Deconvolution of a Causal Signal Roger Barlow

EM simulations and tracking Bunch wake from bunch and wake Need to deconvolute

Problem Noise gives large high frequency components – meaningless Can remove somewhat with filter but – can we be clever?

Extra information We KNOW that the wake function is a ‘Causal Signal’: W(z)=0 for z<0 Can we tell the deconvolution procedure?

Fourier Transform Write series as (r goes from –N to N) d r =b 0 +  a k sin(kr  /N) +  b k cos(kr  /N) Or d=b 0 +S a + C b To have d r =0 for all (-N+1)<r<0 requires a =( 2 / N ) ( S b 0 + S C b) To have d -N =0 requires b 0 =b 1 -b 2 +b 3 -b 4 … For any b, the causality requirement uniquely specifies b 0 and a

Convolution and Deconvolution Unknown function with coeffts (a,b) Known smearing function ( ,  ) Convolved function (A,B) Where B 0 =2  b 0  0 B j =  (b j  j -a j  j ) A j =  (a j  j +b j  j ) d=B 0 +S A + C B = Qb Where Q is made out of known quantities Find b by minimising  2 =  (f r -d r ) 2 Gives equation linear in b – solve it!

Smoothing Result can still be spiky due to noise We know that the wake function should be ‘smooth’, i.e. not jump around from bin to bin Jumps are OK. Peaks are OK. 2 nd differential is ‘obviously’ wrong

Tikhonov Regularisation Add term d T T d to  2 where T is near- diagonal matrix. (Extra terms needed at r=0) is adjusted by hand to give reasonable results

Example…

Future More study needed to understand workings of process and tuning parameters Use to reconstitute delta wakes and apply in Merlin, etc