文档介绍：Outline ? The analytic solution of the signal with synchro-betatron coupling. ? Discuss using ICA to separate synchrtron sidebands. ? Apply this method to the singals with both synchro-betatron coupling and horizontal and vertical betatron / coupling. ? Summary Signal with Synchro-betatron Coupling ? If there is synchro-betatron coupling, the signal is written like this way where is betatron tune, synchrotron tune, a and b are parameters. 1? s? 2 ( / ) ij i M j ? ?? ? i =1, 2, … M, M is the number of BPMs. j=1, 2, … N, N is the number of tracking turns. 1 cos( sin( )), ij ij s ij x a b ?? ??? ? Using the following formulas, We obtain the following analytic result 0 2 1 2 1 1 cos( sin ) ( ) 2 ( ) cos(2 ) sin( sin ) 2 ( )sin(2 1) kkkk z J z J z k z J z k ? ?? ??????? ?? ??? 0 1 1 1 1 { ( ) cos ( )(( 1) cos( ) cos( ) )}. ij ij k k s ij s ij k x a J b J b k k ??? ???????? ?? ???? Where J is the Bessel function. ? Estimate the value of b Betatron tune is given by Compare with the signal, we find out that / . x s b C ? ??? cos( ) sin( ) / x x x x s x x s s dC dC ?? ??????? ??????? ????? ? Use AGS as an example. Read the data from mad file of AGS, 3 4 10 , 10 , C . B s x ? ?? ?? ? ???? Take , the value of b is 1% B ? ??? Compare PCA and ICA ? To see the virtue of ICA, let ’ pare ICA with PCA. ? Choose the following parameters: a=, b=, =, =, M=100, N=1000. ? Use both PCA and ICA to analyze the original data, we get the following two graphs. These graphs are the fft on source signals obtained by PCA and ICA s