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Quantization of time intervals in the registration of a non-stationary poisson process
N U C L E A R I N S T R U M E N T S AND METHODS
QUANTIZATION
123 (I975)
54[--550;
© N O R T H - H O L L A N D P U B L I S H I N G CO.
OF TIME INTERVALS IN T H E REGISTRATION OF A NON-STATIONARY P O I S S O N PROCESS FRANJO JOVIC " Rudjer Bogkovid" Institute, Zagreb, Yugoslavia* Received 21 May 1974
The registration of a non-stationary Poisson process is considered for conversion systems with fixed or variable quantization time. The quantization time is either stochastically or deterministically dependent. Probabilities of registration events are determined for conversion systems with arbitrarily given stochastic quantization time and for conversion systems with linearly dependent quantization time. Data losses in registration systems with fixed dead time and with constant quantization time are given. The analysis and results presented are valid for a registration system with an ideal memory.
1.
Introduction
The time dependence of the density function of a non-stationary Poisson process can be determined from several repetitions of measurements and accumulation of data. Digital registration systems are very efficient for these purposes, especially when connected to a digital computer. Some kind of data quantization should be performed in order to make digital registration systems capable of data sorting and reduction. Two principal methods for the quantization of time intervals have been established during the last few years1): - counting of clock pulses during the time interval under measurement, and - conversion technique, comprising for example, time-to-pulse-amplitude conversion followed by pulse-amplitude-to-digital conversion. The quantization time interval and the resolution are the most interesting values for the registration of a non-stationary Poisson process. The quantization time begins with the end pulse of a quantized interval and ends with the generation of a digital number that represents the quantized value of the time interval. The resolution of the quantization time interval is an important parameter in measuring fast-changing non-stationary processes. The resolution obtainable by the two techniques is of the order of magnitude between nanoseconds and tens of picosecondsl). Regardless of the method used, the quantization of time intervals gives rise to changes in time relations defined by the input process to the conversion system. The input process to the conversion system is a non-stationary Poisson process after detection and shaping have been performed. A block circuit of the registration system is shown in fig. 1. When the inter-arrival time of input events is comparable with the quantization time of the converter in the conversion system, a larger number of converters has to be used. When all converters are occupied, all events newly arrived are refused until the first converter completes the conversion and transfers the datum to the memory of the registration system. The purpose of this paper is to analyse the registration of events, by different conversion systems, assuming that the memory of the registration system is ideally fast and large enough to take over all data processed by it. * Now with the " R a d e Kon~.ar" Institute, Zagreb, Bagtijanova bb, Yugoslavia.
I I II
I
I.put proce~
l Det,'ctinq &
] = l
Co ve ,.o. [ I
Memory
shaplaq sgMem
dala ~orhn9
I Fig. 1. Block circuit of the registration system.
541
542
F. JOVi6
2. Quantization time of different converters As mentioned before, the time interval that begins with the end of the measured time interval and lasts until the end of its conversion period is caned the quantization time. Let us briefly determine the quantization time of some of the main conversion systems used for the registration of a non-stationary Poisson process. The counting technique makes use of either the differential or the integral method of clock-pulse counting. The differential method of counting consists in counting the inter-arrival time of two successive pulses of the input process. The integral method consists in counting the moments of arrival with respect to a fixed pulse. In both methods the quantization time is practically constant, since it is determined by delays in logical circuits of the conversion system. When the difference between a data pulse and the next clock pulse in the counting method is quantized using the conversion technique or time nonius3), then the stochastic quantization time has to be added to the fixed quantization time. However, because of the comparatively long conversion time of the conversion technique, the stochastic quantization time is dominant, so the fixed quantization time may be neglected. Two basic conversion techniques are known: conversion of time intervals into pulse amplitudes followed by conversion of pulse amplitudes into digital numbers, and direct time-to-time conversion. In the latter conversion method, start and stop pulses are propagated along delay lines with different delay time and the coincidence of both pulses is controlled on taps from the line by means of fast coincidence circuits4). For converters of the Wilkinson type 3) the quantization time is given by the expression rw = %0 +tcx,
K>I,
(1)
where Two and ~ are constants, and x is the duration of the quantized time interval. For converters working on the principle of successive approximation the quantization time is constant. Converters with direct time-to-time conversion have a quantization time equal to (2)
r D = TD0 "~ g ,
where ZD0 is a constant and g is a stochastic value. The density function o f # depends on the way of tapping the delay lines of the converter. The constants ZD0 and Two can be neglected, as compared with other values in eqs. (1) and (2). 2.1. TRANSFORMATION OF THE PROCESS IN THE CONVERSION SYSTEM
The process obtained at the output of the conversion system depends on the input process, quantization time, number of converters used, type of conversion performed, as well as on the way of feeding data to the memory. Let us suppose that data are fed to the memory immediately after the end of quantization. Let us also suppose that the input process is a non-stationary Poisson process generated after the passage through detection and shaping circuits which possess a fixed dead time ~. Let us further suppose that only one type of converter is used in the conversion system. When the quantization time is a stochastic variable, as given for example in eq. (2), then its value is restricted within certain limits, i.e., (3)
0 ~ g ~ rma x . I
poi s~on
/~ ~
proce-~s OzA~ ~ w
[ Detect & shape
Cancers~on
I I I I I
Fig. 2. Equivalent circuit of the registration system.
Instont
NON-STATIONARY POISSON PROCESS
543
Tile deterministic part of the quantization time is determined by delays in the logic circuits of the converter and it usually holds TD0/Tmax ,~ 1 . The probability density function of the quantization time can be a positively defined function of any shape. The functional dependence of the probability density function depends on the conversion principle used. Let us suppose that the probability density function g of the quantization time is given by the known functional dependence g = f (x), 0 ~< f ( x ) ~< T m a x . (4) An equivalent circuit of the registration system is schematically shown in fig. 2. The input process is fed to the detection and shaping system with fixed dead time. The conversion system with an arbitrary number of converters and a conversion time in the limits 0 < A t < w is connected to the instant memory, i.e., a memory with a negligible dead time and with an arbitrary number of locations. Let us consider the process at the output of the quantization system in the time interval (t ~, t 1 + r). Let us divide the interval ( t - w , t + r ) into non-overlapping intervals of duration ~. The probability of detection of the time interval in the ith time interval of duration a is given by eq. (5):
w
l.-d s
(5)
2js-~
assuming that -
] I s+~
(x-s)
2(x) dx < 1,
otds
and fl +" ;t(x) dx < 1 . Here s = t - w + ( i - 1 ) ~ and 2(t) is the density function of the input Poisson process. If an event of the input process is detected in the ith time interval of duration ~, then the probability of detection in the next, (i+ 1)th, time interval ~ is given by 5,6) • p,+l_,_w = P [ l ' s + ~ , s + 2 a ] ,
F f~+2~
= 1-exp]-
|
]
2(x) dx .
(6)
Js+k~
L
+ , that this detected If an event is detected in the ith time interval ~ after the moment t - w , the probability P ,,' ,t+, I~
w
LI
V
-I vii
I-
,F
io [ f t÷Z"
7"
._3
Fig. 3. Probability of conversion of the detected event.
544
F. JOWd
event will be converted in the time interval (t, t + z ) is equal to (fig. 3) g dx,
for
0 < i~ ~ ~,
g dx,
for
~ < i~ ~< w,
w - ia t,t+z
w--i~+~
(7) 9 dx,
for
w < i7 < w + r ,
for
u, + , < i~.
The probability krs; , t t+ . . a t an event detected in the time interval (s, s + ~) will be converted in the interval (t, t + r) s+,. . m is equal to the product of probabilities given by eqs. (5) and (7), i.e., Pi-w kp,,t+~ =
I °- la dw
i i w-ia Pt-=
a s , s+=
g dx,
for
0 < i~ ~< r,
g dx,
for
z < i= ~< w,
9 dx,
for
w < i~ ~< w + r .
(8)
,J w - i a + r
PI-~
f w-ia+r do
The probability that one registered datum will be quantized at the output of the quantization system in the time interval (t, t + z) is equal to the sum of probabilities of all events that can separately realize this event, i.e., M kpt,t+r
1a t - w
~
Z
m =
kpt, t+z
i=0
#S,s+i~,
[ b/) q- g
--
I
0~
(9) int"
The probability that two registered events will be quantized at the output of the quantization system in the time interval (t, t + z ) is equal to the probability that the quantization of any two registered events will fall into the interval (t, t + , ) which means that it will be realized with each pair of M possible events, i.e., ~-,-w
=
,-,,, i= 1
~
,.
\j=i+
-,-w+./~,).
(10)
1
Here the expression for the probability of registration in the interval preceding the ~ interval during which the event was registered, is given by eq. (6). 2.2. TRANSFORMATION OF THE PROCESS WHEN THE QUANTIZATION TIME IS DETERMIN1ST1CALLY DEPENDENT The output process y ( t ) from a quantization system which uses converters with deterministically dependent quantization time is given by the deterministic transformation T of the input process, i.e.,
yU) = T{x(t)}.
(ll)
For converters of the Wilkinson type, the operator T is given by eq. (l): the constant two may be neglected as compared with the second member of this expression. For the differential mode of work, the duration of the interval x~ between two successive time intervals xl and xi+ 1 at the input is equal to (fig. 4) x; = (~c+l) Xi+I--KXi,
Xi, Xi+ 1 >1 ~.
(12)
Let us consider the process at the output of the quantization system when converters of the Wilkinson type are used and when the input process is a non-stationary Poisson process after the passage through the system with a fixed dead time ~. Let us suppose that the maximum conversion time is equal to w. Only the data generated in the intervals completed by the time instant t - x w can be quantized in the time interval (t, t + r ) (fig. 5); the
NON-STATIONARY
POISSON
Inpu~ proces5
545
PROCESS
O,lCpu/ p~oees~
It-
Fig. 4. Input and output process to the conversion system with Wilkinson converters. Xs
:" "~,
r
+/
~
x
2 ='x ( g-" ~) - T
Fig. 5. Geometrical interpretation of the probability of conversion with the Wilkinson type of converters. first event of the quantized interval may occur at any m o m e n t t - x , xl > x > x 2
(fig. 5), where
xl = w ( l + ~ : ) ,
(13)
x2 = ~ ( l + K ) -- z.
(14)
The differential of probability that the quantization of a datum will be completed in the interval (t, t + z) is equal to the p r o d u c t o f three probabilities: the beginning o f the interval will be generated in the interval ( t - x , t - - x + dx), no event of the input process will be generated in the intervals ( t - x - ~ , t - x ) and ( t - x + x/[1 + x], t--x+cO, and the end event of the input d a t u m will be generated in the interval ( t - x + x / [ l + x ] , t - x + x / [1 +~c]+~/~c), i.e., (fig. 5)
2 ( t - x ) dx exp " Ap'.'+~
-
2(z) dz -
-
- x-~,
1-exp 1
2(z) dz j,-x+~ 2 ( 0 dt
dt-x+x/(l+r)
(15)
546
F. Jovx d _J o~
o<
~/~
---
--~~
X/.
___
~
~ ---g'~
t ,2-
Fig. 6. Probability of generation of the second event upper limit
Fig. 7. Probability of conversion of two data.
The probability of generation of the second event decreases when x approaches x z, i.e., when x = x2 = ~(1 + x ) - r, keeping its full value until x reaches x~ =e(1 + r) (fig. 6). It follows from the geometric relations in fig. 6 that a value x~ can be taken for the upper limit of x" Xg is equal to xg --- ~(1 + x) - ½z (1 + 1]~;).
(16)
Since the output from the quantization system is realized by any event with the probability given by eq. (15), the probability that one interval at the output of the quantization system will be completed in the time interval (t, t + z) is given by kpt, t+, Kw
2(z-x)
Ixt,
1-exp
dx~
x exp
-
2(z) dz z-x+x/(l
-
2(z) d z -x-~
x
+~)
2(z) dz
dx.
(17)
dz-x+~
The end of quantization of two data in the time interval (t, t + z) realizes two input events which are not necessarily successive, but which should not overlap and the quantization of which ends in the time interval (t, t + z).
NON-STATIONARY
POISSON
PROCESS
547
Intervals A and B given in fig. 7 are successive, while intervals A and C in the same figure are non-successive, but both of them realize two output data in the interval (t, t + r). The differential of the probability that the quantization of two intervals will be completed in the interval (t, t + z) is given by the expression d 2k ~p tt ', tt + _ K~ w
-- -
abcd
2(x) dx + ab 2(x) dx
x-x~(1
+K)
I
a(u) b(u)
2(u) du
(18)
,d x g
where a =exp
-
2(z) d z -
2(z) dz ,
-x-c~
F-
b = 1-exp
L
I-
c = exp
't-x+[x/(1 t--x+[x/(1
L
[x -
F,-x+tx/~+,,)1+~/,,2(z)
dz ] ,
Jt-x+x/(l+K)
+K)]+(t/x)+~t+e 2(z) +K)]+(~/~)+=
1
dz ,
F - i,_x+t~/. +~)j+cz./~)+:+~2(z)
d = 1-exp
e =
dt-x-a
dz ] ,
,)t--x+lx/(l+K)l+(r/~c)+=+e
(x/(1 + x)) -
(~/~) -
~]/(1 + K).
All events that realize the end of quantization o f two data at the output of the quantization system in the interval ~pt, t,*~
40.t~14L-N*(
~ ,I t,Jv
2l"i'-H.,~
4 [-M.C
i
402
2
¢0"3
I
i
I
]
t - b¢.~
I
!
_
/
i0-4 IO ~
20 ~
30 ~
~0~<
50.c
GO~
i
"lO~
i
80~
_+
_
i
gO~
Fig. 8. Probability occurrence of one event and two events at the output of the quantization system; example.
__loose
548
v. J o v [ 6
(t, t + z) are determined by the integration of eq. (18) in the limits t-w(~c + 1) and t - v , where v is given by the expression U =
X g -]- ( T / K )
-~ ( X g / K ' ) ~- ( V / K 2 ) ,
(19)
i.e., t- w(~+ I )
k pt,t +r 2 t,t-~u'
:
I
l(dpt, t + r 2 t,t--Kw *
(20)
Jt--v
Example. Let us determine the probability of occurrence of data at the output of the quantization system in successive time intervals of duration ~, when the probability density function gp(X) of the quantization time is constant and the number of converters is unlimited. Let the maximum quantization time be w - M ~ , M = 10, 20, 30 and 40. Let the density function of the input process be equal to )o(t)= l0 e x p ( - t ) . Let the dead time of the detection system be equal to ~=0.01 (s). The probability of occurrence of one event and two events is given in fig. 8. 3. Data losses in the quantization system with constant quantization time and limited number of converters Data losses occur when all converters in the quantization system are occupied and a new event appears simultaneously at the input. Constant quantization time is a feature common to converters. The advantage of converters with constant quantization time is the simplicity of the quantization process. Let us determine data losses when converters have constant quantization time w, supposing that the number of converters is limited and that w is much larger than the dead time ~ of the primary detection system. Let us suppose that of m converters k converters (m > k) are occupied at a certain moment t, in the interval of measurement t~ ~
p~+l t+~
:
P tk+l{p(o;t,t+c 0
[l_Po(t,t+cO ] + [p(1;t,t+oO] Po(t,t+o~) } +
+ p~+2p(o;t,t+~) Po(t, t+~) + Pkt P(1;t, t + a ) [l - P o ( t , t + : 0 ] .
(21)
The probability that all converters of the quantization system are free at the moment t + ~ is equal to pO+~ = pto P ( 0 ; t , t + c 0 + Pt1 [ P ( 0 ; t , t + c 0 ] Po(t, t+~).
(22)
The probability that all m converters are occupied is m . P,+~ = P,m {P(O,t,t+o 0 [1-Po(t,t+oO] + P ( l ' t , t + a ) } +
+ P t +1 ( P ( 1 ; t , t + ~ ) [ 1 - P o ( t , t+c()]},
(23)
where the probability of data output from the quantization system P0 (t, t + ~) is given by the expression P0(t,t+a) = P{1;t-w,t-w+a}
- gP~....
(24)
where gPt-w is the probability of data loss in the quantization system in the interval ( t - w , t - w + a). The probability of data loss in the interval (t, t+c0, because of the finite number of converters, is equal to up~ = P ( 1 ; t , t + a ) P t [ 1 - P o ( t , t + 0 0 ] .
(25)
Eqs. (21)-(25) are derived under the assumption that the influence of registration probabilities in each ~t interval on the registration in the neighbouring ¢t intervals may be neglected. This assumption is usually fulfilled because the number of converters is much smaller than the number of successive non-overlapping intervals. In this case the number of combinations which realize the occupation of a certain number of converters, with at least one case
NON-STATIONARY
o,g"
POISSON
549
PROCESS
.&
0,8-
0,7-
6o,i.~ 0,6
2A
H= 30
/' \' i
\
,l// i;
,v ~
,. .("
/
~"
~-
" ~
--
"' -
M= 20
.,
"--:~ ~'~'I42
,{ ~ - .
2,'~.,.. ?-'< \~,2c
,~
'. ....
"• .,: :.. ::l,~g ~
~'P,~
" ~'..~'~8-
..~ . .
'" . . .
/
,:::::./, :. ',.,o
0,z
. . . . . " .... ..... ' ...... " ~ " ~
I t0
20
"li/" SO
= "~"
"" ~ , a o
0,,t '"~
~/" ~0
50
60
70
gO
90
I00 i,~
Fig. 9. Probabilities o f occupation o f one, two a n d three converters of the quantization system; probability o f data loss.
of two registered events in two neighbouring ~ intervals, occurs less frequently than the case when registered events are separated by one interval of width ~. Example. Determine the probability of occupation of three converters of the quantization system and the expected number of data losses in the time interval (0,100~) when the conversion time is constant and equal to w = Ma~, M = 10, 20 and 30, provided the detection system has a fixed dead time ~ and the input non-stationary Poisson process has a density function 2 ( t ) = 10 U(t). The quantization system is connected to the instant memory and all converters are free at the initial moment. The result is given in fig. 9.
550
F. J O V I 6
References 1) 2) 3) 4)
E. Gatti, Time coding, Colloq. Intern. l'l~lectronique Nucl6aire, vol. 2 (Versailles, 1968) !3. 79. j. Hahn et al., IEEE Trans. Nucl. Sci. (Febr. 1969) 154. E. Kowalski, Nuclear electronics (Springer-Verlag, Berlin, 1970). M. Feran et al., A weighted chronotron for time-digit conversion, lspra Nuclear Electronics Syrup. (CID Brussels, 1969) pp. 127 135. 5) F. Jovid, Nucl. Instr. and Meth. 111 (1973) 519. ~) A. Papoulis, Probability, random t,ariables and stochastic processes (McGraw-Hill, New York, 1965).
QUANTIZATION
123 (I975)
54[--550;
© N O R T H - H O L L A N D P U B L I S H I N G CO.
OF TIME INTERVALS IN T H E REGISTRATION OF A NON-STATIONARY P O I S S O N PROCESS FRANJO JOVIC " Rudjer Bogkovid" Institute, Zagreb, Yugoslavia* Received 21 May 1974
The registration of a non-stationary Poisson process is considered for conversion systems with fixed or variable quantization time. The quantization time is either stochastically or deterministically dependent. Probabilities of registration events are determined for conversion systems with arbitrarily given stochastic quantization time and for conversion systems with linearly dependent quantization time. Data losses in registration systems with fixed dead time and with constant quantization time are given. The analysis and results presented are valid for a registration system with an ideal memory.
1.
Introduction
The time dependence of the density function of a non-stationary Poisson process can be determined from several repetitions of measurements and accumulation of data. Digital registration systems are very efficient for these purposes, especially when connected to a digital computer. Some kind of data quantization should be performed in order to make digital registration systems capable of data sorting and reduction. Two principal methods for the quantization of time intervals have been established during the last few years1): - counting of clock pulses during the time interval under measurement, and - conversion technique, comprising for example, time-to-pulse-amplitude conversion followed by pulse-amplitude-to-digital conversion. The quantization time interval and the resolution are the most interesting values for the registration of a non-stationary Poisson process. The quantization time begins with the end pulse of a quantized interval and ends with the generation of a digital number that represents the quantized value of the time interval. The resolution of the quantization time interval is an important parameter in measuring fast-changing non-stationary processes. The resolution obtainable by the two techniques is of the order of magnitude between nanoseconds and tens of picosecondsl). Regardless of the method used, the quantization of time intervals gives rise to changes in time relations defined by the input process to the conversion system. The input process to the conversion system is a non-stationary Poisson process after detection and shaping have been performed. A block circuit of the registration system is shown in fig. 1. When the inter-arrival time of input events is comparable with the quantization time of the converter in the conversion system, a larger number of converters has to be used. When all converters are occupied, all events newly arrived are refused until the first converter completes the conversion and transfers the datum to the memory of the registration system. The purpose of this paper is to analyse the registration of events, by different conversion systems, assuming that the memory of the registration system is ideally fast and large enough to take over all data processed by it. * Now with the " R a d e Kon~.ar" Institute, Zagreb, Bagtijanova bb, Yugoslavia.
I I II
I
I.put proce~
l Det,'ctinq &
] = l
Co ve ,.o. [ I
Memory
shaplaq sgMem
dala ~orhn9
I Fig. 1. Block circuit of the registration system.
541
542
F. JOVi6
2. Quantization time of different converters As mentioned before, the time interval that begins with the end of the measured time interval and lasts until the end of its conversion period is caned the quantization time. Let us briefly determine the quantization time of some of the main conversion systems used for the registration of a non-stationary Poisson process. The counting technique makes use of either the differential or the integral method of clock-pulse counting. The differential method of counting consists in counting the inter-arrival time of two successive pulses of the input process. The integral method consists in counting the moments of arrival with respect to a fixed pulse. In both methods the quantization time is practically constant, since it is determined by delays in logical circuits of the conversion system. When the difference between a data pulse and the next clock pulse in the counting method is quantized using the conversion technique or time nonius3), then the stochastic quantization time has to be added to the fixed quantization time. However, because of the comparatively long conversion time of the conversion technique, the stochastic quantization time is dominant, so the fixed quantization time may be neglected. Two basic conversion techniques are known: conversion of time intervals into pulse amplitudes followed by conversion of pulse amplitudes into digital numbers, and direct time-to-time conversion. In the latter conversion method, start and stop pulses are propagated along delay lines with different delay time and the coincidence of both pulses is controlled on taps from the line by means of fast coincidence circuits4). For converters of the Wilkinson type 3) the quantization time is given by the expression rw = %0 +tcx,
K>I,
(1)
where Two and ~ are constants, and x is the duration of the quantized time interval. For converters working on the principle of successive approximation the quantization time is constant. Converters with direct time-to-time conversion have a quantization time equal to (2)
r D = TD0 "~ g ,
where ZD0 is a constant and g is a stochastic value. The density function o f # depends on the way of tapping the delay lines of the converter. The constants ZD0 and Two can be neglected, as compared with other values in eqs. (1) and (2). 2.1. TRANSFORMATION OF THE PROCESS IN THE CONVERSION SYSTEM
The process obtained at the output of the conversion system depends on the input process, quantization time, number of converters used, type of conversion performed, as well as on the way of feeding data to the memory. Let us suppose that data are fed to the memory immediately after the end of quantization. Let us also suppose that the input process is a non-stationary Poisson process generated after the passage through detection and shaping circuits which possess a fixed dead time ~. Let us further suppose that only one type of converter is used in the conversion system. When the quantization time is a stochastic variable, as given for example in eq. (2), then its value is restricted within certain limits, i.e., (3)
0 ~ g ~ rma x . I
poi s~on
/~ ~
proce-~s OzA~ ~ w
[ Detect & shape
Cancers~on
I I I I I
Fig. 2. Equivalent circuit of the registration system.
Instont
NON-STATIONARY POISSON PROCESS
543
Tile deterministic part of the quantization time is determined by delays in the logic circuits of the converter and it usually holds TD0/Tmax ,~ 1 . The probability density function of the quantization time can be a positively defined function of any shape. The functional dependence of the probability density function depends on the conversion principle used. Let us suppose that the probability density function g of the quantization time is given by the known functional dependence g = f (x), 0 ~< f ( x ) ~< T m a x . (4) An equivalent circuit of the registration system is schematically shown in fig. 2. The input process is fed to the detection and shaping system with fixed dead time. The conversion system with an arbitrary number of converters and a conversion time in the limits 0 < A t < w is connected to the instant memory, i.e., a memory with a negligible dead time and with an arbitrary number of locations. Let us consider the process at the output of the quantization system in the time interval (t ~, t 1 + r). Let us divide the interval ( t - w , t + r ) into non-overlapping intervals of duration ~. The probability of detection of the time interval in the ith time interval of duration a is given by eq. (5):
w
l.-d s
(5)
2js-~
assuming that -
] I s+~
(x-s)
2(x) dx < 1,
otds
and fl +" ;t(x) dx < 1 . Here s = t - w + ( i - 1 ) ~ and 2(t) is the density function of the input Poisson process. If an event of the input process is detected in the ith time interval of duration ~, then the probability of detection in the next, (i+ 1)th, time interval ~ is given by 5,6) • p,+l_,_w = P [ l ' s + ~ , s + 2 a ] ,
F f~+2~
= 1-exp]-
|
]
2(x) dx .
(6)
Js+k~
L
+ , that this detected If an event is detected in the ith time interval ~ after the moment t - w , the probability P ,,' ,t+, I~
w
LI
V
-I vii
I-
,F
io [ f t÷Z"
7"
._3
Fig. 3. Probability of conversion of the detected event.
544
F. JOWd
event will be converted in the time interval (t, t + z ) is equal to (fig. 3) g dx,
for
0 < i~ ~ ~,
g dx,
for
~ < i~ ~< w,
w - ia t,t+z
w--i~+~
(7) 9 dx,
for
w < i7 < w + r ,
for
u, + , < i~.
The probability krs; , t t+ . . a t an event detected in the time interval (s, s + ~) will be converted in the interval (t, t + r) s+,. . m is equal to the product of probabilities given by eqs. (5) and (7), i.e., Pi-w kp,,t+~ =
I °- la dw
i i w-ia Pt-=
a s , s+=
g dx,
for
0 < i~ ~< r,
g dx,
for
z < i= ~< w,
9 dx,
for
w < i~ ~< w + r .
(8)
,J w - i a + r
PI-~
f w-ia+r do
The probability that one registered datum will be quantized at the output of the quantization system in the time interval (t, t + z) is equal to the sum of probabilities of all events that can separately realize this event, i.e., M kpt,t+r
1a t - w
~
Z
m =
kpt, t+z
i=0
#S,s+i~,
[ b/) q- g
--
I
0~
(9) int"
The probability that two registered events will be quantized at the output of the quantization system in the time interval (t, t + z ) is equal to the probability that the quantization of any two registered events will fall into the interval (t, t + , ) which means that it will be realized with each pair of M possible events, i.e., ~-,-w
=
,-,,, i= 1
~
,.
\j=i+
-,-w+./~,).
(10)
1
Here the expression for the probability of registration in the interval preceding the ~ interval during which the event was registered, is given by eq. (6). 2.2. TRANSFORMATION OF THE PROCESS WHEN THE QUANTIZATION TIME IS DETERMIN1ST1CALLY DEPENDENT The output process y ( t ) from a quantization system which uses converters with deterministically dependent quantization time is given by the deterministic transformation T of the input process, i.e.,
yU) = T{x(t)}.
(ll)
For converters of the Wilkinson type, the operator T is given by eq. (l): the constant two may be neglected as compared with the second member of this expression. For the differential mode of work, the duration of the interval x~ between two successive time intervals xl and xi+ 1 at the input is equal to (fig. 4) x; = (~c+l) Xi+I--KXi,
Xi, Xi+ 1 >1 ~.
(12)
Let us consider the process at the output of the quantization system when converters of the Wilkinson type are used and when the input process is a non-stationary Poisson process after the passage through the system with a fixed dead time ~. Let us suppose that the maximum conversion time is equal to w. Only the data generated in the intervals completed by the time instant t - x w can be quantized in the time interval (t, t + r ) (fig. 5); the
NON-STATIONARY
POISSON
Inpu~ proces5
545
PROCESS
O,lCpu/ p~oees~
It-
Fig. 4. Input and output process to the conversion system with Wilkinson converters. Xs
:" "~,
r
+/
~
x
2 ='x ( g-" ~) - T
Fig. 5. Geometrical interpretation of the probability of conversion with the Wilkinson type of converters. first event of the quantized interval may occur at any m o m e n t t - x , xl > x > x 2
(fig. 5), where
xl = w ( l + ~ : ) ,
(13)
x2 = ~ ( l + K ) -- z.
(14)
The differential of probability that the quantization of a datum will be completed in the interval (t, t + z) is equal to the p r o d u c t o f three probabilities: the beginning o f the interval will be generated in the interval ( t - x , t - - x + dx), no event of the input process will be generated in the intervals ( t - x - ~ , t - x ) and ( t - x + x/[1 + x], t--x+cO, and the end event of the input d a t u m will be generated in the interval ( t - x + x / [ l + x ] , t - x + x / [1 +~c]+~/~c), i.e., (fig. 5)
2 ( t - x ) dx exp " Ap'.'+~
-
2(z) dz -
-
- x-~,
1-exp 1
2(z) dz j,-x+~ 2 ( 0 dt
dt-x+x/(l+r)
(15)
546
F. Jovx d _J o~
o<
~/~
---
--~~
X/.
___
~
~ ---g'~
t ,2-
Fig. 6. Probability of generation of the second event upper limit
Fig. 7. Probability of conversion of two data.
The probability of generation of the second event decreases when x approaches x z, i.e., when x = x2 = ~(1 + x ) - r, keeping its full value until x reaches x~ =e(1 + r) (fig. 6). It follows from the geometric relations in fig. 6 that a value x~ can be taken for the upper limit of x" Xg is equal to xg --- ~(1 + x) - ½z (1 + 1]~;).
(16)
Since the output from the quantization system is realized by any event with the probability given by eq. (15), the probability that one interval at the output of the quantization system will be completed in the time interval (t, t + z) is given by kpt, t+, Kw
2(z-x)
Ixt,
1-exp
dx~
x exp
-
2(z) dz z-x+x/(l
-
2(z) d z -x-~
x
+~)
2(z) dz
dx.
(17)
dz-x+~
The end of quantization of two data in the time interval (t, t + z) realizes two input events which are not necessarily successive, but which should not overlap and the quantization of which ends in the time interval (t, t + z).
NON-STATIONARY
POISSON
PROCESS
547
Intervals A and B given in fig. 7 are successive, while intervals A and C in the same figure are non-successive, but both of them realize two output data in the interval (t, t + r). The differential of the probability that the quantization of two intervals will be completed in the interval (t, t + z) is given by the expression d 2k ~p tt ', tt + _ K~ w
-- -
abcd
2(x) dx + ab 2(x) dx
x-x~(1
+K)
I
a(u) b(u)
2(u) du
(18)
,d x g
where a =exp
-
2(z) d z -
2(z) dz ,
-x-c~
F-
b = 1-exp
L
I-
c = exp
't-x+[x/(1 t--x+[x/(1
L
[x -
F,-x+tx/~+,,)1+~/,,2(z)
dz ] ,
Jt-x+x/(l+K)
+K)]+(t/x)+~t+e 2(z) +K)]+(~/~)+=
1
dz ,
F - i,_x+t~/. +~)j+cz./~)+:+~2(z)
d = 1-exp
e =
dt-x-a
dz ] ,
,)t--x+lx/(l+K)l+(r/~c)+=+e
(x/(1 + x)) -
(~/~) -
~]/(1 + K).
All events that realize the end of quantization o f two data at the output of the quantization system in the interval ~pt, t,*~
40.t~14L-N*(
~ ,I t,Jv
2l"i'-H.,~
4 [-M.C
i
402
2
¢0"3
I
i
I
]
t - b¢.~
I
!
_
/
i0-4 IO ~
20 ~
30 ~
~0~<
50.c
GO~
i
"lO~
i
80~
_+
_
i
gO~
Fig. 8. Probability occurrence of one event and two events at the output of the quantization system; example.
__loose
548
v. J o v [ 6
(t, t + z) are determined by the integration of eq. (18) in the limits t-w(~c + 1) and t - v , where v is given by the expression U =
X g -]- ( T / K )
-~ ( X g / K ' ) ~- ( V / K 2 ) ,
(19)
i.e., t- w(~+ I )
k pt,t +r 2 t,t-~u'
:
I
l(dpt, t + r 2 t,t--Kw *
(20)
Jt--v
Example. Let us determine the probability of occurrence of data at the output of the quantization system in successive time intervals of duration ~, when the probability density function gp(X) of the quantization time is constant and the number of converters is unlimited. Let the maximum quantization time be w - M ~ , M = 10, 20, 30 and 40. Let the density function of the input process be equal to )o(t)= l0 e x p ( - t ) . Let the dead time of the detection system be equal to ~=0.01 (s). The probability of occurrence of one event and two events is given in fig. 8. 3. Data losses in the quantization system with constant quantization time and limited number of converters Data losses occur when all converters in the quantization system are occupied and a new event appears simultaneously at the input. Constant quantization time is a feature common to converters. The advantage of converters with constant quantization time is the simplicity of the quantization process. Let us determine data losses when converters have constant quantization time w, supposing that the number of converters is limited and that w is much larger than the dead time ~ of the primary detection system. Let us suppose that of m converters k converters (m > k) are occupied at a certain moment t, in the interval of measurement t~ ~
p~+l t+~
:
P tk+l{p(o;t,t+c 0
[l_Po(t,t+cO ] + [p(1;t,t+oO] Po(t,t+o~) } +
+ p~+2p(o;t,t+~) Po(t, t+~) + Pkt P(1;t, t + a ) [l - P o ( t , t + : 0 ] .
(21)
The probability that all converters of the quantization system are free at the moment t + ~ is equal to pO+~ = pto P ( 0 ; t , t + c 0 + Pt1 [ P ( 0 ; t , t + c 0 ] Po(t, t+~).
(22)
The probability that all m converters are occupied is m . P,+~ = P,m {P(O,t,t+o 0 [1-Po(t,t+oO] + P ( l ' t , t + a ) } +
+ P t +1 ( P ( 1 ; t , t + ~ ) [ 1 - P o ( t , t+c()]},
(23)
where the probability of data output from the quantization system P0 (t, t + ~) is given by the expression P0(t,t+a) = P{1;t-w,t-w+a}
- gP~....
(24)
where gPt-w is the probability of data loss in the quantization system in the interval ( t - w , t - w + a). The probability of data loss in the interval (t, t+c0, because of the finite number of converters, is equal to up~ = P ( 1 ; t , t + a ) P t [ 1 - P o ( t , t + 0 0 ] .
(25)
Eqs. (21)-(25) are derived under the assumption that the influence of registration probabilities in each ~t interval on the registration in the neighbouring ¢t intervals may be neglected. This assumption is usually fulfilled because the number of converters is much smaller than the number of successive non-overlapping intervals. In this case the number of combinations which realize the occupation of a certain number of converters, with at least one case
NON-STATIONARY
o,g"
POISSON
549
PROCESS
.&
0,8-
0,7-
6o,i.~ 0,6
2A
H= 30
/' \' i
\
,l// i;
,v ~
,. .("
/
~"
~-
" ~
--
"' -
M= 20
.,
"--:~ ~'~'I42
,{ ~ - .
2,'~.,.. ?-'< \~,2c
,~
'. ....
"• .,: :.. ::l,~g ~
~'P,~
" ~'..~'~8-
..~ . .
'" . . .
/
,:::::./, :. ',.,o
0,z
. . . . . " .... ..... ' ...... " ~ " ~
I t0
20
"li/" SO
= "~"
"" ~ , a o
0,,t '"~
~/" ~0
50
60
70
gO
90
I00 i,~
Fig. 9. Probabilities o f occupation o f one, two a n d three converters of the quantization system; probability o f data loss.
of two registered events in two neighbouring ~ intervals, occurs less frequently than the case when registered events are separated by one interval of width ~. Example. Determine the probability of occupation of three converters of the quantization system and the expected number of data losses in the time interval (0,100~) when the conversion time is constant and equal to w = Ma~, M = 10, 20 and 30, provided the detection system has a fixed dead time ~ and the input non-stationary Poisson process has a density function 2 ( t ) = 10 U(t). The quantization system is connected to the instant memory and all converters are free at the initial moment. The result is given in fig. 9.
550
F. J O V I 6
References 1) 2) 3) 4)
E. Gatti, Time coding, Colloq. Intern. l'l~lectronique Nucl6aire, vol. 2 (Versailles, 1968) !3. 79. j. Hahn et al., IEEE Trans. Nucl. Sci. (Febr. 1969) 154. E. Kowalski, Nuclear electronics (Springer-Verlag, Berlin, 1970). M. Feran et al., A weighted chronotron for time-digit conversion, lspra Nuclear Electronics Syrup. (CID Brussels, 1969) pp. 127 135. 5) F. Jovid, Nucl. Instr. and Meth. 111 (1973) 519. ~) A. Papoulis, Probability, random t,ariables and stochastic processes (McGraw-Hill, New York, 1965).