Model-Independent Analysis of the Fermilab Tevatron Turn-by-Turn Beam Position Monitor Measurements

Coherent transverse beam oscillations in the Tevatron were analyzed with the Model-Independent Analysis (MIA) technique. This allowed to obtain the model-independent values of coupled betatron amplitudes, phase advances and dispersion function around the ring from a single dipole kick measurement. In order to solve the MIA mode mixing problem which limits the accuracy of determination of the optical functions we have developed a new technique of rotational MIA mode untangling. The basic idea is to treat each beam position monitor (BPM) as two BPMs separated in a ring by exactly one turn. This leads to a simple criterion of MIA mode separation: the betatron phase advance between any BPM and its counterpart shifted by one turn should be equal to the betatron tune and therefore should not depend on the BPM position in the ring. Furthermore, we describe a MIA-based technique to locate vibrating magnets in a storage ring. PACS numbers: 29.20.db, 41.85.-p, 29.85.Fj ∗Electronic address: petrenko@ngs.ru 1 FERMILAB-PUB-11-500-AD-APC Work supported by Fermi Research Alliance, LLC under Contract No. DE-AC02-07CH11359 with the US Department of Energy


I. INTRODUCTION
Turn-by-turn readings from beam position monitors (BPMs) in a storage ring are often used for accelerator optics measurements.The main advantage of this approach in comparison with alternative methods (e.g. the orbit response matrix technique [1]) is that it takes very little time to acquire turn-by-turn data.For example, one needs 2 hours of beam time to obtain the full orbit response matrix for the Fermilab Tevatron collider, compared to the 160 ms necessary to register 8000 turns of coherent betatron oscillations excited by a dipole kick.
The conventional harmonic analysis of turn-by-turn measurements is based on the accurate betatron tune determination [2].In the Tevatron collider the betatron tunes are close to the linear coupling resonance and normally synchrobetatron sidebands overlap.Owing to this fact, it is difficult to use harmonic analysis for calculation of the optical functions from turn-by-turn measurements.This triggered our study of the model-independent analysis (MIA) [3,4] as an alternative method.MIA is based on the correlational studies of signals from different BPMs and does not require accurate betatron tune determination.
In order to find the coupled betatron amplitudes and phase advances with MIA it is necessary to solve the MIA mode mixing problem [4].Previously this problem was addressed with two different approaches, namely the rotational mode untangling proposed in [4] and the Independent Component Analysis (ICA) [5] approach based on some statistical criterion of mutual signal independence.Both these methods rely on Fourier analysis as a way to confirm that the mode separation algorithm actually works.However due to the large chromatic tune spread and synchrobetatron sideband overlapping in the case of the Tevatron it is often difficult to determine the quality of MIA mode separation from the Fourier transformations of the modes.In this paper we describe a new method of MIA mode untangling which does not rely on the Fourier spectrum information.Furthermore, we describe a MIA-based technique to locate vibrating magnets in a storage ring.
The paper is organized as follows.In Section II we outline the basics of MIA with some motivational considerations based on the particle tracking simulations.Section III presents the results of MIA applied to the Tevatron turn-by-turn measurements.In Section IV we define the coupled betatron functions and determine their relation to MIA modes.Section V describes a new method of untangling the mixed MIA modes with applications to the Tevatron turn-by-turn measurements and simulation.Here we also compare our approach with other techniques.Section VI describes our MIA-based method to find the locations of vibrating quadrupoles in the Tevatron.Finally, Section VII presents conclusions.

II. MODEL-INDEPENDENT ANALYSIS
Let z tm represent horizontal or vertical beam centroid position recorded at the m-th monitor for the t-th turn.Then B N ×M ≡ (z tm ) is the beam history matrix of N beam turns recorded simultaneously at M monitors (each BPM signal is represented by some column of matrix B).The MIA is accomplished via singular value decomposition (SVD) of the beam history matrix where U N ×M and V M ×M are orthogonal matrices (i.e. the matrices with orthonormal columns) and is the diagonal matrix of declining singular values.Written separately for each BPM signal in the beam history matrix, Eq. 1 reads where v mj are the elements of matrix V and u nj are the elements of matrix U .This sum can be truncated at some point because singular values decline rapidly.The remaining columns of matrix U are the orthogonal basis of source signals also referred to as the temporal modes of MIA.For each temporal mode in U there is one spatial mode represented by the column of matrix V .Spatial mode represents the amplitude variation along the ring for the corresponding temporal mode.Singular value with a proper index gives the overall amplitude of these two modes in the beam history matrix.
The statistical analysis described here is also called the Principal Component Analysis (PCA) [6] and it is particularly known for its ability to compress data and reduce noise.
PCA is used as a first step in numerous Independent Component Analysis (ICA) algorithms [7], where the final signal separation is achieved according to some statistical criterion of mutual signal independence.Previously ICA was successfully applied to the turn-by-turn data analysis at the Fermilab Booster synchrotron [5].At the end of Section V we compare different ICA algorithms as applied to the Tevatron.Fig. 1 shows singular values and fast Fourier transformation (FFT) amplitudes of two simulated turn-by-turn data sets for the Tevatron lattice with different sextupole settings corresponding to small and large chromaticities.In this tracking simulation the closed orbit passes through the centers of all sextupoles and the linear optics remains the same as the sextupole strength is changed.Singular value spectra in both cases are almost identical.In terms of the PCA data compression that means a good approximation exists for the beam history matrix B with 5 temporal and 5 spatial modes in both cases of overlapping and non-overlapping synchrobetatron sidebands.This property of MIA can be understood if we clarify the physical meaning of the leading MIA modes in this case.
Since the single-turn particle motion is dominated by the linear lattice properties, a good approximation for particle (and beam centroid) turn-by-turn positions can be written as where X = (x, p x , y, p y ) T is the vector of particle canonical coordinates in transverse phase space, R(s) is the 4 × 4 transport matrix along the ring from location 0 to location s, D(s) = (D x , D px , D y , D py ) T is the vector dispersion function, and δ(t) = ∆p(t)/p 0 is the relative momentum deviation performing slow synchrotron oscillations (i.e. the synchrotron tune ν s ≪ 1).
Although the exact form of X 0 (t) ≡ X(0, t) − D(0)δ(t) may be complicated due to accumulation of weak nonlinear effects over multiple turns, Eq. 3 means that the particle transverse coordinates x(t, s) and y(t, s) anywhere in the ring can be described with a linear combination of 5 functions x 0 (t), p x0 (t), y 0 (t), p y0 (t), and δ(t).Therefore the leading temporal MIA-modes in this case represent an orthonormal basis in the 5-dimensional space of all linear combinations of these five functions.Corresponding spatial modes represent an orthonormal basis in the space of four linear orbits and dispersion function.Notice that the leading spatial modes are defined here only by the linear optics (up to some arbitrary rotation in the 5-dimensional space).
As we can see the structure of MIA modes near the coupling resonance is simple in spite of the complicated phase space dynamics leading to the turn-by-turn BPM signals with Fourier spectra that are difficult to interpret.FIG.4: FFT amplitude spectra of temporal modes presented in Fig. 3.

III. MIA DECOMPOSITION RESULTS
The Tevatron collider has 118 horizontal and 118 vertical BPMs which can simultaneously record turn-by-turn beam position histories over many thousand turns.Fig. 2 shows typical turn-by-turn readings of the Tevatron BPMs when beam is kicked in the horizontal plane (full beam history matrix file is available in the supplementary material [19]).Fig. 3 shows the MIA modes calculated from this measurement.First four modes correspond to the betatron oscillations.5th mode is the low-frequency mode which is caused by the coherent synchrotron oscillations of the beam.The spatial component of this mode is proportional to the dispersion function.6th mode is caused by the mechanical vibration of one of the final focus quadrupoles in the Tevatron (see Section VI for details).7th and 8th mode is excited by timing errors of BPM electronics with periodicity of five turns resulting in coherent lines at the tunes of 0.2 and 0.4.The spatial components of the last two modes have a typical random pattern.By looking at the spatial component of a particular mode it is easy to tell whether the mode contains some useful information about the beam motion or it is a "noise" signal generated by BPM electronics.
If we retain more singular values then other less significant noise signals and vibrational modes will appear.For example the noise with the tune of 0.4 is below the selected singular value threshold in Fig. 3.
Fourier amplitudes of the temporal modes (see Fig. 4) reveal the residual "mixing" between the modes corresponding to different physical phenomena, i. e. each temporal mode contains low amplitude harmonics that are the leading harmonics for other modes.Such mixing provides limitations for some practical applications of observed MIA modes like dispersion function measurement.There are several ways to deal with this mode mixing.First of all one can apply a Fourier bandpass filter to data before MIA.Fourier prefiltering is useful in order to treat the low frequency and the high frequency signals separately.For example, the low-pass Fourier filter makes it possible to clearly observe another vibrational mode (u 3 in Fig. 9) in the turn-by-turn measurements presented in Fig. 2.
If Fourier filter is not sufficient to remove the mixing between MIA modes, one can try the ICA or the general rotational mode untangling method [4] i. e. the untangled temporal modes can be found as where O is some rotation matrix.In order to find the angles of this rotation one still needs a criterion of mode separation (Fourier amplitudes of temporal modes were used in [4]).
However in the case of overlap between synchrobetatron sidebands all these methods do not provide good betatron mode separation and simply it's hard to tell whether the betatron modes are mixed or not.Therefore we had to develop another model-independent approach which is explained in the next two sections.

IV. COUPLED BETATRON FUNCTIONS
In this section we outline the relation between the spatial MIA modes and the optical functions that are used to describe the coupled betatron oscillations.More detailed analysis is available in [4].
Let us consider single particle betatron oscillations with linear coupling between horizontal and vertical planes.Let L be the ring circumference, then R(L) is the full turn transport matrix i. e.
The R(L) matrix has 4 complex eigenvectors ν 1 , ν * 1 , ν 2 , ν * 2 (where * denotes complex conjugation) and 4 corresponding eigenvalues e ±iµ 1 , e ±iµ 2 [8].Any vector of initial particle coordinates X(0, 0) can be represented as a linear combination of the four eigenvectors Therefore turn-by-turn values of X are where To describe coupled betatron oscillations we introduce the betatron amplitude and phase advance functions as the phases and amplitudes of x and y components of these vector functions, correspondingly, for f 1x we have For simplicity of notation, let us assume that initial conditions of particle motion correspond to a 1 = a 2 = 1.Then x or y component of Eq. 7 reads or in a matrix form The beam history matrix can be written as where Eq. 11 is the exact solution for the beam history matrix in the case of linear single particle betatron oscillations.According to Eq. 3 in the general case of many-particle nonlinear coupled betatron and synchrotron oscillations the turn-by-turn position of beam centroid anywhere in the ring can be written as a linear combination of just 5 functions i. e. the δ(t) and the 4 components of X 0 (t).Using Fourier filter it is easy to filter out the lowfrequency components of BPM signals proportional to δ(t) and leave only the high-frequency components given by X 0 (t).Such filtered BPM signals are given by the x or y component of R(s)X 0 (t).Since the matrix elements R ij (s) and the columns of matrix V f are both linear orbits of the same lattice one can always write any R ij (s) as a linear combination of the columns of matrix V f (because V f represents a complete basis of 4 linear orbits).Therefore the filtered version of matrix B (labeled as B) can be written in the form similar to Eq. 11: where the U f is the generalized version of U f matrix for the case of many-particle beam experiencing nonlinear coupled betatron and synchrotron oscillations.The 4-column matrix U f is difficult to find analytically, however one can obtain it using the particle tracking simulation data as where V † f is the pseudoinverse of V f (i.e. Eq. 15 is the solution of Eq. 14 with respect to U f ).Therefore we can study the properties of U f matrix using tracking simulation.
One important property of U f matrix is that if betatron tunes are far enough from integer, half-integer and coupling resonances, if then U f is proportional to the orthogonal matrix This property (with obvious modification for a 1 = 1 = a 2 ) is still true in the general case for the U f matrix.For example, in both simulations with different chromaticities presented in Fig. 1 the product of U T f U f yields the matrix which is very close to expected diagonal matrix (in particular, the off-diagonal elements in this matrix are about 100 times smaller than diagonal elements).
For large number of monitors V T f V f tends to be diagonal as well, although due to strong modulation of |f 1,2z (s)| near collider interaction points and because typically N ≫ M , the diagonality of U T f U f holds much better than the diagonality of V T f V f .Therefore let us orthogonalize V f using the SVD Now Eq. 11, or similarly Eq. 14, can be rewritten in the form of SVD (see Eq. 1) Finally, we can relate the coupled betatron amplitudes and phase advances to the SVD of beam history matrix B (or B in the general case).is the first column of U f matrix, determined from the SVD of turn-by-turn measurements (see Fig. 2, 3) using the mode untangling technique described in Section V.
where we assume that only first four modes are retained in SVD of B. From the four columns of matrix V f one can easily calculate the coupled betatron amplitudes and phase advances.
In order to obtain V f it is necessary to find the orthogonal 4 × 4 matrix O f .
It is interesting to note that the columns of matrix U f plotted against each other show beam centroid trace in the normalized phase space (see Fig. 5).

V. UNTANGLING BETATRON MODES
The matrix O f describes some combination of rotations and reflections in 4-dimensional space.Reflections only change signs of modes and can be determined if we require that the phase advance ψ 1z (s) and ψ 2z (s) calculated from V f increase with s.Any rotation in 4-dimensional space can be described in terms of the 6 angles, each corresponding to the rotation in the plane of two basis vectors.Two of these angles are not relevant since they describe a mixing between the modes with the same tunes.This kind of rotation results simply in a phase shift, because The remaining 4 angles in O f set the mixing between modes with the different tunes, i. e.
O f can be written as a product of the following 4 matrices cos θ 14 0 0 sin θ 14 0 1 0 0 0 0 1 0 As it was proposed in [4] one can search for these 4 angles by looking at Fourier amplitudes of the columns of matrix U f given by Eq. 20.In the case of linear betatron oscillations the Fourier spectrum of each column in U f has a single peak at one of the two tunes µ 1 or µ 2 .This approach does not work well for the Tevatron turn-by-turn measurements because of significant chromatic tune spread in the beam.Even if there is no overlap between synchrobetatron sidebands (like in Fig. 2

signals) the nonlinearities in betatron oscillations
produce such a distortion of Fourier spectra that MIA modes have residual components which are not possible to separate completely (see the Fourier spectra of u 3 and u 4 modes in Fig. 4).Therefore looking at Fourier spectra is usually not enough for good mode separation.
We have developed another model-independent criterion of mode separation in MIA which makes use of the fact that betatron phase advance over the entire ring is equal to the betatron tune.The idea is to treat each BPM as two monitors separated in a ring exactly by one turn.Thus we can double the number of analyzed signals in the beam history matrix B.
Note that it is not necessary to fully double the number of signals in B. If calculation time is the important issue one can duplicate only a small fraction of all available monitors.
The betatron phase advance between the monitor with readout (z 1m , z 2m , ...) T and its counterpart shifted by one turn (z 2m , z 3m , ...) T does not depend on BPM index m and should be equal to the corresponding tune.Indeed, if we denote the elements of V f as then the following expressions can be calculated at each BPM, and apparently they do not depend on a BPM location s: These expressions give a sensitive criterion of betatron mode separation in MIA.In particular, one may first assume that there is no mode mixing, i. e. O f = I, and calculate betatron tunes at each BPM using the corresponding elements of matrix 2/N V Σ (see Eq. 21).If the resulting BPM-by-BPM tune variation is significant then one can minimize it with a proper O f matrix.
In order to explain our algorithm for the mixing matrix O f determination let us assume that the mixing is small, i. e. O f is close to the unity matrix FIG. 6: Tunes calculated at each BPM using Eq.37 and the similar one for µ 2 .µ 1 (X) -betatron tune µ 1 calculated at horizontal monitors, µ 1 (Y) -the same for vertical monitors.In order to compare these results with the FFT of temporal modes (solid lines) the tunes are plotted along the horizontal f-axis, (vertical s-axis shows the positions of monitors along the ring).Fourier bandpass filter from 0.413 to 0.429 (hard edge) was applied to signals before MIA for better accuracy.
where, for example, Now we can calculate the effect of any rotation on the BPM-by-BPM tune variation given by Eq. 32 and 33, since it can be written as In order to do that let us take a derivative of with respect to θ 13 .Here A 1 (s) and B 1 (s) are defined through with a similar expression giving A 1 and B 1 .The derivative of Eq. 37 yields The same operation for θ 24 gives One may note that as soon as ∆µ = µ 2 − µ 1 is small, is also small, while is approximately twice as big as each of the derivatives.A similar result can be obtained for θ 14 and θ 23 , namely, and Therefore in order to find O f matrix which minimizes the BPM-by-BPM tune variation δµ 1,2 (s) it is more convenient to use the following set of basic rotation transformations The δµ Thereby after the θ 1 and θ 2 angles are determined by minimizing the δµ 1,2 independently in each plane, then usually there is some remaining constant difference between the tunes calculated at horizontal and at vertical BPMs.This difference can be minimized with a proper selection of θ 3 and θ 4 angles.If the mode mixing is strong (i.e. the angles are large) then several iterations of the described procedure are required.In the case of strong mode mixing it is helpful to try several different ICA algorithms as the first step (instead of plain SVD) in the described mode untangling technique.
The smallness of ∆µ is not a critical requirement, we employ it here just in order to explain the selection of the new set of basic rotations.The described algorithm also works for large ∆µ ∼ 0.1.
Fig. 6 shows the result of our method applied to untangle the betatron modes for the Tevatron turn-by-turn measurements presented in Fig. 2. As we can see from the Fig. 6 in this typical case of the Tevatron turn-by-turn measurements it is hard to determine the quality of betatron mode separation looking at the Fourier amplitudes of temporal modes.
On the other hand our new criterion provides clear answer to this question and makes it possible to further improve the mode separation.The initial mode separation provided by the PCA in Fig. 6 is actually quite good, since the Fourier amplitudes of temporal modes before and after rotation are almost identical (although this is not a typical situation).
shown on the right.The same measured turn-by-turn data was used as in the Fig. 2. Model was calibrated with the orbit response matrix technique [13,14].However since we do not know the real optics it is not clear which method (orbit response or MIA) provides the best accuracy.
The coupled betatron functions calculated from the untangled modes (see Fig. 7) show good agreement with the linear model calibrated using the orbit response matrix measurements [13,14].As a measure of quality of optical function determination we will use the RMS-averaged (over all BPMs) difference between the model value of optical function and its value given by the MIA.Fig. 7 may serve as a typical example of how such RMS-average reflects the actual error distribution.
In order to benchmark our method of betatron mode untangling against some ICA algorithms we have made the many-partice tracking simulation of Fig. 2 measurement with similar decoherence of betatron oscillations and similar chromatic tune spread in the beam.
The errors of optical function determination for each algorithm applied to this simulated data are shown in the Table I.Also it was possible to calculate the same optical functions using the conventional Fourier analysis [2] since in this particular case there is no overlap between the synchrobetatron sidebands corresponding to the different tunes.MIA accuracy limit in Table I is the best possible accuracy of determination of optical functions using the 4 linear orbits (4 spatial modes) from the SVD-decomposition of the beam history matrix Notations are the same as in the Table I. "EVD+rotation" means that the EVD algorithm (see appendix) instead of plain SVD was used as initial step in our rotational mode untangling method.
This approach provides the best accuracy.
B. In a tracking simulation this MIA accuracy limit can be easily found since we know the real lattice functions, i. e.
where O best = V T V f is the best possible mode untangling matrix (note that V V T = I but The difference between the model optics V f and the best possible "measured" optics V f, limit is caused by nonlinearities in particle motion as well as by the finite number of beam turns N (which is limited by the decoherence of betatron oscillations in our case).
In real measurements this limit on the MIA accuracy is also worsened by the accuracy of BPM measurements.
As can be concluded from the Table I, our method has the following property when compared to the rest of ICA algorithms: it significantly improves the accuracy of one betatron function determination (i.e.f 2y ) at the expense of some slight increase of the errors for another function (f 1x ).
None of the considered MIA algorithms outperforms the conventional Fourier analysis in  [15].Fast ICA algorithm is from a different package [16].b In the conventional harmonic analysis of turn-by-turn data [2] we used the precise values of tunes obtained with Laskar's Numerical Analysis of Fundamental Frequencies (NAFF) [12] which is implemented in the SDDS-toolkit [10].
Table I.However the Fourier analysis is less accurate than MIA in the case of some overlap between the synchrobetatron sidebands (but when we still can find the central peaks).
In order to investigate how the separation between tunes (i.e. the Eq.16 condition) affects the accuracy of optical function measurements we have performed another set of tracking simulations.The tune separation was regulated with skew-quadrupoles, namely in the beginning the minimum tune separation (the tune split parameter) was corrected to zero and the working point was placed close to the coupling resonance.Then using skewquadrupoles the tune split parameter was gradually increased and at each step the tracking simulation was performed providing the data for the MIA.The results of these tracking simulations are shown in Fig. 8.
FIG. 9: Slow MIA modes.Low-pass Fourier filter (f < 0.004) was applied to BPM readings prior to SVD.

VI. VIBRATING MAGNET LOCATION
To locate the source of the vibrational mode (u 6 in Fig. 3 or u 2 and u 3 in Fig. 9) we use an approach similar to the action and phase jump analysis [9].Since the beam orbit in the region free of vibrating magnets is a free betatron oscillation it can be represented as a superposition of 4 linearly independent orbits.The coefficients of this superposition i. e. 4 initial conditions for the beam orbit can be calculated using data from any 4 nearby monitors (combined function monitor is treated as two separate monitors here).These initial conditions plotted as a function of BPM sequence position along the ring should have sudden change in the point where the beam oscillation is driven by the magnet vibration.In the case of sufficiently weak coupling one can calculate two of the 4 initial conditions with any pair of nearby monitors operating in the same plane.In terms of MIA this is written as where v vibr is the spatial component of vibrational mode, v b1 and v b2 are the two linearly independent orbits in one plane (i.e. v 1 and v 2 modes in Fig. 3).
From the Eq.50 both initial conditions C 1 and C 2 as functions of BPM pair location in the ring can be found.It is more convenient to use the initial phase and amplitude of betatron oscillation defined as arctan(C 1 /C 2 )/2π and C 2 1 + C 2 2 .If there is one local source of vibrational mode then there should be a single jump of the initial phase while amplitude remains constant.Fig. 10 shows the initial phase calculated for the v 2 mode in Fig. 9.
As we can see this MIA mode is driven by a single local source.On the other hand, the same procedure applied to the less significant v 3 mode in Fig. 9 reveals that there are two vibrating quadrupoles located near different interaction points of the collider.v 3 mode is the result of both these vibrations.In fact it was already evident from the Fig. 9 that there are two local sources of this mode: we can see that the amplitude of v 3 mode experience changes in two locations, while the amplitude of v 2 mode remains constant.Note that the amplitude variation of the v 6 mode in Fig. 3 is caused by residual mode mixing.That is why the low-pass Fourier filter applied in Fig. 9 is essential for determination of vibrating quadrupole locations.

VII. CONCLUSION
We applied model-independent analysis to the turn-by-turn BPM measurements of coherent betatron oscillations excited by the transverse kicker in the Tevatron.In order to obtain the amplitudes and phase advances of coupled betatron oscillations, we have developed a new approach to rotational MIA mode untangling, which is based on the assumption that the betatron phase advance calculated between a BPM and the same BPM shifted by one turn is equal to the betatron tune and does not depend on the BPM position in the ring.
The main advantage of this method is that it can be used to untangle the mixed betatron modes in the proximity of the linear coupling resonance when there is an overlap between synchrobetatron sidebands corresponding to the different betatron tunes.However, the obvious disadvantage of our method of betatron function determination in this case is that the temporal modes are assumed to be mutually orthogonal.Therefore the fractional parts of betatron tunes can not be too close because the condition ( 16) should be satisfied.This sets the lower limit on the number of turns to be recorded before the betatron oscillations in the beam decohere, namely, N ∆µ/2π > 4 for typical Tevatron parameters (according to Fig. 8).One can effectively increase N by treating several repeated measurements as a single BPM readout.In particular it is helpful to combine measurements with horizontal and vertical kicks.
Although according to Table I and Fig. 8 in the case of no overlap between synchrobetatron sidebands the accuracy of determination of optical functions via MIA is typically the same or worse than that of conventional Fourier analysis, our algorithm can help to find the central peak which is not always obvious in the Fourier spectrum of betatron oscillations with large chromatic tune spread as can be seen in the Fig. 2 for example.
Another application of our criterion for betatron mode separation may be the identification of malfunctioning BPMs.After the BPM-by-BPM tune spread is minimized via rotation, the tunes calculated at noisy and malfunctioning BPMs deviate from the main distribution.
We have also shown that MIA can be used to locate some unintended sources of transverse beam oscillations like the vibrating quadrupoles in the Tevatron.
So far we have used the turn-by-turn measurements only to verify the predictions of the Tevatron model which was calibrated with orbit response matrix method.However our final goal is to use turn-by-turn data from the Tevatron (as well as from other Fermilab accelerators) for model calibration, similarly to the technique developed at the PEP-II collider [17,18].The model calibration can also provide the BPM tilts and gain factors which should further improve the accuracy of optical functions measurements.

FIG. 2 : 18 π
FIG. 2: Coherent transverse oscillations of proton beam in the Tevatron recorded by horizontal and vertical BPMs (left).At about 900th turn beam was kicked in the horizontal plane.FFT amplitude of the vertical BPM signal is shown on the right.Timing errors of BPM electronics with the periodicity of five turns produce coherent lines at tunes of 0.2 and 0.4.Oscillation amplitude damping is due to nonlinear decoherence of betatron oscillations.Proton beam parameters in the Tevatron: momentum spread (rms) -1.2 • 10 −4 (at 980 GeV), normalized emittance (95%) -18 π mm mrad.

FIG. 3 :
FIG. 3: Temporal (left) and spatial (right) modes of MIA corresponding to the largest 8 singular values.

FIG. 7 :
FIG. 7: Coupled betatron amplitude functions at BPMs (dots) obtained from the columns of matrix V f and compared to the model values (solid lines).The relative difference between the model values of optical functions and their measured values (δ|f 1x | = |f 1x | measured − |f 1x | model ) is

FIG. 8 :
FIG.8: Accuracy of betatron function determination versus tune separation.The turn-by-turn data was obtained via many-particle tracking simulations with different skew-quadrupole settings.

FIG. 10 :
FIG. 10: Vibrating quadrupole location.Initial phase of the betatron oscillations excited by the vibrating quadrupole is shown.The phase was calculated using the pairs of horizontal monitors one after another.
1,2 is much more sensitive to O f 1 and O f 2 than to O f 3 and O f 4 rotation.Typically the most of BPM-by-BPM tune variation can be reduced with O f = O f 1 O f 2 .Because |f 2z /f 1z | and ψ 2z −ψ 1z tend to be approximately constant all over the ring, O f 3 and O f 4 add a constant shift to δµ 1,2 though this shift has a different value for horizontal and for vertical BPMs.

TABLE I :
Our method compared to several ICA algorithms and conventional harmonic analysis of turn-by-turn data.This table shows the errors of determination of betatron amplitudes via MIA.For example, δ|f 1x | = |f 1x | MIA − |f 1x | model (RMS average over horizontal BPMs).Very similar results are obtained for the accuracy of phase advance determination.The turn-by-turn data was generated via tracking simulation with parameters similar to the measurement presented in Fig. 2.
a Matlab implementations of JADE, AMUSE, and EVD (see appendix) algorithms are taken from the ICALAB package