Linear optics measurement and correction - II 139
matrix W that joint diagonalizes the time shifted covariance matrix of z for
a few selected time shifts,
C z (τ ) = WC ˆ
s (τ )W
T , for τ = τ 1 , τ 2 , · · · , τ k ,
(5.41)
where C ˆ
s (τ ) are diagonal matrices as they are the covariance matrices of the
scaled source signals, ˆ s = Λ
−1 s. The choice of the time shifts may depend on
the frequency contents of the source signals. For turn-by-turn data, typically
no special care is needed. A simple choice such as τ = 0, 1, 2, 3 would work
for most cases. Because of noise in the data, joint diagonalization can be
achieved only approximately. The numeric algorithm for approximate joint
diagonalization is found in Ref. [18]. The source signals and the mixing matrix
are then determined by
A = U P Λ P WΛ
−1
P ,
(5.42)
s = Λ P W
T z = Λ P W
T Λ
−1
P U
T
P X.
(5.43)
When the horizontal and vertical data are processed together in one data
matrix, there are two pairs of betatron modes. From the mixing matrix, the
beta functions and betatron phase advances can be calculated,
β i = a(A
2
i,c + A
2
i,s ), ψ i = tan
−1 A i,s
A i,c
,
(5.44)
where subscripts c and s indicate the cosine and sine modes for the betatron
motion, respectively, i is the BPM index, and a is an overall scaling factor
which can be approximately determined by requiring the average beta function
over all BPMs to be the same as the model value. There is only one synchrotron
mode because typically the phase of synchrotron motion has little change over
one turn. The dispersion function can be determined by
D i = bA i,d ,
(5.45)
where subscript d indicates the synchrotron mode and b is a scaling constant.
Figure 5.7 shows two of the ICA modes for the tracking data plotted in
Figure 5.4 where PCA fails to separate the horizontal and vertical betatron
oscillations due to their equal variances. With ICA, the betatron modes are
now completely separated out.
ICA is also applied to the data shown in Figures 5.5 and 5.6. The results
are shown in Figure 5.8 and 5.9, respectively. In both cases, the betatron
modes are now separated from the contaminating signals or the synchrotron
motion.
To evaluate the accuracy of phase advance determination of the ICA
method, the betatron phase advances determined from the ICA modes of the
simulated data in Figure 5.7 are compared to the values calculated with the
lattice model. Betatron phase advances obtained with the harmonic analysis
method (using NAFF for tune determination) and the direct sinusoidal fitting
matrix W that joint diagonalizes the time shifted covariance matrix of z for
a few selected time shifts,
C z (τ ) = WC ˆ
s (τ )W
T , for τ = τ 1 , τ 2 , · · · , τ k ,
(5.41)
where C ˆ
s (τ ) are diagonal matrices as they are the covariance matrices of the
scaled source signals, ˆ s = Λ
−1 s. The choice of the time shifts may depend on
the frequency contents of the source signals. For turn-by-turn data, typically
no special care is needed. A simple choice such as τ = 0, 1, 2, 3 would work
for most cases. Because of noise in the data, joint diagonalization can be
achieved only approximately. The numeric algorithm for approximate joint
diagonalization is found in Ref. [18]. The source signals and the mixing matrix
are then determined by
A = U P Λ P WΛ
−1
P ,
(5.42)
s = Λ P W
T z = Λ P W
T Λ
−1
P U
T
P X.
(5.43)
When the horizontal and vertical data are processed together in one data
matrix, there are two pairs of betatron modes. From the mixing matrix, the
beta functions and betatron phase advances can be calculated,
β i = a(A
2
i,c + A
2
i,s ), ψ i = tan
−1 A i,s
A i,c
,
(5.44)
where subscripts c and s indicate the cosine and sine modes for the betatron
motion, respectively, i is the BPM index, and a is an overall scaling factor
which can be approximately determined by requiring the average beta function
over all BPMs to be the same as the model value. There is only one synchrotron
mode because typically the phase of synchrotron motion has little change over
one turn. The dispersion function can be determined by
D i = bA i,d ,
(5.45)
where subscript d indicates the synchrotron mode and b is a scaling constant.
Figure 5.7 shows two of the ICA modes for the tracking data plotted in
Figure 5.4 where PCA fails to separate the horizontal and vertical betatron
oscillations due to their equal variances. With ICA, the betatron modes are
now completely separated out.
ICA is also applied to the data shown in Figures 5.5 and 5.6. The results
are shown in Figure 5.8 and 5.9, respectively. In both cases, the betatron
modes are now separated from the contaminating signals or the synchrotron
motion.
To evaluate the accuracy of phase advance determination of the ICA
method, the betatron phase advances determined from the ICA modes of the
simulated data in Figure 5.7 are compared to the values calculated with the
lattice model. Betatron phase advances obtained with the harmonic analysis
method (using NAFF for tune determination) and the direct sinusoidal fitting