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Track Fitting using Kalman Filter and Smoother. S. Gorbunov DESY Zeuthen I. Kisel KIP , Uni-Heidelberg E. Kryshen SPbSPU, St. Petersburg. KIP. CBM Collaboration Meeting, GSI March 9-12, 2005. Contents. Analytic formula of track extrapolation Kalman Smoother routine.

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Track fitting using kalman filter and smoother

Track Fitting using Kalman Filter and Smoother

S. Gorbunov DESY Zeuthen

I. KiselKIP, Uni-Heidelberg

E. KryshenSPbSPU, St. Petersburg

KIP

CBM Collaboration Meeting, GSI

March 9-12, 2005


Contents
Contents

  • Analytic formula of track extrapolation

  • Kalman Smoother routine

S. Gorbunov, I. Kisel, E. Kryshen


Extrapolation in magnetic field
Extrapolation in Magnetic Field

Differential equation of charged particle motion in a magnetic field

Along Z direction :

S. Gorbunov, I. Kisel, E. Kryshen


Analytic expression
Analytic Expression

Coefficients:

Equation of motion:

Extrapolation error:

S. Gorbunov, I. Kisel, E. Kryshen


Analytic expression features
Analytic Expression Features

  • Features :

  • The precision of extrapolation does not depend on a shape of the magnetic field.

  • One can cut off the higher-order terms in the series.

S. Gorbunov, I. Kisel, E. Kryshen


Analytic 3
Analytic 3

Example: Third order extrapolation in a homogeneous By field

Small parameter:

S. Gorbunov, I. Kisel, E. Kryshen


Coefficients in the analytic formula
Coefficients in the Analytic Formula

Analytic Light

Analytic 1

Analytic 2

Analytic 3

S. Gorbunov, I. Kisel, E. Kryshen


Performance of extrapolation
Performance of Extrapolation

S. Gorbunov, I. Kisel, E. Kryshen


Kalman filter scheme
Kalman Filter Scheme

Final result

rn,Cn

Initialization

r0,C0

Filtering

Kk= CkHTk(Vk+HkCkHTk)-1

rk+1= rk+Kk(mk-Hkrk)

Ck+1= (E-KkHk)Ck

Extrapolation

rk = Ak-1 rk-1

Ck = Ak-1Ck-1ATk-1+Qk

rk,Ck

Measurement

mk, Hk, Vk

Noise

Qk

rk,Ck

S. Gorbunov, I. Kisel, E. Kryshen


Kalman smoother scheme
Kalman Smoother Scheme

Final result

r0n,C0n

Extrapolation

r0k= Ak-1r0k+1

C0k= Ak-1C0k+1A-1Tk

Smoothing

Kk= Qk(Ck+Qk)-1

r0k= r0k+Kk(rk-r0k)

C0k= KkCk+(E-Kk)C0k(E-Kk)T

Noise

Qk

r0k+1,C0k+1

S. Gorbunov, I. Kisel, E. Kryshen


Kalman smoother
Kalman Smoother

  • Optimal track parameter estimation at each measurement

  • Optimal track parameter estimation between measurements

  • Detection and suppression of outliers

S. Gorbunov, I. Kisel, E. Kryshen


Summary
Summary

  • An analytic formula of extrapolation in magnetic field has been derived

  • A simplified formula is used in the KF fitter and CA track finder

  • Kalman smoother is ready for track fitting in presence of noise hits

  • Tracking is well tested and available

S. Gorbunov, I. Kisel, E. Kryshen


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