Problems of obtaining random quantities with counters

PROBLEMS OF OBTAINING RANDOM WITH COUNTERS*

QUANTITIES

M. DRIML and Z. KOUTSKII Prague (Received 27 January 1962) § 1. INTRODUCTION

The method of summation with respect to the modulus k is very convenient for the formation of random quantities with physical devices. The essence of the method consists in summing a sequence of random quantities {X(o) t)},Z’, which take integral non-negative values, with respect to the modulus k, and investigating the limiting distribution of this sum?. In other words it is necessary to determine lim P { we.p X(W,~) = j(modk)) “-rdD t=1

forj=O,l,...,k-1

The necessary and sufficient conditions for the uniformity of the limiting distribution of the sum studied were given in the papers [l]-[7]. Thus the conditions under which the equation lim P { 0’ .~x(w,i)-j(modk))=~fori=o,1,...,

n-co

k-l.

t=1

is satisfied are investigated. If the effect of the properties of the technical apparatus is neglected, the method of summation with respect to the modulus k leads, under sufficiently general conditions, to a uniform limiting distribution. But if this boils down to registration of pulses with a random amplitude and arriving at the inlet of the counter at random moments of time, the technical defects of the counter distort the input process, which again can affect the uniformity of the limiting distribution of random numbers, as shown in [4]. Here we propose a simple method of removing these shortcomings by reducing somewhat the rate of formation of the random numbers. 5 2. DEFINITIONS

AND

ASSUMPTIONS

We shall first define a counter. By counter we shall mean a device capable of determining (perhaps inaccurately) the number of pulses appearing at the inputg. If the device can determine the exact * Zh. vych. mat. 2: No. 3, 475-481, 1962. t Here w denotes an elementary event from a probability space (Q, S, P). 5 The definition given here differs from the usual definition -a pulse detector. 497

498

M. DRIML and Z. KOUTSKH

number of pulses at the input, it is called an ideal counter. On the other hand, the counters used (mechanical, electromagnetic, electronic etc.) are not, generally speaking, ideal. Thus, for example, the defects of electronic pulse counters containing bistable chains are revealed in the asymmetric operation of these chains, which have different dead times and different sensitivities to the amplitude of the input pulses. To be more precise, the time after the registration of the pulse, during which the counter does not register any more pulses at the input, and the magnitude of the pulse necessary for passing from one position to another, depend on the position of the chain at the given moment. Counters of other types have similar defects. A mathematical model of a counter was constructed in [4] for the detailed investigation of these phenomena. Here we shall describe this model in a somewhat simplified form, convenient for our purpose. First we shall give some definitions. As is usual in present-day literature on counter-detectors, we shall distinguish two types of counters. In counters of the first kind pulses appearing at the input while registration is impossible do not affect the length of this interval. In counters of the second kind every pulse blocks the device afresh for a certain time and thus prolongs the interval for which pulse registration is impossible. This is clearly illustrated with graphs in [5]. We shall say that a counter is in the statej(j = 0, 1, . . . , k- 1) if it has registered n = j(mod k) pulses before a given moment. We shall differentiate between dead time (in the proper sense of the word) and real dead time. Dead time (in the proper sense of the word) is the time during which the counter cannot register the first pulse after the one just registered. Real dead time is the time during which the counter cannot register any pulses of the given process after the one registered. In counters of the first kind the two terms coincide, but in counters of the second kind they differ. In the latter case the length of real dead time depends essentially on the properties of the process at the input of the counter. We can now describe a simplified mathematical model of the counter. Let pulses from m sources arrive. at the input of a counter with k states. We assume the following : 1) Each of the m sources produces pulses similar in amplitude. But different sources may have pulses of different amplitude; 2) the activity of the lth source can be described by a Poisson process with the value of the parameter A1; 3) the sources are stochastically independent; 4) after the registration of the pulse there sets in a dead time, which depends on the state of the counter at the moment of pulse registration.* * In [4] it is assumed that dead time is a random quantity, the distribution of which depends on the state mentioned of the counter. Here we shall confine ourselves to a special case, in which further assumptions of stochastic independence mentioned in [4] are fulfilled automatically.

Obtaining

random

quantities

499

with counters

Deviations of the limiting distribution from uniform distribution are established in [4]. The author uses the following scheme of formation of random numbers: / Sources I----$deal

detectorl----_,I

counter /

Scheme I

At the output of the counter we obtain a random number determined by the state of the counter after a definite, sufficiently long, working time. Uniform limiting distribution can be obtained even when the counter has technical defects by suitably changing scheme 1. The altered scheme is as follows : / Sources /-+/Ideal

detector/-+/Bistable

chainI----

Scheme 2

If it is a case of the output of noughts and ones (and we shall consider this case only, because it can be generalized to the case of k states without difficulty), the second scheme is, in essence, supplemented by a detector of “1” + “0” transitions, because here the counter and bistable chain are equivalent. $ 3. PROBABILITY CALCULATION

OF THE STATE “1”

FOR FINITE t IN SCHEME 2

We introduce the notation & and & for dead time (in the proper sense of the word) of the counter in the states “0” and “1” respectively, /I,-,and ,$ for dead time (in the proper sense of the word) of the bistable chain in the states “0” and “1” respectively. We shall assume

which is fulfilled without difficulty in technical apparatus. The probability that the counter in scheme 2 is in state “1” is equal to the probability that the bistable chain has registered n pulses, and n s 2 (mod 4) or II 3 3 (mod 4). We now denote by p&i) the sum of the parameters of sources with an amplitude large enough to produce the transition “0” -+ “1” (“1” + “0”) in the bistable chain. Let Hi(r) be the distribution function of real dead time of a bistable chain in the state j(j = 0, 1). If the bistable chain has the properties of a counter of the first kind, then Hi(t) has a unit discontinuity at the point ~j; if the bistable behaves like a counter of the second kind, the following equation is satisfied Hj(t)=O Let Fj(t) denote the distribution

for t
(j=O,l).

function of the random quantity yj,n(W), the

500

and Z. KOUTSKII

M. DRIML

time of stay of the bistable chain in the state j, when it is in it for the nth time for all n = 1,2, . . . . and j = 0,l; except for Y~,~(o), for which we have P{o: yoJ(O

< t} = c,(t) = 1 - e-rot.

Let us introduce the functions c,;t> = 1 - e-rlt,

G(t, = F,(t) * F,(t),

where F,(t) *F,(t)

= ‘s F,(t - x, dF(x) = 5 F,(t - x)dF,(x), 0

0

and the notations Qlo(s) = 19

CJJ~(S> = 5 e-“‘dF1(t),

q(s) = 7 eMSrdG(t)

0

0

for all complex numbers s with a non-negative Evidently the following equation is true:

real part.

E;(t) = Hjct) * Cj(t’*

In [4] it is proved that Laplace transformation

of the function

P{t, 2p +j>, p = 0, 1) . ..) j = 0, 1) which expresses the probability that the bistable chain will register in a time interval (0, t} not greater than 2p +j pulses is given by the formula L(P{t,2p

+j))

= 7 eWS’P{t, 2*9+j}dt

= $

1- *

IV(s)YYj(S)}.

c

0

It is also proved that, if P’(t, 2p+j} denotes the probability that the bistable chain will register exactly 2p+ j pulses in the time interval (0, t}, the following equations are obtained :

W’{c 0)) =

Q&, p=

UP’{& 2r + 1)) = +*

[p,(41P[1 - %(41?

Denoting by F{t, l} the piobability “1” at the moment t, we obtain

+

[cp (s)]2(pll!s>-

1,2 9 ***,

p=o,1,2

(3.1)

) ... .

that the counter in scheme 2 is in state

6.7
[P’(Wl - CPIW) + ...I

Obtaining random quantities with counters

and thus (3.2) For a bistable chain with the properties of a counter of the first kind we have y,(s) =

ps, uople-

Substituting

(3.3)

s(8o+Pd

qJw = (s+Po)(s+fh)

’

(3.3) in (3.2) we obtain (3.4)

Using the expansion of this function in a power series, we can find its original by term by term conversion of the series. § 4. LIMIT OF THE FUNCTION

ij(t, I} AS t + co

Since there are serious difficulties in the calculation of the values of the function p”(t, l} for finite t, we shall study limj{tj t+cC

l}.

With the help of theorem 14 in [8] (see chap. VIII) it can be shown that lim P”(t, l}+. r-1oJ But the proof that the assumptions of this theorem are fulfilled is rather complicated. We shall therefore prove the convergence of the function F{ t , l> by estimating the integral which determines the conversion of the Laplace transformation. We write (4.1) Since the function R(s) is continuous at the point 0, we have limR(s) = +. S-+0

Formula

~4.2)

(3.4) can now be written as (4.3)

502

M. DRIMLand Z.

KOUTSKII

To prove the convergence it is sufficient to show that

L-1(;*

[R(s? -+7(l),

(4.4)

since it follows from this that ~{t,l}~o(l)

+ *(l

- eerot)

lim F(t, 1) = -+.

(4.5)

t-cc According to Theorem shoti that the integral

1 of [9], chapter XV, to prove (4.4) it is sufficient to

(4.6) converges absolutely.

It is clear that lim R(iy) = 0, Y-+W

and therefore there exists a Y > 0, such that jR(iy) - +I< 1 for all y > Y. It follows that co

eitY -~1

sl iv

Y

PO

iv+irb

R(Q) -

1 dy, +)ldyd POT’ y y I/y"+cca2

and the last integral converges. The absolute convergence of integral (4.6) for the negative part of the y-axis is verified in the same way. Using the fact that there- is a limit lim _I

R(iy'_

y+s!Yl

’

LI

21’

we infer that the integral (4.6) does converge absolutely, which was to be proved. 5 5. THE CASE OF A COUNTER

OF THE

SECOND

TYPE

KIND

In exactly the same way as in the case of a bistable chain of a type of counter of the first kind we can prove that lim E;{t, l} = -k f-rcu

(5.1)

for the case when a bistable chain behaves like a counter of the second kind. For this it is sufficient to take

~o(l~e-Bl(s+~l)-Bo(s+Po) yJ (4 = (s+poe-80(s+Po) )(s+ple-Bl(S+P1).)

and substitute these expressions in (3.2).

Obtaining

random

quantities

with counters

503

5 6. CONCLUSIONS

(4.5) and (5.1) show that by making the counter more complicated it is possible to remove errors caused by technical defects in the counters (for example, asymmetricity of bistable chains). Certain technical difficulties concerning, above all, an ideal detector of the transfer of the bistable chain from one state to another arise in the construction of such an apparatus. Pulses arriving at the inlet of the bistable chain, which may not transfer the chain to another state but can, being amplified, pass on to the subsequent elements of the counter, must be removed. Another disadvantage of the scheme is that the rate of formation of random numbers is reduced (approximately by half). In spite of the difficulties mentioned, we can still consider this extended scheme suitable for generators of random numbers that have to produce a uniform distribution of random numbers very accurately. Equations

Translated by PRASENJITBASJ

REFERENCES 1. DVORETZKY, A. and WOLFOWITZ, J., Sums of random integersreduced modulo m. Duke, Math. J. 18: No. 2, 501-507, 1951. 2. HGRNER, S., Herstellung von Zufallszahlen auf Rechenautomaten Z. angew. Math. und Phys. No. 1, 26-52, 1957. 3. KOUTSKY, Z., Einige Eigenschaften der Module k addierten Markowschen Ketten. (Transactions of the Second Prague Conference on Information Theory, Statistical Decision Functions, Random Processes.) 263-278, Prague 1960. 4. KOUTSKY, Z., Theorie der Impulszlihler und ihre Anwendung. Apl. mat. 7: No. 2, 116-140, 1962. 5. TAKACS,L.,On a probability problem arising in the theory of counters. Proc. Cambridge Philos. Sot. 52: No. 3, 488-498, 1956. 6. ULLRICH, M., and URBANM, K., A limit theorem for random variables in compact topological groups. Colloquium math. 7: No. 2, 191-198, 1960. 7. GNEDENKO, B. V., On the theory of Geiger-Muller counters. Zh. exp. teor. fiz. 11: 101-106, 1941. 8. GARDNER, M. F., and BARNES, J. L., Transients in linear systems. Vol. 1, New York 1954. 9. DOETSCH, G., Handbuch der Laplace-Transformation. Band I, Base1 1950.