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Additional resources for Beam Diagnostics in Superconducting Accelerating Cavities: The Extraction of Transverse Beam Position from Beam-Excited Higher Order Modes (Springer Theses)
3 Singular Value Decomposition 51 Fig. 9 Mean spectrum of the 92 calibration samples The size of the matrix Acal svd is significantly smaller ( p + 1 columns, representing p SVD modes and 1 intercept term) compared to the original spectra matrix Acal ex p (6242 columns, representing 6241 sampling points and 1 intercept term). Replacing cal cal Acal ex p by Asvd in the regression (Eq. 3), the linear system composed by Asvd and cal is now over-determined, and has a best solution in a least square sense.
It can be seen that the spectrum is well represented by using only the first 17 SVD modes, which is also illustrated by r 2 ≈ 1. The choice of p will be discussed later. Fig. 7 Spectra reconstructed from each of the first nine SVD modes. The first six SVD modes plots were reprinted with permission from . 11) Acal svd, f ull = Aex p · V = (A1 , A2 , · · · ). Aisvd ∈ Rm×1 contains the mode amplitude for each beam position of the ith SVD mode. 8a shows the mode amplitude of all 92 SVD modes with non-zero singular values obtained from the calibration samples.
Reprinted with permission from . Copyright 2012, American Institute of Physics Fig. 13 Prediction errors of the beam position for calibration (white) and validation samples (black). The applied method is SVD with the first 17 SVD modes. Reprinted with permission from . Copyright 2012, American Institute of Physics The rms errors remain small and comparable for calibration and validation samples as shown in Fig. 2. Being a row-based method, SVD is able to effectively cut the signal size down by two orders of magnitude while retaining the concealed beam position information.