Application of independent component analysis to Fermilab Booster

Autocorrelation is applied to analyze sets of finite-sampling data such as the turn-by-turn beam position monitor (BPM) data in an accelerator. This method of data analysis, called the independent component analysis (ICA), is shown to be a powerful beam diagnosis tool for being able to decompose sampled signals into its underlying source signals. We find that the ICA has an advantage over the principle component analysis (PCA) used in the model-independent analysis (MIA) in isolating independent modes. The tolerance of the ICA method to noise in the BPM system is systematically studied. The ICA is applied to analyze the complicated beam motion in a rapid-cycling booster synchrotron at the Fermilab. Difficulties and limitations of the ICA method are also discussed.

Figure 1

The top and the second row are ICA modes of the linear coupling (LC) model. Model parameters are ${\nu }_x=6.74149$, ${\nu }_z=6.69149$, $C=0.05$, $x_0=y_0=1.0$ $A_1=1.0682$, $A_2=-0.0682$, $B_1=0.0791$, $B_2=0.9209$. Top row: the spatial function and FFT of one of the $+$ mode. The second row: the spatial pattern and FFT of one of the $-$ mode. The horizontal (solid) and vertical (dash) curves in left plots are spatial functions. The right plots show FFT spectra of the temporal functions. The third and the bottom rows: modes 1 and 3 of the LC model using the PCA analysis. The SVs of the modes 1 and 3 are 122.0 and 98.8, respectively.

Figure 2

PCA modes of the LC model with a localized bad-BPM signal. Model parameters: ${\nu }_x=6.74149$, ${\nu }_z=6.69149$, $C=0.05$, $x_0=1.0$, $y_0=0.0$, $A_1=0.9945$, $A_2=0.0055$, $B_1=0.0736$, $B_2=-0.0736$. A harmonic oscillation signal is added to BPM V37 with tune $\nu =0.57545$ and amplitude $D=0.4$. The top row: mode 3 with $\mathrm{SV}=9.4$. The second row: mode 5 with $\mathrm{SV}=7.5$. These two modes are mixed. the third and the fourth rows are ICA modes of the LC model with a bad-BPM harmonic oscillation signal. The localized “bad-BPM” mode is completely separated.

Figure 3

The top and the second row are PCA modes of the LC model with a localized bad BPM with Gaussian white noise signal. Model parameters: ${\nu }_x=6.74149$, ${\nu }_z=6.69149$, $C=0.05$, $x_0=1.0$, $y_0=0.0$ $A_1=0.9945$, $A_2=0.0055$, $B_1=0.0736$, $B_2=-0.0736$. The signal added to BPM V37 is white Gaussian noise. The top row corresponds to mode 3 with $\mathrm{SV}=8.4$, and the second row is the mode 5 with $\mathrm{SV}=6.1$. The two PCA modes are mixed. The third and the fourth rows are ICA modes of the LC model with a localized bad-BPM Gaussian white noise. The localized “bad-BPM” mode is completely separated.

Figure 4

Estimation of errors of ICA (cross) and PCA (square) methods with various random noise levels in the LC model. The model parameters are the same as Fig. 2. Data of 1000-turn are used to calculate ${\sigma }_{\beta }/\beta$ (left plot) and ${\sigma }_{\psi }$ (bottom plot). The estimation at each noise level ${\sigma }_{\mathrm{noise}}$ ($x$ axis) is made by repeating the measurement of $\beta$ and $\psi$ 10 times with white Gaussian random noises added to each BPM.

Figure 5

Effect of number of the sampled turns ($N_t$) in the LC model with ${\nu }_{+}=6.7447$ and ${\nu }_{-}=6.7372$. Top: off-diagonal elements of source signals $C_s(1,3)$. Bottom: ${\sigma }_{\beta }/\beta$ vs $N_t$ for ICA (solid) and PCA (dashed).

Figure 6

Third order resonance signals corresponding to $-{\nu }_x+2{\nu }_z-1=0$ in tracking data (500 turns) of the booster. The tune of the signals is $0.02107$, while ${\nu }_x=6.65753$, ${\nu }_z=6.83929$. The currents in sextupole families are $\mathrm{ISEXTL}=20\mathrm{A}$, $\mathrm{ISEXTS}=5\mathrm{A}$. Left plot: Amplitude of the resonance signal at horizontal (solid cross) and vertical (dash square) BPMs. Right plot: the FFT spectrum of the two resonance signals.

Figure 7

Two modes of horizontal signal are shown on the left plots, and the corresponding FFT spectra are shown as the horizontal betatron tune. The ICA extracts a single betatron mode from data of all BPMs. The betatron mode is a purer sinusoidal signal and the tune evaluation method in Ref. 18 can be used to achieve higher precision. Note that the decoherence is not an independent signal, and thus cannot be isolated.

Figure 8

Using the spatial function of two horizontal modes, one can calculate the betatron amplitude function at each BPM and phase advance between BPMs. The measured BPM is compared with the MAD model. The error bars were estimated with the standard deviation of the betatron function derived from the four data sets.

Figure 9

The horizontal (square) and vertical (cross) betatron tunes in a booster cycle. Tunes calculated by MAD model are compared to measurements (solid and dashed lines).

Figure 10

The measured dispersion function is compared with the MAD modeling (dotted line) based on a realistic Fermilab Booster lattice. The error bars were estimated with the standard deviation of the dispersion function derived from the four data sets.

Figure 11

The synchrotron tune in a booster cycle. The squares are measured from turn-by-turn data with ICA method. The crosses are measured from phase signal with synchrotron phase detector (SPD) as in Ref. 9. Note that the SPD method has difficulty in measuring the synchrotron tune above the transition energy at around the 14.5 ms.

Figure 12

The turn-by-turn data, divided by the dispersion function, of the synchrotron mode is plotted at each BPM for a total of 50 revolutions, starting from the revolution number 3001. The normalized vertical axis is equivalent to the fractional off-momentum variable. Each square box represents one revolution.

Figure 13

Synchrotron modes in the middle of a booster cycle. The turn range is 3001 to 3400 (see Fig. 12). The temporal signals provide us the synchrotron tune shown in Fig. 11. The spatial pattern does not resemble the dispersion function. Furthermore, the spatial function crosses zero. The effective amplitude of the off-momentum coordinate is about ${10}^{-5}$. This may result from the longitudinal damper or mismatch between the beam energy and the main dipole field.

Figure 14

The spatial function of the synchrotron modes shown in Fig. 13 divided by the dispersion function. The resulting function can be thought of as the amplitude of the off-momentum deviation at each BPM location. The locations of rf cavities are shown as dots on the horizontal axis.