Xiaoying PangIndiana UniversityMarch. 17th , 2010

Independent Component Analysis for Beam

Measurement

Outline

Introduction to ICA Application to linear betatron motionStudy of nonlinear motion – 2x modesBeam-based measurement of sextupole

strength

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Black Source Separation

Without knowing the positions of microphones or what any person is saying, can you isolate each of the voices?

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source1

source2

source3

source4

mixture1

mixture2

mixture3

mixture4

Mixing

Source signals

Mixing Measured signals

Demixing ?

Mixing

Mixing

Mixing

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BPM dataTurn-by-turn BPM signals, x(t) is usually a

mixuture of betatron motion, synchrotron motion, nonlinear motion and noise.

For a turn-by-turn measurements of M-BPMs and N-turns, we construct BPM data matrix:

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Principal Component Analysis

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x : measured signals (mixtures of sources signals)W: demixing matrixY: hopefully the source signals

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ij

ji

Y = Wx

Cov(Y) =

Larger variance modes contain more information.Source signals should have zero correlation.

TTTTTTTTTT DDWUWUWUVDWUDVWWxxYY ,

IYYUW TT 2/1

Singular Value Decomposition (SVD)

Independent Component AnalysisRequirement: ICs have different

autocovariance. Autocovariance: the covarience between the

values of the signals at different time points.For one signalFor two different signalsAll these autocovariances for a particular time lag

can be grouped into an autocovariance matrix

Due to the independence of the source signals, the source signal autocovariance matrices Cs

, t = 0,1,2… should be diagonal.

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BPM data

X =

x11 x12 x13 X1,997 X1,998 X1,999 X1,1000x14

xm1 xm2 xm3 Xm,997 Xm,998 Xm,999 Xm,1000xm4

x21 x22 x23 X2,997 X2,998 X2,999 X2,1000x24

x31 x32 x33 X3,997 X3,998 X3,999 X3,1000x34

AutoCov(X) = T

Turn number

BPMnumber

ICA using time structure

s1

X =

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0

0S1

S2

S3

s2 s3

CsE{s(t)s(t-)T} is diagonal !

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ICAOne time lag and the AMUSE (Algorithm for Multiple

Unknown Signals Extraction) algorithm Consider the whitened data z(t), with the

separating matrix W, the source signals s(t) can be found as:

Slightly modified time-lagged covariance matrix:

The new time-lagged covariance matrix is

symmetric. So the eigenvalue decomposition is well defined and easy to compute.

W can be obtained by SVD of

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ICA (cont’)Drawbacks of the AMUSE algorithm Requirement: the eigenvalues of matrix

have to be uniquely defined. The eigenvalues are given by , thus the source signals must have different autocovariances. Otherwise, ICs can not be estimated.

We can search for a suitable time lag so that the eigenvalues are distinct, but this is not always possible if the source signals have identical power spectra, identical autocovariances.

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ICA (cont’)Using several time-lags An extension of the AMUSE algorithm that improves

its performance is to consider several time lags instead of a single one. Then the choice of the time lag is a less serious problem.

Using several time lags, we want to simultaneously diagonalize all the lagged covariance matrices. This joint-diagonalization cannot be perfect, but we can define a quantity to express the degree of diagonalization and try to find its minimum/maximum.

Minimizing off(M) is equivalent to diagonalizing M.

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PCA to Linear Betatron motion

Consider a simple sinusoidal model With M BPMs in the accelerator and N

turns, the (i,j) element of the turn-by-turn BPM data matrix X is

SVD of x is: Spatial proper

ty

Temporal

property

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PCA Linear Betatron motion(cont’)

Now consider the betatron motion:

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ICA vs. PCA on Linear Betatron motion(cont’)

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ICA vs. PCA on Linear Betatron motion(cont’)

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ICA vs. PCA on Linear Betatron motion – effect of BPM noise

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Study of nonlinear motion -- 2x modeAGS lattice with 12

superperiod of FODO cells

Add sextupoles in the lattice.

Particle tracking was carried out and the data were analyzed by PCA and ICA.

We found totally 6 important modes. We only consider the 3rd and 4th modes at the tune of 2x

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Study of 2x mode (cont’)Equation of motion of 2x mode

Hill’s eqn:For a short sextupole, use the localized kickFloquet transformation:

whereSolution: Get the particular solution

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Study of 2x mode (cont’)

Closed Orbit

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Study of 2x mode (cont’)

Closed Orbit= x-x-x2

Simple betatron oscillation !21

Study of 2x mode (cont’)

AGS lattice with two sextupleslocated at 185m and 420.37m, with strength K2L = 1m-2 and -1.5m-2.

Black lines indicate the locations of two sextupoles.

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ICA vs. PCA on 2x mode Compare the spatial

function obtained by ICA and PCA

After ICA processing, the normalized spatial wave functions of the 3rd and 4th modes have simple linear betatron motion outside the sextupole.

The spatial function of the 4th mode obtained by PCA preprocessing is messy, but still important in a proper ICA analysis.

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Beam-based measurement of sextupole strength

BPM1

BPM2

BPM3

SXT

SXT

3x̂

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3

sin

2s

ss

LxKx

With single sextupole in the lattice, very point corresponds to one turn of tracking, totally 1000 turns.

The slope indicates strength of the sextupole.

The slope can be accurately determined by the centroid of each bin of

The band width is proportional to noise level.

This method can also be used for other higher order non-linear elements

2sx

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1 experiment

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With 12 sextupoles in the lattice

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ConclusionBasic idea of ICAWe have developed ICA for both linear and

nonlinear betatron motion, particularly beam-based measurement of nonlinear sextupole strength.

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Thank you.

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