130 Beam-based Correction and Optimization for Accelerators
broad meaning and could lead to misunderstanding, in the following we will
refer it as the PCA method.
The PCA method can fail to separate the beam motion harmonics when
the variances of the harmonics are nearly equal. This could happen when
there are bad BPMs, or when there are contamination signals leaked into the
BPM electronics. The independent component analysis (ICA) [58] method
uses additional features of the underlying beam signals to achieve successful
separation despite the above scenarios. Hence it is a much more reliable and
robust method to process the BPM data for the global analysis.
In this section we will describe both the PCA and ICA methods and compare the performances.
5.2.1 A model for the turn-by-turn BPM data
BPMs are designed to record the transverse positions of the beam centroid. In
a storage ring, ideally, the beam is centered on the closed orbit and there is no
orbit change on the turn-by-turn basis. When the beam is displaced from the
ideal orbit, the beam will be “excited” and starts to oscillate. The turn-byturn orbit readings on the BPMs are discrete samples of a continuous motion.
Depending on the lattice conditions and the way the beam motion is excited,
the beam motion may contain various components, including synchrotron oscillation, betatron oscillations of both transverse planes, and motion from
nonlinear resonances. These components are considered source signals. The
source signals represent the different physical processes that drive the beam
motion. They are considered independent from each other. All BPMs detect
the same motion and hence the BPM data contain the same source signals;
but the strengths and phases of each component vary with the BPM location.
Considering the orbit signals as linear combinations of the source signals,
the raw signal of a BPM can be decomposed as
x i (t) = a ij s j (t) + ξ i (t),
(5.19)
where subscript i indicates the i’th BPM, j = 1, 2, · · · , P with P being the
number of source signals, and ξ i is the random noise on the BPM. The orbit
oscillation is centered on the closed orbit, which is not of interest here since
it is the orbit variation that contains information about the dynamics of the
beam motion. For this reason the closed orbit is subtracted from the orbit
signal, such that the ensemble average of many turns, x i (t), is zero. The a ij
coefficients can be scaled up while scaling down s j (t) by the same factor. The
ambiguity can be eliminated by requiring that the row vector a j = (a 1j , a 2j ,
· · · , a P j ) to have a unit Euclid norm, i.e., ||a j || = 1.
The raw signals and source signals can be arranged in vectors, respectively.
In a matrix form, Eq. (5.19) becomes
x(t) = As(t) + ξ,
(5.20)
broad meaning and could lead to misunderstanding, in the following we will
refer it as the PCA method.
The PCA method can fail to separate the beam motion harmonics when
the variances of the harmonics are nearly equal. This could happen when
there are bad BPMs, or when there are contamination signals leaked into the
BPM electronics. The independent component analysis (ICA) [58] method
uses additional features of the underlying beam signals to achieve successful
separation despite the above scenarios. Hence it is a much more reliable and
robust method to process the BPM data for the global analysis.
In this section we will describe both the PCA and ICA methods and compare the performances.
5.2.1 A model for the turn-by-turn BPM data
BPMs are designed to record the transverse positions of the beam centroid. In
a storage ring, ideally, the beam is centered on the closed orbit and there is no
orbit change on the turn-by-turn basis. When the beam is displaced from the
ideal orbit, the beam will be “excited” and starts to oscillate. The turn-byturn orbit readings on the BPMs are discrete samples of a continuous motion.
Depending on the lattice conditions and the way the beam motion is excited,
the beam motion may contain various components, including synchrotron oscillation, betatron oscillations of both transverse planes, and motion from
nonlinear resonances. These components are considered source signals. The
source signals represent the different physical processes that drive the beam
motion. They are considered independent from each other. All BPMs detect
the same motion and hence the BPM data contain the same source signals;
but the strengths and phases of each component vary with the BPM location.
Considering the orbit signals as linear combinations of the source signals,
the raw signal of a BPM can be decomposed as
x i (t) = a ij s j (t) + ξ i (t),
(5.19)
where subscript i indicates the i’th BPM, j = 1, 2, · · · , P with P being the
number of source signals, and ξ i is the random noise on the BPM. The orbit
oscillation is centered on the closed orbit, which is not of interest here since
it is the orbit variation that contains information about the dynamics of the
beam motion. For this reason the closed orbit is subtracted from the orbit
signal, such that the ensemble average of many turns, x i (t), is zero. The a ij
coefficients can be scaled up while scaling down s j (t) by the same factor. The
ambiguity can be eliminated by requiring that the row vector a j = (a 1j , a 2j ,
· · · , a P j ) to have a unit Euclid norm, i.e., ||a j || = 1.
The raw signals and source signals can be arranged in vectors, respectively.
In a matrix form, Eq. (5.19) becomes
x(t) = As(t) + ξ,
(5.20)