Many beam-dynamical phenomena are studied, experimentally or computationally, by means of spectral analysis of a time-series of values of a dynamical variable. When the underlying dynamics is regular, the frequencies appearing in the spectrum are integer combinations of a small set of basic frequencies, e.g., the three tunes in the case of single-particle orbital dynamics. For well-known reasons, identification of the frequencies can be ambiguous or subjective in practice. We present an algorithm that overcomes these difficulties by exploiting theoretical bounds on the spectral power density to transform time series into sets of labelled resonance lines. In our examples, the time series are orbits obtained by tracking single particles from many initial conditions. The method has been applied to off-momentum LHC injection optics. This is a deterministic Hamiltonian system. A second application, to orbits with strong quantum fluctuations in LEP2, shows that it also works well in a noisy, dissipative system

Burgess, R T

Jowett, John M

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EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCHCERN - SL DIVISIONMany beam-dynamical phenomena are studied, experimentally or computationally, by means of spectralanalysis of a time-series of values of a dynamical variable.  When the underlying dynamics is regular, thefrequencies appearing in the spectrum are integer combinations of a small set of basic frequencies, e.g.,the three tunes in the case of single-particle orbital dynamics. For well-known reasons, identification ofthe frequencies can be ambiguous or subjective in practice.  We present an algorithm that overcomesthese difficulties by exploiting theoretical bounds on the spectral power density to transform time seriesinto sets of labelled resonance lines.  In our examples, the time series are orbits obtained by trackingsingle particles from many initial conditions. The method has been applied to off-momentum LHCinjection optics. This is a deterministic Hamiltonian system. A second application, to orbits with strongquantum fluctuations in LEP2, shows that it also works well in a noisy, dissipative system.CERN-SL-2000-025 paper presented  at  EPAC2000, 26-30 June 2000, Vienna, AustriaGeneva, Switzerland 12.07.00SL/Div RepsPhase-space analysis by multiple resonance-frequency identification: Applicationsfor the LHC and LEPAPR.T. Burgess, J.M. JowettPhase-space Analysis by Multiple Resonance-Frequency Identification:Applications to the LHC and LEPR.T. Burgess* J.M. Jowett CERN, Geneva, SwitzerlandAbstractMany beam-dynamical phenomena are studied,experimentally or computationally, by means of spectralanalysis of a time-series of values of a dynamical variable.When the underlying dynamics is regular, the frequenciesappearing in the spectrum are integer combinations of asmall set of basic frequencies, e.g., the three tunes in thecase of single-particle orbital dynamics. For well-knownreasons, identification of the frequencies can beambiguous or subjective in practice.We present an algorithm that overcomes thesedifficulties by exploiting theoretical bounds on thespectral power density to transform time series into sets oflabelled resonance lines.  In our examples, the time seriesare orbits obtained by tracking single particles from manyinitial conditions.The method has been applied to off-momentum LHCinjection optics. This is a deterministic Hamiltoniansystem. A second application, to orbits with strongquantum fluctuations in LEP2, shows that it also workswell in a noisy, dissipative system. *1  INTRODUCTIONAnyone who has looked at the tune spectrum of a storedbeam will appreciate that unambiguous identification ofthe lines that appear is not always easy and may be partlysubjective. In this paper, we develop an algorithm to solvethis problem (in certain conditions) and apply it to particletracking data.Our multiple resonance frequency identification(MRFI) algorithm exploits bounds on the spectral powerdensity given by KAM theory [1,2] in an heuristicmanner, to return quantitative evaluations of the integercombinations, so identifying the spectral lines.  Apreliminary version of the algorithm was used in [3];related theoretical background and methods are discussedin [4].2  PRINCIPLES OF MRFIThe discrete Fourier transform maps a time-series of Nvalues of a dynamical variable onto a spectrum over a setof frequencies ( )2121 ,,,,0 NN .  In the following, wemainly consider the corresponding power spectrum.The MRFI algorithm has two parts: first the spectrumof a dynamical variable is transformed into a set of peaks(frequencies, amplitudes and possibly widths); secondly,each peak is assigned to one of the resonance frequencies.                                                          * Technical student from the University of Bath, UK.The appropriate peak-identification method depends onthe nature of the dynamical system.  Two illustrative casesare treated below.The second part of the algorithm is motivated by resultsfrom KAM theory. When the underlying dynamics of asystem is quasi-periodic, it is well known that thefrequencies appearing in the spectrum are integercombinations of a small set of basic frequencies. Forparticle motion in an accelerator (3 degrees of freedom),the basic frequencies (in “tune” units) are ( )321 ,, qqq=q ,and peaks appear at the integer combinations qk. ,( ) 3321 ,, Zk ∈= kkk . KAM theory [1,2] suggests thedefinition of a function:( ) ( )}{1,}{min,, 4 qkqkkqk ⋅+−⋅−= fffc (1)Here, the resonance order 321 kkk ++=k  and }{xdenotes the fractional part of x. Moreover, the amplitudesof the corresponding spectral components are bounded bykλ−Me  for some constants 0, >λM .Knowing the tune q, the function c is a measure of thedistance from the frequency of some peak, f, to theresonance line labelled by k. This distance is minimisedover some ( ) { }rr ≤∈⊆ kZkK :3 .  (The inclusion, ⊆ ,allows for the possible application of a selection rulewithin the octahedral set of all resonances up to the orderr.)  The minimum gives the resonance *k most likely tohave generated the peak:( ) ( ) ( )qkqqkKk,,min,,,)(* fcfCfcr∈== (2)For illustration, Figure 1 plots ( )q,fC , for( )12.0,2.0,35.0=q  and ( ) { }2:2 3 ≤∈= kZkK , i.e., allresonances up to the second order.The first order resonances are visible as minima in( )q,fC  at 0.35 0.2, ,12.0=f .  The comparativelyweaker influence of the second order resonances isindicated by the nine much narrower troughs at thepositions of each resonance.   3 LHC AT INJECTIONThe LHC injection optics as a function of the centralbeam momentum is well corrected for shifts in centralmomentum of 002.0± [7]. Hence a scan of phase spacepresents few peaks within the Fourier spectra of particles.To produce a more complex phase space and show theneed for the b5 correctors, we consider the example of theLHC optics V6.0 with b5 correctors switched off. In oursl-2000-02525/06/2000example, particles were tracked in 4D using MAD8 [5]for 1000=N  turns with a shift in central beammomentum of +0.002. We illustrate the application ofMRFI to the spectrum of the normalised co-ordinate  xn.Spectral peaks are first identified to accuracy 1/N  inthe Fourier spectra.  Since the system is deterministic,their frequencies can be resolved more accurately with thehelp of the NAFF method [6]; for quasi-periodic data theintrinsic error is 4−∝ N  and the MRFI method then worksbetter.Initially, single particle spectra were investigated. Atsmall-to-moderate amplitudes, the two tunes were taken tobe near the frequencies of highest peak occurred in theFourier spectra of the xn, yn co-ordinates; see Figure 2.The frequency and amplitude of peaks found by thealgorithm match the original spectrum well. In a well-corrected machine spectra would exhibit only one mainpeak, with possibly another of very small amplitude.To study the appearance and disappearance ofparticular resonances along a line in phase space, a set ofparticles with initial action m 1082.7 10−×=xI , andincreasing Iy, were tracked through the above-mentionedoptics. The peaks and associated k for each particle werecalculated as before.At large initial Iy other peaks can be greater inmagnitude than the “tune” peak. To ensure that the correctfrequencies were attributed to the tunes, the values foundby the algorithm were compared with those evaluated forthe previous particle and the identity of the principal tunepeak retained by continuity as far as reasonably possible.Figure 3 tracks all resonances up to the 6th order as theinitial action Iy co-ordinate is increased; the comparisonwith the background spectral density shows that MRFIprovides a good quantitative representation of the mainfeatures.Particular resonances may also be tracked through aplane in phase space. As before, it is necessary to trackthe tunes from the smallest amplitude outwards whilstperforming MRFI on all particles that survived 1000 turnswithin that plane of phase space. A set of MRFI plots, each depicting the amplitude of adifferent k found, at all points in phase space, produce acomplete picture of the resonances affecting any region ofphase space tracked.A combination of 3 MRFI plots, for k =(0,3,0), (2,1,0)and (5,0,0) in the Ix-Iy plane is shown in Figure 4. Eachcolour denotes one k. The shading of the plot representsthe amplitude of the resonant peak at that point in phasespace.  In this case, the fainter the shade on the plot thesmaller the amplitude of the resonance at that pointrelative to the highest amplitude measured for thatparticular resonance.0.1 0.2 0.3 0.4 0.5f0.020.040.060.080.10.120.14CHfLFigure 1: The distance C to 1st and 2nd order resonances. 0.1 0.2 0.3 0.4 0.5f-35-30-25-20-15 81, 0, 0<80, 1, 0<80, 2, 0<8-1, 1, 0< 80, 4, 0<80, 3, 0<81, 1, 0<8-1, 2, 0<81, 2, 0<82, -1, 0<Figure 2 (colour): The original spectrum (black), theresulting peaks and associated k, coloured by theirresonance order for a single particle with initial actionsm 1082.7 10−×=xI , m 1027.17−×=yI .0 0.1 0.2 0.3 0.4 0.5f01 10-72 10-73 10-74 10-75 10-76 10-77 10-7I y 8-2, 4, 0<8-1, 1, 0<8-1, 2, 0<8-1, 3, 0<8-1, 5, 0<80, 0, 0< 80, 1, 0<8 , 2, 0<80, 3, 0<80, 4, 0<80, 5, 0<81, 0, 0<81, 1, 0<81, 2, 0<81, 3, 0<82, -1, 0<82, 0, 0<82, 1, 0<Figure 3  (colour): Tracking various resonances on thebackground power spectra (lighter corresponds to higherspectral density) as the action Iy, is varied. The resonancesare overlaid as labelled lines, coloured to indicate theresonance order.0 110-7210-7310-7410-7510-7Ixm0110-7210-7310-7410-7510-7610-7IymThus, MRFI provides an additional tool to map out thefeatures of the phase space of a Hamiltonian system.4 LEP2Single-particle motion in LEP2 is dominated bysynchrotron radiation whose quantum fluctuationsintroduce a strong stochastic element [8,3].  The NAFFalgorithm is not applicable to this noisy and dissipativesystem.  Nevertheless the orbit may contain informationabout resonances present in the underlying Hamiltoniansystem (that includes the non-dissipative part of theradiation defining the closed orbit).  To apply MRFI weneed a different peak-finding algorithm.Compared to the smooth spectra exhibited by the LHCmodel, the power spectra of a particle’s displacementfrom the closed orbit contains much more noise. Theapplication of a “moving average” such that every point isreplaced by the average position over some n surroundingpoints reduces this level of noise and the true spectralpeaks emerge more clearly.Our method works by sorting the points within thespectrum in order of power density. The first point in thislist is taken to be the tip of the first peak; subsequentpoints are tested in turn to see whether they lie within acertain frequency range of the peak. If so they are taken tobe included within that peak; otherwise they are taken tobe the tips of new peaks. This continues until all data hasbeen sorted, leaving a list of peak objects, with specificfrequencies, heights and widths. These peaks may then beanalysed using the MRFI method described in Section 2.This procedure has been applied to a number ofdifferent tracking cases in LEP2.  As an example, Figure 5shows the analysis of the Fourier spectrum of thenormalised yn co-ordinate of a particle started on theclosed orbit and tracked for 104 turns (some 200 dampingtimes).  The beam energy was 96 GeV, the phaseadvances per cell were ( ) ( )°°µµ ,90102=, yx .  Aftersimulation of typical operational correction procedures,the particular imperfect machine chosen had a verticalemittance of nm 78.0=ε y and a well-corrected verticaldispersion ( m 03.0 RMS =yD ).The MRFI method was applied to find resonances up tofifth order, using a 50 point moving average. The originalspectrum, its moving average, and the labelled resonancelines produced by the algorithm are depicted in yellow,blue and red respectively.The labelled resonances are in good agreement with thepeaks in the moving average of the spectrum.  Comparedwith the cases shown in [3] the absence of second-ordersynchro-betatron sidebands is consistent with the smallvertical dispersion and relatively small vertical emittance.MRFI remains applicable despite the loss in accuracy dueto the intrinsic nature of the system that requires thedifferent peak finding method and the moving average.5 CONCLUSIONSThe multiple resonance frequency identification (MRFI)method is a useful quantitative tool for detailed insightinto the resonance terms influencing particle motionthroughout phase space. Unlike some other methods, it isapplicable both to deterministic Hamiltonian systems suchas tracked orbits in a proton storage ring and to noisydissipative systems like electron orbits with quantumfluctuations.  It could, of course, also be applied to otherproblems inside and outside the accelerator physicscontext.  The algorithms are implemented in Mathematicapackages, available from [9].REFERENCES[1] V.I. Arnol’d, Russ. Math. Surv. 18:6 (1963) 85-191[2] J.M. Jowett, AIP Conf. Proc.  150, (1986) 374[3] J.M. Jowett, CERN-SL/97-06 & 98-006 (1997-98)[4] R. Bartolini, F. Schmidt, Part. Acc. 59 (1998) 93[5] H. Grote, F.C. Iselin,CERN SL/90-13 (AP) (1996)[6] J. Laskar, et al, Physica D 56 (1992) 253[7] R.T Burgess, CERN LHC Project Note to appear.[8] F. Barbarin et al, Proc. 4th EPAC, World Scientific,Singapore, 1994, p. 193.[9] http://wwwslap.cern.ch/~jowett/Madtomma/Figure 4 (colour): The resonances (0,3,0), (2,1,0) and(5,0,0) in yx II -  phase space (in purple, blue and greenrespectively.)0.1 0.2 0.3 0.4 0.5f-32.5-30-27.5-25-22.5-20-17.5-1580, 1, 0<81, 0, -1<80, 1, 1<80, 0, 2<81, 0, 0<80, 1, -1< 80, 0, 1< 81, 0, 1<Figure 5 (colour): Analysis of a LEP2 positron spectrumwith quantum fluctuations.

Phase-space Analysis by Multiple Resonance-Frequency Identification: Applications to the LHC and LEP

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