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Track Fitting using Kalman Filter and Smoother

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

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  1. 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

  2. Contents • Analytic formula of track extrapolation • Kalman Smoother routine S. Gorbunov, I. Kisel, E. Kryshen

  3. Extrapolation in Magnetic Field Differential equation of charged particle motion in a magnetic field Along Z direction : S. Gorbunov, I. Kisel, E. Kryshen

  4. Analytic Expression Coefficients: Equation of motion: Extrapolation error: S. Gorbunov, I. Kisel, E. Kryshen

  5. 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

  6. Analytic 3 Example: Third order extrapolation in a homogeneous By field Small parameter: S. Gorbunov, I. Kisel, E. Kryshen

  7. Coefficients in the Analytic Formula Analytic Light Analytic 1 Analytic 2 Analytic 3 S. Gorbunov, I. Kisel, E. Kryshen

  8. Performance of Extrapolation S. Gorbunov, I. Kisel, E. Kryshen

  9. 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

  10. 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

  11. 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

  12. 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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