[Halld-cal] [New Logentry] BCAL time-walk corrections

[elton at jlab.org]

Logentry Text:
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I have started thinking again about functional forms for the time-walk, in particular comparing your functional form from Shaun Krueger's thesis (6.2) with my eq. 5 in GlueX-doc-2618.

>From GlueX-doc-2618, Eq, 5 and 6:

t^w = t - f^w(P/T) + f^w(P_0/T), where
t^w is the walk-corrected time,
t is the time-over threshold for a fixed discriminator threshold T (in units of ADC counts)
P is the pulse height
P_0 is the reference pulse height that determines the time reference.

f^w(P/T) = c1/(P/T)^c2

Note that I am using the pulse height P instead of area A. If the pulse height is available, we should use it, as we know reflected pulses distort the pulse shape as a function of the position z along the module. Of course, we can always use A~15P, but this is an approximation which could be avoided.

>From Shaun Krueger's thesis, we have

F(P) = a + (b/(P+c))^d.

If we make the following identifications:

a = -(b/P_0+c))^d

c1 = -b^d

c2 = d

With these definitions, we see that the two formulas are the same, except for the following:
1. The function f^w is a function of f^w(P/T), and F is a function of F(P+c) -- that is scale vs offset adjustments. f^w gives the explicit dependence of the time-walk relative to thresholds, so it can easily be applied to different threshold conditions. This function is also only valid for P/T > 1, so the range of validity is clear. The parameter T has a well-defined physical interpretation, which c does not. 
2. We also need to define the time reference, i.e. the time when the time-walk correction is zero. The reference chosen for the FADCs is the time at half-height. This corresponds to setting P_0=2T in t^w. It is not so clear to me how to set this offset in F(P). 

It is likely that in practice the two forms will yield similar performance, but for the reasons given I think we should use Eqs 5 and 6 from GlueX-doc-2618 as we proceed to determine these constants.

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