COUNTING STATISTICS
This section contains a discussion on some of the properties of a counting channel comprising a detector, logarithmic countrate meter, and period meter. The properties considered are:
a. Variance of the countrate indication,
‘ b. Probability of false trip from the countrate indication,
c. Variance of the period indication, and
d. Probability of false trip from the period indication.
The derivations are based on the assumption of constant average counting rate (infinite period), but the results should be valid for finite periods greater than the longest timeconstant in the system.
2. 1 VARIANCE OF THE COUNTRATE INDICATION
The logarithmic countrate meter considered is of the multiplediode pump type (Cooke^ Yarborough), of which one section is delineated in Figure 21, and in which R is the same for
Figure 21. One Section of MultipleDiode Pump Type of Log CountRate Meter
each section, C^/C ^ is the same for each section, and of any section is — Jq ^f °* preceding section. That is,
(Rcf)k = 10(RC()(k+D
and
(RCT)k — 10(RCT)(k,
where к is the section number and starts at 1 for the section with the lowest frequency breakpoint. All sections of the multiple pump are driven from the output of a scaleoftwo circuit.
Let the average counting rate, r, at the input of the scaleoftwo be such that the output of the k— diode pump is onehalf of its saturation value; i. e.,






Then individual pulses from the (k1)^, k—, and (k +l)^ pumps are






and






respectively, where Tk ^ ^RCf]k, тк = (RCjk , and V is the voltage swing of the output of the scaleoftwo. To obtain these expressions, note that the charge delivered to C, each time the scaleoftwo output swings in the positive direction, is
R (1 + 0. 5 rT)
and this charge leaks off through R with a timerconstant, r, hence,
» . VT—eVr.
t (1 + 0.5 ГТ) ‘
By combining Equation (27) with the relations,
. 05 r T(k_ 1} » 1 ,
0. 5 r Tk. = 1 ,
, . 0 5 rT(k+ 1) <<1 ’
Equations (24), (25), and (26) are obtained.
Now rewrite Equations (24), (25), and (26) as
An individual pulse from the log countrate meter is a fraction, A, of the sum of these three pulses:
AvfeV (213)
(Actually, the total pulse is a fraction of the sum of the outputs of all of the individual sections; however, the terms below (к — 1) and above (k+ 1) contribute an insignificant amount of energy to the total pulse and can be neglected. )
2
Now the variance, a„ , of a voltage, S, that consists of a linear superposition of randomly •Э (11
arriving pulses of the form v (t) and an average arrival rate of 0. 5 r is given by’ ‘
(Equation (214) is true for pulses whose arrival satisfies Poisson’s distribution; pulses from a scaleoftwo do not. However, it is shown in Appendix A that the variance obtained by the use of Equation (214) is in error by less than a factor of 2 and is on the large side.) Substitution of Equation (213) into Equation (214) yields the variance,
and this can be combined with Equation (23a) to obtain
The average voltage from one diode pump is given by
0. 5 r VTt 1+ 0. 5 r Tt ’
It should be remembered that the manner in which Equation (218) was obtained imposes the following restrictions:
a. It is valid when the counting rate is such that the k— pump is half saturated,
b. It is approximately valid for noninteger values of k,
c. It is too large by a factor less than /2 because of the slight regularity in the output of the scaleoftwo, and
d. It is invalid for к = 1 or к = N (where N is the number of pumps in the circuit).
For к = N (high count rate, N— pump half saturated, all other pumps saturated), the
standard deviation is
The average output voltage is
S
and the fractional standard deviation is
This expression is subject to restriction c. of the preceding paragraph.