## MATHEMATICAL DERIVATIONS

Consider the system shown in Figure 3-1. It consists of four components: a detector, an amplifier, a squaring circuit, and an averaging (or integrating) circuit of time-constant T.

The detector and amplifier are linear, and somewhere in the system before the squaring circuit (i. e., in the linear part of the system) there is a component to remove the d-c, Such as a series capacitor or a shunt inductor. The time-varying voltage at the output of the amplifier is designated as V(t). At the output of the squaring circuit the voltage is AV (t), where A is the squaring-circuit constant; i. e., the ratio between its output voltage and the square of its input voltage. The voltage at the output of the integrating circuit, which will be called the signal, is S(t), and is related to the output of the squaring circuit by the differential equation of an integrating circuit of time-constant j

. . dt

Due to the random nature of the input, the value of the voltage or current at any point in the system at time t cannot be predicted or expressed mathematically; however, if simultaneous observations were made at the same point and at the same time, t, in a large number of identical systems, the results could be averaged. This ensemble average is called the expected value of that voltage or current at time t, and can be predicted and expressed mathematically. Also, the square of the difference between each result and the expected value could be averaged to obtain the variance of that voltage or current at time t; this variance can also be predicted and expressed mathematically. .

We will use the brackets < > to indicate the expected value of a voltage or signal.

О

It chould be noted that Equation (3-1) applies to the expected values of S(t)and AV (t) as well as to their instantaneous values; i. e.,

<*<S(t) > + <S(t) > = <AV2(t) > (3_2)

dt t t

Since the system is linear up to the input of the squaring circuit (output of the amplifier), the total effect at any point in this part of the circuit is just the sum of the effects caused by the detection of each neutron. Hence, if the detection of a neutron at time zero produces a voltage pulse v(t) at the output of the amplifier, then the voltage at time t due to the detection of a neutron at time t^ is v(t-t^), and the total voltage is the sum of the individual voltages produced by all previously-detected neutrons.

t

V(t) =53 v(t-tk), tk = — oo

where the kin particle arrives at t^..

This equation can be written in a form suitable for statistical analysis in the following manner. Divide the time axis into small equal intervals of length At, then the detection of a neutron during the nth interval of time will produce a voltage at time t of v(t-nAt), and the total voltage due to all of the previously-detected neutrons will be:

V(t) = ^3 v(t-n л 0, n

where the summation includes only those intervals during which a neutron is detected. If we define a random variable 77 that equals one if a neutron is detected during the n*1*1 time interval and equals zero if no neutron is detected during the n n time interval, then the total voltage at the output of the amplifier can be written as,

. t/д t •

V(t) = 23 ‘vn v(t’n At)’ . (3’.3)

. n = -00

where the summation now includes all the time intervals that precede t.