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Reconstructing Delta wakes from Bunch wakes

This presentation discusses the reconstruction of delta wakes from bunch wakes, including the convolution of bunch shape with the delta-wake function and the deconvolution process. It also explores the incorporation of causal functions and the use of Tikhonov regularization for smoothing. The presentation concludes with the potential applications and future work in this area.

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Reconstructing Delta wakes from Bunch wakes

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  1. Reconstructing Delta wakes from Bunch wakes Roger Barlow COLSIM meeting CERN October 15th 2007

  2. EM Simulation programs ECHO2D Simple Taper m=1 mode m=2 mode

  3. Points • Wakefield(s) follow bunch shape • Wakefield(s) do not (quite) follow bunch shape • Wakefield shape is made by convolution of bunch shape (Gaussian) with delta-wake function: effect of single (leading) particle on another (trailing) particle. • Delta-wake function needed for general study of wake effects, i.e. for bunches of different shapes

  4. Extraction: use Fourier Bunch wake = bunch shape  delta wake Delta wake = bunch wake  bunch shape • Take FT • Divide by FT of appropriate Gaussian • Take Back transform • Filter to reduce high frequencies

  5. Bunch wakes: typical collimator

  6. Reconstructed delta wakes

  7. But… Reconstruction introduces wakefield at t<0 (Appears at large t due to periodicity) Totally unphysical! ‘Causal function’ f(t)=0 for t<0 Can we incorporate this in the deconvolution Yes!

  8. FT of causal functions Write fr=b0+ak sin(kr/N) +  bk cos(kr/N) Must be zero for all r<0 For r>0 ak sin(kr/N) = b0 +  bk cos(kr/N) S a = b0 + C b (Skj=sin(kj /N) etc) a =(2/N) S(b0 + C b) given b0 and b this ensures all fr=0 for all r<0… except f-N That is ensured by b0 = b1 - b2 +b3 … So: work with b . The above equations specify b0 and a

  9. Manipulations Causal function given by f=(SM+A+C)b Aij=(-1)j Mij=(2/N) k SikCik-Sik(-1)j Unfolding: FT of Gaussian coeffts ,  (known) FT of signal a,b (unknown) FT of observations A, B (measured) B0=20b0 Bj=(bjj-ajj) Aj=(ajj+jbj)

  10. Solution Combine convolution equation and causal requirement to get p=Qb Q made from S,C, ,  Use to minimise 2=(dj-pj)2 Justification? Yes Linear Equations: exact solution for b

  11. Smoothing Solution still has spurious high frequency components Add info: we know that the solution does not jump around much Tikhonov regularisation (2nd order seems natural and sensible) d2f/dx2 = fi+1- 2fi + fi-1 2 -1 0 0 0 … T= -1 2 -1 0 0 … 0 -1 2 -1 0 … ( )

  12. Nearly there Add smoothness term for the desired delta wake: Minimise 2 +  bTLTTTTLb (L=SM+A+C) Adjust  to remove spurious high frequencies without blurring out real detail

  13. And it works…

  14. Work in progress • Spike at zero – finally removed • Need to polish technique – make robust • Need more consistency checks • Apply to more cavities and more simulation programs • Produce library of tabulated delta wakes

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