Model Independent Analysis of HERA Proton Tune Spectra

1. Model Independent Analysis of HERA Proton Tune Spectra (work by Steve Herb and T.L.) based on image processing using techniques wavelet Thomas Lohse Humboldt-Universität zu Berlin

2. Tune Contol Multiknobs • Q_x • Q_z Quadrupole correction coils on superconducting magnets in arcs • S_x • S_z Sextupole correction coils on superconducting magnets in arcs • Qc_1 • Qc_2 Warm skew quadrupole correctors in straight section (west)

3. ProtonBeam Pickup MHz Kicker 2.1 MHz Band Pass kHz Alias Low Pass kHz 208 MHz (RF) 8.33 MHz FFT Audio Chirp 10-20 kHz 208 MHz (RF) 34 ms S. Herb, Kerntechnik 56 (1991) 229

4. Variable tune signal Variable tune signal Variable tune signal Noise Variable coupling tune signal Variable coupling tune signal Variable coupling tune signal Variable background shape Variable background shape Variable background shape Variable background shape Artefacts Artefacts Tune FFT Spectra featuring...

5. Tune Position Problem:No general model is known for fitting the tune spectra Information to be extracted ... Relative FFT Amplitude, Cross Plane Coupling Tune Spread, Tune Height, Chromaticity Tune

6. This is what WAVELETS are made for!! The mathematical tasks • Remove large scale smooth background • Filter out smallest scale noise • Identify sharp structures: • measure the scale • measure the position simultaneously !!

7. support width scaling function smooth: wavelet detailed: What are Wavelets?

8. How does a basis look like? Smooth Scale 1 Detail Scale 1/2 Detail Scale 1/4

9. Example: Step Function (Haar Wavelet)

10. Fourier transform: Windowed Fourier transform: zoom in, resolution up Wavelet transform: Fourier transform vs. Wavelet analysis

11. Wavelets in Science and Nature • Human ear (hardware) => wavelet trafo • Subband filtering in electrical engineering (QMF) • Image processing and data compression • Analysis of seismic data • Quality control of synthetic tissues • Inversion of large matrices • Solution of integral- and differential-equations • Coherent states in quantum mechanics • Decomposition of operators, quantum field theory • .........

12. Daubechies wavelets (orthonormal bases of ) • Compact support • Maximum number of vanishing moments, high regularity • Identical forward/backward filter coefficients • Haar wavelet ( N=1 ) • support width = 1 • integral = 0 • Daub4 wavelet ( N=2 ) • support width = 3 • integral = mean = 0

13. Properties of DaubechieswaveletsI. Daubechies, Comm. Pure Appl. Math.41 (1988) 909. • Compact support • finite number of filter parameters / fast implementations • Maximum number of vanishing moments • high compressibility • fine scale amplitudes are very small in regions where the function is smooth / sensitive recognition of structures • good, but not quite optimal regularity • Identical forward / backward filter parameters • fast, exact reconstruction • very asymmetric

14. Alternative wavelets with compact support • Optimal regularity (I.Daubechies, SIAM J. Math. Anal.24 (1993) 499.) • larger support width or smaller number of vanishing moments • Smaller perceptible effects from image compression • Symmetry • there is only one symmetric orthonormal basis: the Haar wavelet • biorthogonal wavelet bases: (A. Cohen et al., Comm. Pure Appl. Math.45 (1992) 485.) different forward / backward filters with different support widths, less regularity per support width • image compression less perceptible by human visual system • ... and many other intermediate wavelets Plus many wavelets with infinite support: Meyer, Battle-Lemarié, ...

15. Daub4 wavelet decomposition of FFT spectra Raw spectra smooth part of background Signal plus some small scale structures in background Mostly noise

16. Partial Daub4 wavelet reconstruction Smoother: retain largest x% wavelets distorted acceptable smoothness Filter: take out smallest scale wavelets too noisy

17. raw spectrum processed spectrum => candidates for maxima biased towards small scales good compromise: sharp minima => peak boundaries large scale structures wash out peak boundaries => useful for very wide peaks ? Tune peaks in detail wavelets

18. Extraction procedure for tune parameters • processed spectra => max. candidates • detailed spectra => find side dips • subtract background • refine maximum => local parabola fit • interpolate side bins => fwhm

19. Systematic performance studies • gaussian background shape • gaussian direct/coupled tune signals • gaussian noise at fixed percentage of (bkg+sig) Tune reconstruction

20. Systematic performance studies Height reconstruction for tune signals

21. Systematic performance studies Width reconstruction for tune signals

22. Systematic performance studies Dependence on noise level

23. Systematic performance studies Dependence on tune separation

24. dynamic range Systematic performance studies Reconstruction of line width saturation at width of scale 8 wavelets

25. Tune tracking April 20, 17:23 April 21, 20:08 coupled main

26. Examples for proplematic cases (1)

27. Examples for proplematic cases (2)

28. Tracking of FFT power: hor => vert vert => hor April 20, 17:23 April 21, 20:08

29. The coupling ratio April 20, 17:23 April 15, 20:08

30. April 20, 17:23 April 15, 20:08 Tracking of tune spread

31. The next steps... • Online implementation (S. Herb) • Gain operational experience • Provide useful online plots for manual control • Study of tune correction knobs • Work towards a feedback system (?)