The Wiener Filter

OutlineLecture7 IntroductiontotheWienerfilter 1 Thesignalinnoiseproblem 2 TheWienerfilter 1 Derivationinfrequencydomain 2 Derivationintimedomain 3 Wiene...

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Summary of lecture 6



Model estimation problem: For a given model structure, solve the optimization problem ˆN = arg min 

  

X N

(y (t)

yˆ(t jt

1; ))2

t=1

Linear model estimation: For example AR. The optimization problem’s solution is given by the normal equations. ar Non-linear model estimation: For example ARMA and state space. Non-linear optimization requires numerical algorithms. armax, pem ˆ N ) ! (0 ; 0 ) as N ! 1, or Statistical properties: (ˆN ;  ˆN



1 N

! arg min E 

h

(y (t)

yˆ(t jt

1; ))2

i

as N

!1

Data pre-processing and model validation F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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Outline Lecture 7

Introduction to the Wiener filter 1 2

The signal in noise problem The Wiener filter 1 2 3 4 5

Derivation in frequency domain Derivation in time domain Wiener-Hopf equations Non-causal Wiener filter Examples

F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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Wiener filter: Background



Named after American Norbert Wiener, prof of Mathematics at MIT.

Uses include

  

Target tracking Image noise suppression Denoising audio in speech recognition F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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Example: target tracking Problem: track position of moving target, e.g. a pedestrian. 60

Model of problem: Position changes like a random walk

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pd (t + 1) = pd (t) + Twd (t); py(t)

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and is measured in white noise

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yd (t) = pd (t) + vd (t): Var(wd (t)) = Q, Var(vd (t)) = R, and d = fx; yg.

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0

−10 −10

0

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px(t)

F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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Example: target tracking In filter form the model is (for d = fx; yg) 0

q

1

wd (t) + vd (t) |Hnc (ω)|

yd (t) =

10

T

The signal and noise spectra are Φpp (z) =

2

QT 2 z z

1

;

−1

10

Φvv (z) = R

Q/R = 4 Q/R = 0.2 Q/R = 0.01 −2

10

−3

10

The non-causal Wiener filter is Φpp (z) = Hnc (z) = Φpp (z) + R

F. Gustafsson (LiU)

R

−1

0

10

10

ω [rad/s]

QT 2 R QT 2

−2

10

+2

z

z

1

The filter depends on 2 quantity QT R , related to the SNR.

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Example: target tracking Results: Estimates pˆx (t) = Hnc (q)yx (t) and pˆy (t) = Hnc (q)yy (t) 50

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py(t)

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py(t)

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py(t)

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0

0

0

−10 −10

0

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−10 −10

px(t)

0

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−10 −10

0

px(t)

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px(t)

We can observe the following:

 

Q =R too large: the estimated trajectory is too noisy. Q =R too small: too much smoothing of the estimated trajectory.

For this problem the non-causal Wiener filter has a low-pass filtering effect on the measurements. F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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Summary of Lecture 7

  

Optimal linear filter: The filter that minimizes the expected value of the estimation error. Wiener-Hopf equations: Results from finding the filter that minimizes the expected value of the estimation error. Wiener filter: The solution to the Wiener-Hopf equations, a linear filter that removes noise from a signal. Comes in different variants, one is the non-causal Wiener filter.

F. Gustafsson (LiU)

Digital Signal Processing, Lecture 7

2017

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