FIR filtering theory is described in Lars presentation but other documents and presentation from Doris and Kazu are available at Indico search: FIR
It is applied in a way as close as possible to a TELL1 realistic scenario (thanks to Guido, which designed me this algorithm):
You want to apply a filter of order say 5. For every channel, you have to compute:
newADC[i] = g[-2] * ADC[i-2] + g[-1] * ADC[i-1] + g[0] * ADC[i] + g[1] * ADC[i+1] + g[2] * ADC[i+2] this can be trivially written newADC[i] = 1/128 * ( g[-2]*512 * (ADC[i-2]/4) + g[-1]*512 * (ADC[i-1]/4) + g[1]*512 * (ADC[i+1]/4) + g[2]*512 * (ADC[i+2]/4) ) + ADC[i] g[0] is assumed to be one.FIR coef inside the TELL1s must have a binary representation. The real values of the g[k] are of the order of the x-talk coefficients, that is O(0.05). Storing g[k]*512 on a 8 bit signed integer allows to explore the range +127/512 = 24% to -24% by step of 0.2%. Dividing the ADC value by 4 allows to drop the least significant bits which we do not want to enter the computation.
A VeloTELL1CableFIRFilter instance has a given N (x-talk order, defined by the amount of Kmi vector you give in options (see below)) and M (FIR order, given by the M
option). Both have to be 3,5,7 or 9 (means you act on 1,2,3 or 4 chip channel neighbors).
The FIR filter acts only on a per-analog-cable basis, it cannot correct for a pulse that spreads over the boundary of two cables.
InputDataLoc
and OutputDataLoc
tells the algorithm where to read and write the data.
M
is the FIR order, the default value is 9, you acts on -4, +4 neighbors.
DebugCable = x;
prints for the given cable the ADC values before and after the FIR filter.
Tell1List
and coefficients options Tell1List = { t1,t2,t3,...tn };
is the list of TELL1s to analyze the order matter. All the other TELL1s see their data duplicated to the OutputDataLoc
.
Km4, Km3, Km2, Km1, Kp1, Kp2, Kp3, Kp4
are the x-talk (not FIR) real value coefficients per TELL1s and analog cable. For example Km2 defines all the h[-2] coefficients.
Kxy = { t1_cable0_coef_xy, ..., t1_cable63_coef_xy, t2_cable0_coef_xy, ..., t2_cable63_coef_xy, ..., tn_cable0_coef_xy, ..., tn_cable63_coef_xy };the t1, t2, ..., tn being ordered like in the Tell1List vector. This is of course awful to write by yourself, but VeloXTalkComputer spits you a file with these lines out.
Basic equation to determine the transfer function of the filter to cancel the x-talk : d[n] = sum_-inf^+inf { g[k]*h[k-n] } where d[n] is the kronecker delta g[n] is the transfer function of the FIR, i.e the coefficients for the filter. h[n] is the transfer function of the link, i.e the measured x-talk if the g[n] is non-zero for only M terms, which means FIR filter of order M, then the sum is reduced to: d[n] = sum_(-M-1)/2}^{(M-1)/2 { g[k]*h[k-n] } This can be considered as an infinite series of equations, one for each value of n, that will give the relation between g[n] and h[n]. These equations will have non-trivial solution only for values of n where h[k-n] are not zero for all terms in the sum. Assuming that the x-talk only extends (N-1)/2 bins forwards and backwards, the requirements on n for non trivial equations are: -(M-1)/2 - n <= (N-1)/2 (only first term in sum non-zero) (M-1)/2 - n >= -(N-1)/2 (only last term in sum non-zero) This gives N+M-1 equations, -(M+N)/2+1 <= n <= (M+N)/2-1, that defines the relation between g[n] and h[n]. Writing out the equations explicitly for M=3 and N=3: k=-1 k=0 k=1 n=-2 d[-2] = g[-1]*h[ 1] + 0 + 0 n=-1 d[-1] = g[-1]*h[ 0] + g[ 0]*h[ 1] + 0 n= 0 d[ 0] = g[-1]*h[-1] + g[ 0]*h[ 0] + g[1]*h[ 1] n= 1 d[ 1] = 0 + g[ 0]*h[-1] + g[1]*h[ 0] n= 2 d[ 2] = 0 + 0 + g[1]*h[-1] This system of equations can be put in matrix form Y = (d[-2], d[-1], d[0], d[1], d[2])^T = ( 0, 0, 1, 0, 0 )^T G = (g[-1], g[0], g[1])^T A = { (h[ 1], 0, 0 ), (h[ 0], h[ 1], 0 ), (h[-1], h[ 0], h[ 1]), ( 0, h[-1], h[ 0]), ( 0, 0, h[-1]) } this give the equation AG = Y with M unknowns (the g[n]) and N+M-1 equations that can be solved by the least square method.
-- Main.jborel - 30 Jun 2007