BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20091 Fourier transforms of wall impedances N. Mounet, B. Salvant and E. Métral

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20092 Fourier transforms of wall impedances To get a wake field we need to compute (e.g. in transverse) We can do that using a discrete fast Fourier Transform with evenly spaced z and . Problem: a (resistive) wall impedance can span many decades in frequency  We need a lot of points (under Matlab, limitation to 10 9 points for now).

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20093 Wall impedance Example: transverse wall impedance of an LHC graphite collimator at injection energy → a trade-off between low and high frequencies is not easy to find, and is at the expense of using a huge number of points.

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20094 A “new” Fourier transform technique Ideas (some of them inspired from a similar technique in Numerical Recipes in C, but there only even sampling is considered):  For a relatively smooth function in log scale, an accurate description can be obtained using a piecewise polynomial interpolation on points distributed with irregular spacing, e.g. logarithmic,  On each subinterval the Fourier transform (FT) of the interpolating polynomial can be exactly computed,  We “only” need to sum all the FT on the subintervals, and to add some correcting terms to take into account the “tails” of the function toward infinity.

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20095 Computing a Fourier integral Given a function, we want to compute We first replace this integral by a sum of two terms: (if f is unknown in zero,  min >0 but chosen “small enough”). Introducing the interpolation by a piecewise polynomial on the intervals [  i,  i+1 ], 1≤ i ≤ n-1 p i (  ) being the polynomial approximating on [  i,  i +1 ].

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20096 Computing the Fourier integral of piecewise polynomial cubic interpolation p i (  ) is given by a linear combination of the two polynomials whose arguments is eitheror On each subinterval the Fourier integral is given by a sum of four integrals involving  and , that can be analytically computed (once for all) as functions of time. There was a big issue: under Matlab® numerical errors give chaotic results when computing the analytical expressions found  we solved it by reprogramming those functions, using their series decomposition.

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20097 Computing the “tail” of the Fourier integral This is often crucial to get an accurate final result. Use a first order expansion: Ref.: Press, Teukolsky, Vetterling and Flannery, Numerical Recipes in C, Cambridge University Press, 1999, p.590 The term in was always useless in the FTs we computed.

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20098 Some results Check the algorithm on something whose FT is known:

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/20099 Some results SPS’s beam pipe (1m) transverse impedance at injection energy (  =27.7, pipe radius=2cm, wall 2 mm thick of stainless steel 304L)

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/ Some results Same wake, before the bunch → the long range behaviour before the bunch seems more accurate with this new method

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/ Some results LHC’s collimator (1m) at injection energy (  =479, pipe radius=2mm, infinite wall of graphite) Oscillations due to the relaxation time of graphite But we need to be much more precise when interpolating the high frequency peak (still only about 1300 frequency points used) Before bunch On both sides After bunch

BE/ABP/LIS section meeting - N. Mounet, B. Salvant and E. Métral - CERN/BE-ABP-LIS - 07/09/ Advantages and drawbacks of this new FT technique Advantages  We are (almost) no longer limited in the frequency window we choose → it’s an FT on a really infinite domain,  We can compute the Fourier transform at any time t, i.e. we can use logarithmic spacing or even irregular spacing in t. In particular, accuracy is better before the bunch, compared to classic DFT.  Provided the function is smooth enough, we may compute FT with very few points even in a very large frequency domain (we typically used several hundreds sampling frequencies). Drawback  The method is inefficient if the function has a lot of peaks, such that we need many points for the interpolation (with a lot of points the algorithm is not as fast as FFT, obviously). → problem e.g. for the wall impedance of a dielectric CLIC collimator