134 Beam-based Correction and Optimization for Accelerators
0
20
40
60
SV index
10
-3
10
-2
10
-1
SV
0
20
40
60
BPM index
-0.2
0
0.2
u
1
, u
2
0.05 0.1 0.15 0.2
tune
0
5
10
|F(v)|
0
100
200
turn
-0.1
0
0.1
v
1
Figure 5.3 PCA for the simulated vertical turn-by-turn BPM data on 57 BPMs
over 256 turns for SPEAR3. The SVs (top left), leading spatial patterns (top right),
first temporal pattern (bottom left), and the FFT power spectrum of the temporal
pattern (bottom right) are shown.
offset of 1.2 mm (w/ β y = 9.1 m). The data matrix of the vertical centroid
position on 57 BPMs over 256 turns is analyzed with SVD. Random BPM
noise is added to all data points, with a noise sigma of 10 µm. There are only
two SVs that stand out from the continuous band of small SVs. The small
SVs are due to random BPM noise. The two leading SVs correspond to the
two orthogonal modes for the betatron motion, which may be referred to as
the sine and cosine modes, respectively.
The horizontal and vertical BPM data matrices can be stacked to make a
2M × T overall matrix. With betatron oscillations in both planes, the data
matrix will have four non-zero singular values. If the variances in the horizontal
and vertical oscillations are not close to equal, the four SV modes consist of
two pairs of normal modes - each normal mode represents the motion in a
single frequency. In the case of weak linear coupling, the two normal modes
can be identified as the horizontal and vertical betatron motions according to
their frequencies. The spatial patterns of the horizontal modes contain mostly
large amplitudes on horizontal BPMs, and the spatial patterns of the vertical
modes mostly on vertical BPMs.
However, if the variances of the two normal modes are about equal, the normal modes may be mixed. This is because if two eigenvalues of the covariance
matrix are equal, the corresponding eigenvectors are not uniquely determined.
Given the existence of random BPM noise in the data, the normal modes can
be more easily mixed in the SVD modes. For example, when the horizontal
0
20
40
60
SV index
10
-3
10
-2
10
-1
SV
0
20
40
60
BPM index
-0.2
0
0.2
u
1
, u
2
0.05 0.1 0.15 0.2
tune
0
5
10
|F(v)|
0
100
200
turn
-0.1
0
0.1
v
1
Figure 5.3 PCA for the simulated vertical turn-by-turn BPM data on 57 BPMs
over 256 turns for SPEAR3. The SVs (top left), leading spatial patterns (top right),
first temporal pattern (bottom left), and the FFT power spectrum of the temporal
pattern (bottom right) are shown.
offset of 1.2 mm (w/ β y = 9.1 m). The data matrix of the vertical centroid
position on 57 BPMs over 256 turns is analyzed with SVD. Random BPM
noise is added to all data points, with a noise sigma of 10 µm. There are only
two SVs that stand out from the continuous band of small SVs. The small
SVs are due to random BPM noise. The two leading SVs correspond to the
two orthogonal modes for the betatron motion, which may be referred to as
the sine and cosine modes, respectively.
The horizontal and vertical BPM data matrices can be stacked to make a
2M × T overall matrix. With betatron oscillations in both planes, the data
matrix will have four non-zero singular values. If the variances in the horizontal
and vertical oscillations are not close to equal, the four SV modes consist of
two pairs of normal modes - each normal mode represents the motion in a
single frequency. In the case of weak linear coupling, the two normal modes
can be identified as the horizontal and vertical betatron motions according to
their frequencies. The spatial patterns of the horizontal modes contain mostly
large amplitudes on horizontal BPMs, and the spatial patterns of the vertical
modes mostly on vertical BPMs.
However, if the variances of the two normal modes are about equal, the normal modes may be mixed. This is because if two eigenvalues of the covariance
matrix are equal, the corresponding eigenvectors are not uniquely determined.
Given the existence of random BPM noise in the data, the normal modes can
be more easily mixed in the SVD modes. For example, when the horizontal