 4 Measurement update: (t t) =(1 1) (t) (1) P 1 ˆ 1 P 1 1 ˆ 1 1 (1) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
12 4 (t) P 1 1 ˆ 1 1 P 2 1 ˆ 2 1 Time update: (t t 1) =(2 1) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
13 4 Measurement update: (t t) =(2 2) (t) (2) P 2 1 ˆ 2 1 P 2 2 ˆ 2 2 (2) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
14 4 (t) P 2 2 ˆ 2 2 P 3 2 ˆ 3 2 Time update: (t t 1) =(3 2) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
15 4 Measurement update: (t t) =(3 3) (t) (3) P 3 2 ˆ 3 2 P 3 3 ˆ 3 3 (3) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
16 4 (t) P 3 3 ˆ 3 3 P 4 3 ˆ 4 3 Time update: (t t 1) =(4 3) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
17 4 Measurement update: (t t) =(4 4) (t) (4) P 4 3 ˆ 4 3 P 4 4 ˆ 4 4 (4) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
18 4 (t) P 4 4 ˆ 4 4 P 5 4 ˆ 5 4 Time update: (t t 1) =(5 4) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
19 4 Measurement update: (t t) =(5 5) (t) (5) P 5 4 ˆ 5 4 P 5 5 ˆ 5 5 (5) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
20 4 (t) P 5 5 ˆ 5 5 P 6 5 ˆ 6 5 Time update: (t t 1) =(6 5) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
21 4 Measurement update: (t t) =(6 6) (t) (6) P 6 5 ˆ 6 5 P 6 6 ˆ 6 6 (6) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
22 4 (t) P 6 6 ˆ 6 6 P 7 6 ˆ 7 6 Time update: (t t 1) =(7 6) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
23 4 Measurement update: (t t) =(7 7) (t) (7) P 7 6 ˆ 7 6 P 7 7 ˆ 7 7 (7) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
24 4 (t) P 7 7 ˆ 7 7 P 8 7 ˆ 8 7 Time update: (t t 1) =(8 7) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
25 4 Measurement update: (t t) =(8 8) (t) (8) P 8 7 ˆ 8 7 P 8 8 ˆ 8 8 (8) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
26 4 (t) P 8 8 ˆ 8 8 P 9 8 ˆ 9 8 Time update: (t t 1) =(9 8) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
27 4 Measurement update: (t t) =(9 9) (t) (9) P 9 8 ˆ 9 8 P 9 9 ˆ 9 9 (9) 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
28 (t) ˆ(t t) P(t t) Result 4 4 F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
29 Comparison to Wiener filter I Problem: track position of moving target, e.g. a pedestrian. Model of problem: Position changes like a random walk p d (t + 1) = p d (t) + Tw d (t); and is measured in white noise d (t + 1) = p d (t) + v d (t): p(t) 4 Var(w d (t)) = Q, Var(v d (t)) = R, and d = f; g. 4 p(t) h T State space model: (t) = p (t) p (t)i ; A = " 1 1 # ; Q = " T 2 Q T 2 Q # ; C = " 1 1 # ; R = " # R R F. Gustafsson (LiU) Digital Signal Processing, Lecture 9 17 / 14
30 Comparison to Wiener filter II Top row: n-c Wiener filter. Bottom row: Kalman filter. Same Q= R p(t) p(t) p(t) 4 p(t) 4 p(t) 4 p(t) p p p 4 p 4 p 4 p Smoothing vs. filtering. Not shown here but equall important: P(tjt). F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
31 Peer review You will peer review each others reports, i.e. read and give constructive criticism. Wh? Good for learning, both to review and be reviewed. Important not just to be able to do DSP, but also to be able to communicate what ou have done. Practice for Master s Thesis writing. How? See instructions in lab-pm. Submit our report via the web interface. See course homepage, Don t hesitate to ask if ou have an questions regarding the lab, the report, the review process, or anthing else! F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
32 Summar of Lecture 9 Dnamic equation: Describes how the state evolves over time. Measurement equation: Describes how the measurements are related to the state vector. A model of the sensor. Innovation: The new information in the measurement, i.e. information that is not available in the predicted estimate. Time update: The part of the Kalman filter that uses the dnamic equation to predict the state estimate and covariance. Measurement update: The part of the Kalman filter that uses the innovation and the measurement equation to update the state estimate and covariance. F. Gustafsson (LiU) Digital Signal Processing, Lecture / 14
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