I have tried to examine the different timing resolutions (as a result of

using RAPCAL in conjunction with either Andrew Laundrie's or Gerald

Przbylski's cable filter circuit).

(note, Andrew Laundrie emailed Nobuyoshi that we might be better off if we

used his specified component values, instead of using components which

were more readily available).

Nevertheless, here is a brief outline of what I do, followed by the results

to date.

The setup is described at

http://www.amanda.wisc.edu/krasberg/rapcal/index.html

Example plots are located at the bottom of this webpage.

I perform individual time calibrations using RAPCAL (500 time calibrations

per experiment).

The raw output which each time calibration produces is then read in and

analyzed. One of the things the RAPCAL code produces is the raw round-trip

time for the DOR card (in clock ticks).

Additionally, RAPCAL produces a histogram of a square-wave pulse.

I use the first 20 bins of each RAPCAL calibration to calculate the

"event pedestal". Each RAPCAL histogram is then followed (after the pedestal)

as it rises to its maximum, and when the pulse starts falling the bin number

is saved (one bin after the maximum). Also, the bin number is saved one

bin before the minimum is reached.

The part of the RAPCAL histogram bordered by these two bins is then

used in a 3rd degree polynomial fit. The fitted curve parameters are

then used to solve for the "fractional bin value" which corresponds to the

moment when the fitted curve crosses the "event pedestal".

This fractional bin value (which represents a time in DOR clock tick units) is

then added to the round-trip DOR time (which is also in DOR clock tick units)

to produce the "DOR round trip sum". This number might not have too much

meaning, except when it is compared to other similarly produced values.

(I calculate the "DOR round trip sum" because the round-trip DOR time is

not always the same for each time calibration, and when it shifts one

way often the "fractional bin value" position shifts the other way - for

this study it is necessary to add the two together.

The mean of the "DOR round trip sum", sigma (the error in a single time

calibration), and the standard deviation from the mean (the error

in the mean for the 500 events) are then all calculated.

This procedure is repeated for each cable filter (Andrew's and Jerry's

filters, and also no filter inserted) and also for each available cable

length (a short 10 meter quad

cable and also a medium length 500 meter quad cable were used). Additionally,

a 2-meter quad cable extender (which is actually a different type of quad

cable from the other two quad cables) is used to increase the length of

each cable and the tests are repeated.

Here are the results:

cable	extender?	filter?	DOR round trip sum	sigma	sigma-bar	difference (expected=0.374)
short	no	andrew	405.756	0.290	0.013	0.351 +/- 0.019
short	yes	andrew	406.107	0.299	0.013
medium	no	andrew	509.923	0.502	0.022	0.370 +/- 0.031
medium	yes	andrew	510.293	0.465	0.021
short	no	jerry	399.902	0.291	0.013	0.390 +/- 0.018
short	yes	jerry	400.291	0.285	0.013
medium	no	jerry	NO COMMUNICATIONS
medium	yes	jerry

RESULTS:

In the table you can see the effect of the 2-meter extender cable in DOR

card clock ticks. Also, you can see (in DOR card clock ticks) how accurate

(with this method) an individual time calibration is (in DOR card clock ticks).

If I understand correctly how long a DOR card clock tick is then I expect

the timing difference when a 2-meter extender is used to be 0.374 clock ticks,

and the data seems to agree with this (it looks like I could have used

more than 500 events each time I ran RAPCAL, to pin the errors down a little

better).

The things Nobuyoshi and I were interested in is how Jerry's filter

circuit compared to Andrew's circuit. I think it is fair to

say that we expected Andrew's circuit to do better because in the

example plots

( http://www.amanda.wisc.edu/krasberg/rapcal/index.html )

Andrew's circuit results in a much wider curve and I for

one thought that the polynomial fit would therefore be quite a bit better

(since more bins would be used in the fit than when Jerry's circuit is used).

In the table above Jerry's and Andrew's circuits can only be compared

using the short quad cable (data using the medium length

quad cable in conjunction with Jerry's circuit could not be obtained).

It turns out that the sigmas for Andrew's and Jerry's circuits are

roughly the same.

How can this be?

I believe the following table provides the answer:

cable	extender?	filter?	sigma	approx # of bins used in the fit	approx vertical displacement during the fit
short	no	andrew	0.290	9-10	620
short	yes	andrew	0.299
medium	no	andrew	0.502	12-13	400
medium	yes	andrew	0.465
short	no	jerry	0.291	6-7	1000
short	yes	jerry	0.285

In the two filter circuits we have, Jerry's circuit does a much better

job at maximizing the vertical displacement (saturation occurs), while

Andrew's circuit instead does a much better job at maximizing the total

number of bins used in the fit.

Consequently, from the available data, the timing resolution is roughly

the same for the two circuits.

To make the timing resolution as small as possible we want to maximize both the

number of bins used in the fit and also the total vertical displacement

of the fit.

Perhaps one of the circuits can be modified to accomplish this?