Smooth profile vs binned numerical profile -> numerical noise added

For impedance calculation, this noise is injected into the model

• At what frequencies this noise becomes important? Can lead to
  • Extra blow-up for microwave instabilities
  • Earlier onset of instabilities
• What is the Signal To Noise (STN) ratio for a stationary distribution?
  • Start from binomial profile
  • Spectrum follows Bessel functions -> make an asymptotic approximation (maximum frequency, without dips)
• Adding artificially white weighted noise
  • Make a histogram of the noise amplitude spectrum
  • Figure-of-merit: mean of the noise amplitude (if we want a single number)
  • Compare smooth spectrum with noisy spectrum -> get SNR ~ sqrt(Np)/bunch_length^(mu' + 1)
• SNR interpretation
  • Doesn't depend on bin size -> for a single impedance at a given frequency, noise is not increased or decreased
    • For high-frequency impedance, need to increase the bin size!
    • Space charge or diverging impedance sources: too small binning adds large noise!
  • For a parabolic bunch profile, doubling the bunch length requires x16 more macro-particles!
    • How can we model the tails better?
    • Core + tail distributions? 
    • Constant bin density like in pyheadtail?
  • Criterion for Np using SNR > 1
    • SPS injection (Run II): min. 1 M/bunch
    • SPS extraction (Run II): min 250 k/bunch
    • SPS long bunches: min 39 B/bunch
    • PS transition crossing, Gaussian: tails in measurement not real -> replace with Gaussian
    • PS cycle: 126 k/bunch -> 420 M/bunch spanned