• Including strip-to-strip cross-talk
  • The external pulser data taken at SX5 (see note [4] by S. Durkin et al.) allow a determination of the cross-talk parameters. The shapes of the observed pulser signals are basically the Buckeye amplifier's response to a delta function input charge. To use them to measure the cross-talk for real signals, the pulser data were convoluted with the ion drift distribution and the distribution of the individual cascading electrons (see details in [4]). Then the resulting shape of the signal on the pulsed strip was fitted to the semi-Gaussian distribution [4]. It was later suggested (S. Durkin, private communication) to fit the cross-talk on adjacent strips with the function (s+Z1)/(s+P1)/(s+P0)**5, which just represents the capacitance-resistance circuit for the semi-Gaussian signal on the pulsed strip. The corresponding functional form is given by the function Fc(t):
    
    Fc(t) = buckeye_pulse_full(t, P0, P1, Z1) / N,
    
    where buckeye_pulse_full is a function from the library in [3] and N normalizes Fc(t) to 1 at its maximum for a given set of of P0, P1 and Z1 values. Then the pulser data from the pulsed middle strip and the side strips with the cross-talk can be fitted with the following functions:
    
    

    Q_left(t)		= 0				+ Ct * Fc(t) * Q
    Q_middle(t)		= Q		* S(t)		+ 0	(3)
    Q_right(t)		= 0				+ Ct * Fc(t) * Q

    
    where Q is the charge on the pulsed (middle) strip, which was normalized to 1 at the maximum of Q_middle(t) in the convoluted pulser data provided by S. Durkin. The fitted parameters are the following:
    
    Q - the charge on the pulsed (middle) strip;
    T0, Ts - the peaking time and pulse arrival time of S(t);
    Ct - the cross-talk coefficient;
    T0c, P1 and Z1 - the parameters in buckeye_pulse_full(t, P0, P1, Z1), P0 = 1/T0c;
    Tsc - the pulse arrival time for the cross-talk,
    
    We use t = t - Tsc for the time t in buckeye_pulse_full(t, P0, P1, Z1). In the test-beam and simulation data, the analysis pulse arrival time for the cross-talk was set to the same as for the signal (Tsc = Ts).
    
    There are a total of 8 fitted parameters for 119 points of pulser data, taken in steps of 6.25 ns. The results of the fit are shown in Fig. 2 . The residuals are < 0.035 for the signal and < 0.01 for the cross-talk. The cross-talk coefficient, Ct, was found to be ~ 0.1. The fitted value of the parameter Q ~ 0.98 is slightly biased, probably due to the use of the semi-Gaussian S(t), which does not model well the tails of the central peak (see [4]). The other fitted parameters: P0 = 1/T0c = 0.0334, P1 = 0.03392, Z1 = 0.00512 and the corresponding N = 0.458079, were fixed in the following after their evaluation using the test-beam and simulation data.
    
    The simple (and at the same time tricky) fit of the cross-talk data above was done in ROOT using the TGraphErrors::Fit method with an arbitrary error of 0.01 applied just to illustrate the use of the buckeye_pulse_full(t, P0, P1, Z1) function. Therefore, the errors on the parameters in Fig. 2 should be ignored. For future implementations of the functions (2) and (3), a more advanced fitting method can be used. Also, instead of S(t) and Fc(t), the specialized functions developed by S. Durkin (from the library in [3]) should be used to describe the signal and cross-talk with better accuracy, not only in the pulser data but in the test-beam and simulation data, as well.
    
    
  • The stability of the fitted parameters T0, Q and Ct in (3) was studied using the same pulser data but sampled with a period of 50 ns, shifted 8 times in steps of 6.25 ns in such a way that the maximum of the central peak remained within the same 50 ns long time bin. The parameters P0, P1, Z1 and N in the cross-talk function Fc(t) were fixed. Again, as for the test-beam and simulation data, only four time bins were used in the fit. Results of the fit for each peak position are in Fig. 3 and the values of the parameters Q, Ts, T0 and Ct versus peak position are in Fig. 4 . Note that the pulse arrival time, Ts, follows the peak position almost linearly, while Q, T0 and Ct remain approximately constant (Q = 0.98 - 0.99, T0 = 34.1 - 36.0 and Ct = 0.095 - 0.098).