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
