138 Beam-based Correction and Optimization for Accelerators
The time shifted covariance of two source signals is a way to characterize
their spectral correlation. It is defined as
C 12 (τ ) = s 1 (t)s 2 (t + τ ) =
s 1 (t)s 2 (t + τ )dt,
(5.37)
where τ is the time shift. For continuous signals, we have
C 12 (τ ) =
1
2π
F 1 (ω)F
∗
2 (ω)e
iωτ dω,
(5.38)
where F 1,2 (ω) are the Fourier transforms of s 1,2 , respectively, and
∗ indicates
the complex conjugate. Clearly, if the power spectra of the two source signals
have no overlap, C 12 (τ ) = 0 for any time shift τ . If s 2 = s 1 , C 12 (τ ) becomes
the auto-correlation of the source signal, which is equal to the inverse Fourier
transform of its power spectrum. Similar results are valid for discrete samples
of the source signals. In this case, the time shifted covariance is evaluated by
shifting the series of one source signal by τ units and summing up the product
s 1 (k)s 2 (k + τ ).
From Eq. (5.20), we have
C x (0) = AC s (0)A
T + Σ n ,
(5.39a)
C x (τ ) = AC s (τ )A
T , for τ = 0,
(5.39b)
where the first equation is the ordinary covariance matrix we have seen in
Eq. (5.26), the second equation utilizes the fact that the time shifted covariance of the white noise is zero for τ = 0. As discussed in the above, the
time shifted covariance matrices of the source signals are diagonal matrices.
Eq. (5.39b) provides additional information regarding the connection between
the data signals and the source signals, which states that the mixing matrix
A not only diagonalizes the usual equal-time covariance matrix, but also diagonalizes all time shifted covariance matrices of the data signals. This extra
condition can be used to determine the mixing matrix. Mathematically, the
challenge becomes finding a matrix that simultaneously diagonalizes a few
time shifted covariance matrices with selected time shifts.
Tools for joint diagonalization of a few matrices have been developed and
applied for identification of source signals from mixed data samples. An ICA
algorithm, called second order blind identification (SOBI) [13], utilizes this
approach and was found to be effective for the application to turn-by-turn
BPM data in accelerators. The algorithm consists of two steps. The first step
is data whitening, in which PCA is performed to achieve dimension reduction
and a normalized, orthogonal representation of the data. The new data matrix
becomes
z = Λ
−1
P U
T
P X,
(5.40)
where P SVD modes are kept. The new data matrix satisfies zz
T = I P , with
the P -dimensional identity matrix I P . The second step is to find an orthogonal
The time shifted covariance of two source signals is a way to characterize
their spectral correlation. It is defined as
C 12 (τ ) = s 1 (t)s 2 (t + τ ) =
s 1 (t)s 2 (t + τ )dt,
(5.37)
where τ is the time shift. For continuous signals, we have
C 12 (τ ) =
1
2π
F 1 (ω)F
∗
2 (ω)e
iωτ dω,
(5.38)
where F 1,2 (ω) are the Fourier transforms of s 1,2 , respectively, and
∗ indicates
the complex conjugate. Clearly, if the power spectra of the two source signals
have no overlap, C 12 (τ ) = 0 for any time shift τ . If s 2 = s 1 , C 12 (τ ) becomes
the auto-correlation of the source signal, which is equal to the inverse Fourier
transform of its power spectrum. Similar results are valid for discrete samples
of the source signals. In this case, the time shifted covariance is evaluated by
shifting the series of one source signal by τ units and summing up the product
s 1 (k)s 2 (k + τ ).
From Eq. (5.20), we have
C x (0) = AC s (0)A
T + Σ n ,
(5.39a)
C x (τ ) = AC s (τ )A
T , for τ = 0,
(5.39b)
where the first equation is the ordinary covariance matrix we have seen in
Eq. (5.26), the second equation utilizes the fact that the time shifted covariance of the white noise is zero for τ = 0. As discussed in the above, the
time shifted covariance matrices of the source signals are diagonal matrices.
Eq. (5.39b) provides additional information regarding the connection between
the data signals and the source signals, which states that the mixing matrix
A not only diagonalizes the usual equal-time covariance matrix, but also diagonalizes all time shifted covariance matrices of the data signals. This extra
condition can be used to determine the mixing matrix. Mathematically, the
challenge becomes finding a matrix that simultaneously diagonalizes a few
time shifted covariance matrices with selected time shifts.
Tools for joint diagonalization of a few matrices have been developed and
applied for identification of source signals from mixed data samples. An ICA
algorithm, called second order blind identification (SOBI) [13], utilizes this
approach and was found to be effective for the application to turn-by-turn
BPM data in accelerators. The algorithm consists of two steps. The first step
is data whitening, in which PCA is performed to achieve dimension reduction
and a normalized, orthogonal representation of the data. The new data matrix
becomes
z = Λ
−1
P U
T
P X,
(5.40)
where P SVD modes are kept. The new data matrix satisfies zz
T = I P , with
the P -dimensional identity matrix I P . The second step is to find an orthogonal