- Email: [email protected]
The application of time-variant filters to time analysis
NUCLEAR
INSTRUMENTS
AND METHODS
66 (i968)
181-192; © NORTH-HOLLAND
PUBLISHING
CO.
THE A P P L I C A T I O N OF TIME-VARIANT FILTERS TO TIME A N A L Y S I S T. D. DOUGLASS and C. W. WILLIAMS
ORTEC Inc., Oak RMge, Tennessee, U.S.A. and J. F. PIERCE The University of Tennessee, Knoxville, Tennessee, U.S.A. Received 26 August 1968 Timing with semiconductor nuclear radiation detectors, particularly Ge(Li) detectors, is increasing in importance in the field of nuclear research. In determining the time of occurrence of a nuclear event from such a detector, the present limitation by current timing methods is charge collection variations in the detector. Tile ultimate limitation of timing unczrtainty, however, m any semiconductor detector is given by the ratio of noise to signal slope at the point of discrimination. For a minimum wdue of timing error, tl~erefore, a low noise level and fast rise time signal are necessary. Lowest noise and fastest rise time are contradictory requirements; and therefore, an optimum filtt:r and a minimum timing error can be determined. The optimum filter is normally non-realizable; however, the minimum tinting error associated with the optimum filter provides a good basis for comparison of realizable filters.
For a given noise spectrum and signal spectrum, an examination of the effect of certain realizable filters onAtiming error is presented. The filters included are RC low-pass, RC high-pass, and the combination of RC low-pass, RC high-pass. Both the time-invariant and time-variant cases were examined, and theoretical data is compared for each case indicated to demonstrate the advantages of the time-variant filter for timing. The advantages of the time-variant filter are lower discriminator levels, allowing the reduction of the effect of charge collection variations on timing error, and lower noise/slope error at low discriminator levels. A time measurement on 511 keV gamma rays using a 20cm 3 coaxial GelLi)detector and time-variant filtering on the preamplifier output signal resulted in a timing error spectrum with fwhm equal to 5.63nsec and fw(.0.1)m equal to 10.58 nsec.
1. Introduction
2. The signal and noise from a charge-sensitive preamplifier
Recently, much emphasis has been placed on the m e a s u r e m e n t of time associated with signals from the charge-sensitive preamplifier used with Ge(.Li) detectors. The accuracy of time m e a s u r e m e n t is limited by charge collection variations in the detector and noise from tile detector and preamplifier. The use of filters on the preamplifier signal affects both of these limitations. The effect of charge collection variations is very dep e n d e n t o n the p a r t i c u l a r detector. In addition, charge collection variations affect the time m e a s u r e m e n t spect r u m in a n o n - G a u s s i a n m a n n e r p r o d u c i n g a very irregular spectrum. Thus, a theoretical description of the effect of filters o n charge collection variation~ for a general detector would be very difficult, if not impossible, to accomplish. The effect of filters o n the error in time m e a s u r e m e n t due to noise is less difficult to analyze. In addition, since the error due to noise is the present ultimate l i m i t a t i o n for time m e a s u r e m e n t of signals from a charge-sensitive preamplifier, an analysis of the effect of filters on this error should provide a s t a n d a r d for c o m p a r i s o n . Also, the e x a m i n a t i o n of particular filters may lead to conclusions a b o u t the possible a p p l i c a t i o n of these filters to reduce errors due to charge collection variations. JANUARY 1969
The simplified equivalent circuit of the semiconductor detector, charge-sensitive preamplifier system is shown in fig. 1 ~}. A l t h o u g h this analysis is for the charge-sensitive preamplifier, the results are valid for any integrating preamplifier. The a s s u m p t i o n of a single pole for the Laplace t r a n s t o r m of the active portion response is made. Thus,
G(.s) = - K / ( T , s +
I),
(l)
where K is the voltage gain, T~ is the time c o n s t a n t associated with the impulse response of G(s), a n d s is the Laplace t r a n s f o r m variable. F o u r circuit equations can be written to find the i n p u t to o u t p u t transfer function. These equations are
and
Id(S)= It(s)-Io(0,
(2)
If(s) = l E e ( s ) - E o ( O ] s G ,
(3)
Ic(s) = - E~(s) s( C + Ca),
(4)
Eo(s ) = G(s)E~(s).
(5)
Solving eqs. (1)-(5) simultaneously, the transfer function is
e o ( s ) / l d ( s ) = -- K {s(C + Cd + Cf + K C f ) ( T o s + 1)}-1 , (6) 181
T.D. DOUGLASS et al.
182
sure of the charge collection time. The total area of id(t) is equal to Q as required. The preamplifier output signal is found from the current signal and the input to output transfer function. The Laplace transform of id(t ) is
)[ cfs Nb(O~) = Nd(tO) + Ni((.O)
Ec(S) Id(S)
If(s)
'S' 1 s(c~d/
I El(S)
%L
[
No( i)
la(s ) = Q/(T~s + 1).
Substituting eq. (11)into eq. (8), solving for Eo(s), and defining A equal to -Q/C•,
Eo(s) = A{s(Tos+ I)(Tcs+ 1)}-' Fig. 1. A simplified equivalent circuit of the semiconductor detector, charge-sensitive preamplifier system. where
To = T a ( C . - [ - C d - i - C f ) / ( C ' - b C , l q - C f ' - l - K C f ) .
(7)
If K is assumed to be very large, eqs. (6) and (7) simplify to
Eo(s)/I,l(s) = -- {sCf(Tos + 1)} -~,
(8)
To = Ta(C + Ca + Cf)/(KCf).
(9)
ia(t) = ( Q / T c ) e x p ( - t/T¢),
(10)
where Q is the total charge collected and Tc is a mea-
(12)
The determination of the noise power spectral density at the output requires that the proper transfer function be found for each noise source. For Nb(co), the power transfer function of voltage-out to current-in at the noise source is I Gb(co)l2 -- {C2co2(T02co2+1)} - '
(13)
At the output the effect of Nb(co) is
Nob(co) = N~(co)/ {C~co~(Tbco ~ + l)}.
and The detector current signal, ia(t), is a function of several variables. These variables include the energy of the absorbed radiation, the location of the charge created in the detector, the electric field distribution in the detector, the hole and electron mobility, and the detector geometry. For a particular detector and monoenergetic radiation all of these variables are constant except for the location of the charge created in the detector. A variation in the charge location is caused by a difference in the point radiation enters the detector, by a variation in the range of the radiation, and by the possibility of complete absorption of the radiation from a multiple interaction process of absorption as opposed to a single interaction. A variation in the charge location results in different charge collection properties and, therefore, different detector current signal shapes. The total charge collected and the total area of the current signal, however, are equal for monoenergetic radiation. The determination of the average detector current signal shape for a single detector is very difficult. Moreover, establishing a signal shape that can be used in a theoretical analysis for a general detector is formidable without making some compromise. A first approximation to the average current signal shape of a general detector is an exponential, or
(11)
(14)
The power transfer function for a voltage source at Na(co) is I Ga(co)[2 = (Ca-l- C + Cf) 2 / {Cf2(To2co2 + 1)}.
(15)
Thus, the effect of Na(co) at the output is Noa((D) = [(C d -1- C -[- Cf) 2 / {C2(T02co2 + l)}]X,(co).
(l 6)
Nb(e)) and Na(co) are both white noise sources; therefore, the total output noise power spectral density is
No(co) = [b / {co2(r2co 2 + 1)}3 + [a/(T2co 2 + l)], (17) where a and b are constants given by
b = Nb(CO)/C 2,
(18)
a = {(Cd+C+CO2/C~}N,(CO).
(19)
and The two terms of No(c0) are known to be equal in the range of 105-106 rad/sec for current detector-preamplifier systems. In time measurement using Ge(Li) detectors, the measurement is made at a time much less than the charge collection time. Thus, only a knowledge of the high frequency properties of the signal and noise are necessary. Nob(co) is practically negligible as compared to No,(co) in the frequency range above 10 7 rad/sec. Therefore, in this study, the output noise power spectral density that will be used is
No(co ) = a/(T~co 2 + 1).
(20)
3. The optimum filter and time measurement
From geometrical considerations, the timing error,
TIME-VARIANT
aT, due to noise is seen to be related to the noise and signal by
ar = e./[de(t)/dt],
183
FILTERS
G m Eo w o)eJOtdo) -~
(21)
where e. is the noise level and de(t)/dt is the signal slope at the time of discrimination1). The relationship of eq. (21) is true only if the slope and noise do not change appreciably over the period in time of a T. For an arbitrary signal and noise, an optimum filter for minimizing a T in eq. (21) can be found. For a signal and noise spectrum of Eo(o)) and No(o)) into a filter with an arbitrary transfer function, G(o)), the signal slope and mean square noise voltage out are
I G o)
o) do)_-<
-oo
<
(IEo(o))12/go(o))}o2do),
(30)
--o9
and substituting eq. (25) into eq. (30),
~__< (2~) -~
{le, o(o))[2/No(o))}o)2dm.
(31)
~-oo
The maximum value for p~ is, therefore,
Pr2 (Max) = ( 2 0 - '
-oo
{IEo(o))12 / No(o)))o)2 do); (32)
and the maximum value is obtained only when
del(t)/dt = (2zr) -1
Eo(o))G(o))jo)eJ~"do),
(22)
and 2 enl
= (27t)-'
No(o))lG(o))12do).
(23)
f ~ --00
u(o)) = kv*(co),
where k is some constant and v*(o)) is the complex conjugate of v(o)). By substitution of eqs. (26) and (27) into eq. (33) and solving for G(o)),
G(o)) = {kE*(o))/No(o))}(-jo))e -j°~'.
The mean square error in time measurement is then a~ = 27z
//[ f]
(24)
Maximizing 1/a~- is equivalent to minimizing a~-. Thus, by defining Pr = a t - 1,
p-Sr=If~oEo(o))G(o))jo)eJ'°'do)l:' / No(o))lG(o))12do).
(25)
By defining
u(o)) : G(o))N~(o))
(26)
v(o)) = {Eo(o))/ No(o))}jo)e j°'
(27)
and
and using the Schwarz inequality,
f:(,()d o) vo)
co < --o5
luo) (), 2do)
s
Ivo) ( , 12do, (28)
the expression, 2
i.i
o) o o9 ¢oeJ'°t do)
<=
I -vr:
o)
o) o) -o0
{lEo O) 2/No
(1)
[f
{IEo(o))12/No(o))}o)2do) -o~
1-1
(35)
A similar analysis for the optimum filter for the measurement of time was performed by Balland2). Eqs. (34) and (35) relate the optimum filter and the minimum error in time measurement for an arbitrary signal and noise and, also, for an arbitrary measurement time. The optimum error in time measurement is independent of the time the measurement is made; however, the optimum filter is a function of this time. Thus if the noise and signal spectra are known, the optimum error in time measurement is known; and if, in addition, the measurement time is chosen, the optimum filter can be described. To simplify the description of the optimum filter, the filter can be broken down into two parts. The first filter, the whitening filter, changes the input noise spectrum into white noise. The necessary transfer function is [ G,(o))[ = [W/No(o))] ~, (36)
coZdo),
(29) is obtained. Rearranging eq. (29),
a~ (opt.) = 2~
where W is a constant. The signal and noise spectrum at the whitening filter output are
o(
< --
(34)
Eq. (34) is a representation of the optimum filter for timing, and the minimum error in the measurement of time is the inverse of eq. (32),
No(o))[ G(o)) 21do)
2re
(33)
Ew(o) ) : E o ( O ) ) G I ( O ) )
(37)
Nw(o)) = W.
(38)
and
T.D. DOUGLASS et al.
184
The second filter operates on the signal and noise f r o m the whitening filter in order that the transfer function of both filters is equivalent to the o p t i m u m filter described by eq. (34). The transfer function of the second filter or matched filter is G2((D
kE*(oo)(-j~o)e -j'°'',
) =
(39)
where t a is the m e a s u r e m e n t time. Analysis of Gz(~o) reveals that the impulse response of the matched filter is the mirror image of the signal slope from the whitening filter, delayed by the m e a s u r e m e n t time, t~. The advantage of using two filters to describe the optim u m filter is that the first filter, the whitening filter is a function of the input noise alone and the second filter, the matched filter, is a function of the signal from the whitening filter alone. Using eq. (12) and (20), an o p t i m u m filter and minim u m error in time m e a s u r e m e n t for the signal and noise from a charge-sensitive preamplifier can be found. If a whitening filter with a transfer function, Gl(co ) = jooT0 + 1,
ew(t)
(42)
The impulse response of G2(oJ) is p r o b a b l y the most revealing description of the filter. As described previously, the impulse response of G2(o)) is the mirror image of the signal slope from the whitening filter delayed by the measurement time, 11. In fig. 2, the procedure for determinmg the impulse response is outlined for the signal and noise from a charge-sensitive preamplifier. F o r a realizable filter, the impulse response must be zero for t less than zero3). Thus, the o p t i m u m filter represented by the impulse response in fig. 2 is not physically realizable. The m i n i m u m error in the measurement of time is found by the substitution of eqs. (12) and (20) into eq (35). Performing these substitutions and simplifying,
A2/(2aa
o-~.(opt.)=
((~2T2+1) --,7,
a 2, (opt.) = 2aTc/A 2.
]
ld{~)
. (43)
c~r (opt.) = (2aTc)~/A.
(45)
Since the o p t i m u m filter is not physically realizable, I I
= Signal
(44)
The root mean square (rms) value for the m i n i m u m error in time m e a s u r e m e n t is
(41)
The matched filter portion of the o p t i m u m filter has a (a)
G2(w ) = k[A / { - j e o ( - j o , r c + 1)}] ( - j o , ) e-J'"n.
Evaluating the integral,
is applied, the noise and signal f r o m the whitening filter are N,.(~o) = a (40) and Ew(o) ) = A/{jco(jcoT~+ 1)}.
transfer function described by
dew (t} dt
=
S ig[lal
sl~pe
A 'Pc
e
'
c
i - e -t/Tc)
/
IL t]
f
D = time
dew{-t )dt
of m e a s u r e m e n t
F -l [- }tUEw* (~)1
t
tI
(d
,r Itl
A -(1 I t ) I ' "t/-'
Tc
/ I I -t: 1
t 1
Fig. 2. An outline for the determination of the impulse response of the optimum filter, a. Signal from the noise whitening filter; b. Signal slope from the noise whitening filter; c. Signal slope mirror image; d. Optimum filter impulse response.
TIME-VARIANT
the minimum error represented by eq. (45) is impossible to attain. Although neither the optimum filter nor the minimum error in time measurement are realizable in this case, useful information is still available from a knowledge of their values. For example, the minimum error is a good standard for the comparison of realizable filters. In addition, the impulse response of the physically non-realizable optimum filter may provide insight into the design of a filter that is realizable. 4. Time-invariant and time-variant RC filters
A group of filters that have particular usefulness in nuclear instrumentation are the resistance-capacitance filters. These filters in the simplest forms consist of RC low-pass, RC high-pass, and the combination of RC low-pass, RC high-pass circuits. The application of these filters in reducing the error in time measurement is the object of this section. The most straightforward use of these filters is as a time-invariant, linear, passive filter. The procedure for determining the error in time measurement will be the -2-~-±2 and same for each filter. The rms noise voltage, {en} the signal slope, del(t)/dt, at the filter output for each filter will be found. From the values for e n2 and del(t)/dt, the rms error in time measurement is 2 • a,r = (e,,)~ / [de,( t) / dt].
(46)
To perform an evaluation of each filter, a r normalized to the optimum value is plotted as a function of the measurement time, q, and the filter time constant, T~, for each of the filters examined. In order to do this, particular values of To and T~ must be chosen; or the variables, t~ and Tt, may be normalized in some manner. The most general approach is to normalize the variables. The normalization used in-this analysis is to define T o equal to T~ and to normalize the filter time constant, T~, and the measurement time, q, to To or T~. For a filter transfer function of G(s) and the noise from the charge-sensitive preamplifier into the filter, the rms noise voltage and the signal slope at the filter output are
-[
2 ~ (e,,)~= ( 2 ~ z )1
and
f
{a/(T2~j)2+l)}]G(~o)]Zd~o
de~(t)/dt = L-I[AG(s)/(Ts+ 1)z],
7
(47) (48)
where T i s equal to T o and T~ and L - I implies the inverse Laplace transform. Substitution of eqs. (47) and (48) into eq. (46) gives the expression for the rms error in time measurement as a function of the particular time-invariant filter and the measurement time,
FILTERS
185
Time-variant filters have received considerable attention in the improvement of pulse amplitude measurement recently. For several reasons, time-variant filters appear to possibly have application in time measurement. For example, the response of a time-variant filter may more closely approximate the optimum filter response than that of the time-invariant filter. The noise/slope error is decreased in this case. In addition, the steady-state noise level of a time-invariant filter which limits the minimum discriminator level is not a limitation using time-variant filters. The error in time measurement due to pulse shape variations can, therefore, be reduced in this case. Using the RC lowpass, RC high-pass, and the combination of RC lowpass, RC high-pass filters and a linear gate, certain time-variant filters are examined. A filter whose impulse response varies as a function of the time the impulse is applied can be classified as a time-variant filter. The time-variancy may be due to a variation in the element values of the filter as a function of time, a variation in the gain of the filter as a function of time, or a combination of the two variations. A special family of time-variant filters exists in which the filter is composed of a linear gate or switch followed by a time-invariant linear network. Nowlin and Blalock 4) have examined this family of time-variant filters for the general case of the gate opening and closing at arbitrary times with white, 0)2, and 1/co2 noise at the gate input. A simplification to this family of time-variant filters is for the gate to open on the arrival of the signal and to close after the measurement time. The transfer function of this filter is
G2(s),
to <=t <=t , ,
Gg(s) =
(49) t0,
t+
where G2(s) is the transfer function of the time-invariant filter following the gate, t o is the time of arrival of the signal, and ti'- is some time after the measurement is made. Since the gate is open to the signal for t o < t < ti i , the effect of filtering on the signal is time-invariant. Thus, the signal slope for some G(s) is given by eq. (48). The effect of the filter represented by eq. (49) on noise, however, is time-variant. Lampard 5) gives equations adequate for the special case of noise given by eq. (20) that describe the output noise as a function of time and filter. The expression for the rms value of noise at the output of the time-variant filter represented by eq. (49) is {~,2(t)}~ =
If, f, 0
0
g2(u)g2(v)qS~,(u- v)dudv
7
, (49a)
186
T.D. DOUGLASS et al.
where (g~t) is mean square transient noise, g2(t) is the impulse response of the filter after the gate, q~a(z) is the autocorrelation function of noise at the input to the gate, and u and v are dummy variables. The autocorrelation function's relationship to the preamplifier output noise power spectral density is 4~a(z) = F-'[No(cO) I G,(co) 12],
(50)
where F-1 is the inverse Fourier transform and Gl(cn) is the transfer function of the filter between the preamplifier and the gate. From eqs. (49) and (50) the transient noise of a time-variant filter can be found for particular filters before and after the gate. With this value for noise and the slope given by eq. (48), the error in time measurement can be determined for the timevariant filter• RC low-passfilter: T h e transfer function of the timeinvariant R C lowpass filter is G(s) = ( T , s + 1 ) - ' ,
(51)
where T 1 is equal to R C . T h e rms, steady-state, output noise voltage and output signal slope are (e2) ~ = [½a I(T, + T)] ~
(52)
and
noise is { ~ ( t ) } ~ = { ( a / T ) ( T ~ / T 2 - 1) I [-~(Ti/T+ I) × x(1-e-Z'l'r')+e-t/re
-'/T'--I]} ~
By defining X = T , / T and Y = t/T, substituting eqs. (53) and (58) into eq. (46), and normalizing to the optimum value aL={(1--X) 2[(X+l)(1-e-zr/x)+2e-re-r/x-2]1} • {2(X 2 - 1): [X e - fix _ ( _ X + Y - X Y ) e - r] }- ,,
O(s) = T l s / ( T , s + 1).
(60)
The rms, steady-state output noise voltage and output signal slope are (e2)~ = [ ( ½ a / T ) ( T , / T ) / ( T , / T + I ) ]
(61)
½,
and d e l ( t ) / d t = ( A / T ) ( T / T 1 - 1)-2"
•
{_ (TIT,)e-'/~,+r(rlr,)--{1--(r, lr)}tlrde -"T} (62)
- {e-'/T(1 - t i T ) l ( 1 - T i l T ) } +
-T,/Ty}].
(53)
By defining X = T , / T
and Y = t / T and substituting eqs. (52) and (53)into eq. (46),
<,~ = A-I{½aT/(1 + X)}&[{X /(1 - X y } e - ' ' < (54)
The timing error for the optimum filter is a T (opt.) = (2aT)¢/A.
½(1+ x )
-+ [ { x / ( 1 - x y } e
- { e - r(1 - r ) / O
-x)}
(55)
-1
(63)
{eV2(t)} ~- = (a/T) ~ [½(1 - T i l T ) -1 {e -2tiT' - 1} -
+(TilT+l)-'
{e-tlr'e-tlr-1}+½]
+t.
(64)
~. = {(1 - X) z [(1 + X) (e- 2fix _ 1) -
-xy)
e - r ] - i,
(56) where by definition ar = a t ~ a t (opt.)
•[-e-rlX+(l-Y+Xr)e-r]
The normalized transient timing error is
-r/~-
+ ((1 - 2 x ) / ( i
a . = ~ { X l(1 + X ) } + [ ( 1 - X ) Z l X ] •
- {1 - ( T i l T ) Z } - ' {e -'/r' e-'/T-- I}+
By normalizing eq. (54) to the optimum, ~ =
The normalized timing error with X and Y as defined previously is
For the R C high-pass filter after a linear gate, the rms, transient, output noise is
- { e - r(1 - r ) / ( 1 - x ) } +
+ {(1 - 2 x ) / ( 1 - x ) 2 } e - r] - ,.
•
(59)
where a L is the normalized timing error of the timevariant R C low-pass filter. In fig. 3, crL and 6 L are shown as a function of Y for several values of X. RC high-pass filter." The transfer function of the timeinvariant R C high-pass filter is
dei( t) / dt = ( A / T ) [ {(T1/T ) e -tIT'/(1 - T , / T ) 2 } -
+ ((1-2T,/T)e-'/~/(1
(58)
(57)
for the time-invariant R C low-pass filter. With the R C low-pass filter after a linear gate as described previously, a time-variant filter is produced. The signal slope is given by eq. (53) since the filter is time-invariant to the signal. The rms, transient, output
- 2X(e- rlx e- y - 1) + (1 - X2)] :~} • • {2x(a - x z ?
[ - e - r/x + 0 -- r + X r ) e - Y]) - ' . (65)
In fig. 4, an and 5n are shown as a function of' Y for several values of X. RC low-pass, RC high-?ass filter: The transfer function of the time-invariant R C low-pass, R C high-pass filter is G(s) = T 2 s i { ( T l s + l ) ( T a s + l ) } , (66)
TIME-VARIANT
where T~ is the R C time constant of the low-pass network and T 2 is the R C time constant of the high-pass network. Two special cases for the relationship of T1 to ~ were examined to demonstrate the effect of this filter. The first case is with equal time constants for T 1 and 72. The rms, steady-state, output noise voltage and output signal slope for T2 equal to T 1 are
187
FILTERS
The normalized timing error is O'HL= [(X-{-0.1) -1 -(X-{-|)-13½ / ( 2 ( 9 9 ) {[--~(1 - X ) / ( X -
+ ~y(1 + 8 X ) / ( X - 1)23 e- r _ - x[(10x-
(eZ~)~ = ( a / T ) ' [ ¼ ( T a / T ) / ( T , / T + 1)23~,
(67)
and de l( t) / dt = ( A / T ) { ( T I l T ) l ( 1 - T~ /T)2 } • • [e -tIT {1 - ( t / T ) + 2 / ( T / T 1 - 1)} +
+e-tiT'{1 - ( t / T ~ ) - 2 / ( l
(68)
-TILT)}].
The normalized timing error is + Y)]-I. (69)
gate with 7"2 equal to T 1, the rms, transient, output (O,2(t)}~ = (a/T) t { [ ¼ ( T J T ) / ( T , / T + 1)2] + + e- 2t/r, [ _ ¼(T~ IT) I(Y, IT - 1) 2 +
+½(t/T)/(T1/T- 1) 2 - ½ ( t 2 / T 2 ) / ( T 1 / T e - t / r , [ ( 1 + T / T , ) -2 ( T , / T - I )
-(t/T)(T,/T-1)-Z(T,/T+I)-']}
- 1)] + -2 -
(70)
~.
The normalized transient timing error is 2x
x e - 2r/x [ _ X + 2 Y - ( 2 Y 2 / X 2) ( X - 1)3 + +4e-r e-r/x[x2-
Y ( X + I)]} ~x
x ( ( 2 x / 2 ) X ( X + l) 2 [ e - r ( l
[ 8 . 1 ( l - 10X)] -1 e- l°r}.
(74)
For the R C high-pass filter with T2 equal to 0.1 T b e fore the linear gate and the RC low-pass filter with T~ variable after the linear gate, the rms, transient, output noise is
- Y+X+XY)+
+e-r/x(-1-r/x-x+r)]}-l.
(71)
In fig. 5, aLn and a L . are shown as a function of Yfor several values of X. The second special case of the RC low-pass, RC highpass filter examined was for T 2 to equal 0.1 T a n d for T 1 to vary. The rms, steady-state, output noise voltage and output signal slope are (e2.)~ = (a/99T) ~ [½(TI/T + O. 1)-1 _ ½(T1/T + 1)- 1]¢ (72)
and d e l ( t ) / d t = ( A / T ) {[-~-(1 - - t / r ) / ( T 1 / r 1)+ +~T(1 + ST,/T) / ( T 1 / T - 1 ) 2] e -t/r -
-(T,/T)(IOT,/T- 1)-' ( T , / r
+ e-1 o,/r e-t/v' _ 1] - ~ [ ( T , / T ) 2 - 1] - ' x x [½(T~/T + 1)(1 - - e - 2 t / T ' ) +
+ e -tIT e-'Iv' - 1]} ~.
(75)
The expression for O'nL is found by dividing eq. (75) by eq. (73) and normalizing to the optimum value given in eq. (55). In fig. 6, aHL and a . L are shown as a function of Y for several values of Jr.
noise is
3 {X(X-1)2+(X+I)
1)23 - ' e - r / x -
x [ ½ ( I O T 1 / T + 1)(1 - e - 2,/r' ) +
For the R C low-pass, R C high-pass filter after a linear
~LH = ( l - - X )
-
l)(X-
{~2(t)}¢ = ( a / T ) " { 9 - ~ [ ( I O T 1 / T ) 2 + 13 -1 x
crLn = [X ~ / {(2x/2) (1 + X)}] [(1 - X ) 3 / X ] • '[e-r(1- Y + X + XY)+dr/x(-1-Y/X-X
+e-,/r
1)+
-
x e -'/r' -[8.1(I -IOTI/T)]-'e-,o,/r}.
×
(73)
5. Conclusions In this study, an examination of the effect of filtering on the measurement of time has been performed. The optimum filter and the minimum noise/slope error were determined for both an arbitrary signal and noise and for the signal and noise from a charge-sensitive preamplifier. The error in the measurement of time using several R C filters, both time-invariant and time-variant, was calculated; and this error was compared to the error using the optimum filter. Experimental verification of the results obtained theoretically was performed for the signal, noise, and filters described. An evaluation of the filters examined reveals that for the measurement of time of signals from Ge(Li) detectors, time-variant filters have two distinct advantages over time-invariant filters. Lower discriminator levels or earlier measurement times is the first advantage. The lower discriminator levels are possible because the noise level from the gate is very low in comparison to the noise level from any of the time-invariant filters. This first advantage allows the reduction of the effect of charge collection variations• The second advantage is the lower noise/slope error at low discriminator levels for most of the time-variant filters. This advantage decreases the discriminator level at which the noise/slope error becomes dominant•
T . D . DOUGLASS et al.
188 o-~ ,oo
~,, 1 [ I k l
-
~,>
°,,
t
NIX kllkL\
el\
0o
. . . . . . .
k x L&k \
,~kxl I \ xi
"~o
__
e r
_
'°
1 I Ill
tlAII
\t
Y
\
f
- 1//I
\ ~
\
i li
//II1,1
(~)
\~ XI\ !! \\',,k~\~",L'K~, lX,l.'kl'i,, N.q.\ll,
'~
I l
\
li
I I _
-
\
I
.~X_~X_J~\
\
IX
NIX
lj
l\
i
k
I
= 2
.
3 Y
4
.
. ~
.
\
'
\\1
YltlX J,
. o
•
2
8
3
4
5
e 91o
t/T
=
\
I I
I I
] %,
] 1
11/)
,%
%,
IV li
ir//t /;
(b)
// I // I //
\\%. \\'%.
1 1
/ t/)'Yl
:://
)4
J/, V
.01
02
03
04
.05
06
oo
1
~
3 Y=
4
5
6
.7
8
~) 1
2
3
I
I [ I/I
,)
5
6
7
0
~ 10
t/T
FJg. 3. T h e effect o r RC low-pass filters on the m e a s u r e m e n t o r time. a. Timcdnvariant
filters; b. T h ] ] e - v a r i a n t filters.
TIME-VARIANT
189
FILTERS
cT.
(a
2
.3 Y=
4
5
~
7
e
9
1
.4
5
.e
7
e
9
1
t/T
(b
01
02
03
o4
os
oe
oe
1
.2
3 Y~.
2
3
4
s
6
7
e
s 10
t/T
Fig. 4. T h e effect of R C high-pass filters on the m e a s u r e m e n t of time. a. Time-invariant filters; b. Time-variant filters.
190
D O U G L A S S et al.
T.D.
C~L H 90
I\ Io~
X=TI/T=O~
I
lick 2d
\
O2
\\
eo
\
\
\
\
\ Ca)
\
\
\
\ \\ \
\
\
\ r~
\
,,
\ \
X
\
\
\
\ \
\
\
~
\
\
\\'
\ \1
\
\
I
\\
\
~
I
,,'~
",, \ \ , , / ' , \
/
/
/
X
3
y =
/ ,/I
/
,~N< .2
/ '/
,,.
4
15
6
7 8 91
2
3
4
5
6
7 8910
.4
.5
.e
.7 .8 .g 1
2
3
4
5
~
7
t/T
(bl
.O1
,02
,03
.04
.05 ,0~
.0~
.1
.2
.3
Y=
t/T
Fig. 5. The effect of R C low-pass, R C high-pass filters on the measurement of time for Tz = T2. a. Time-invariant filters; b. Time-variant filters.
El g 10
TIME-VARIANT
191
FILTERS
CTHL
(a)
ol
.o2
03
04
05
.os
08
1
.2
3 Y=
4
5
e
4
5
e
7 8
9
1
t/T
~L
Cb
01
02
03
04
os
.o(s
08
1
2
3
y=
7
8
G) 1
2
3
4
5
o
7
t//T
Fig. 6. T h e effect o f R C high-pass, R C low-pass filters on the m e a s u r e m e n t o f time for 1"2 a. T i m e - i n v a r i a n t filters; b. Time-variant filters.
=
0.l T.
B 9 10
192
T.D. DOUGLASS et al.
The use of the filters described previously in a Ge(Li) detector, preamplifier system resulted generally in much i m p r o v e m e n t in c o m p a r i s o n with the time measurem e n t on the unfiltered preamplifier signal. The detector was a 20 cm 3 true coaxial detector detecting 511 keV g a m m a rays. The best t i m e - i n v a r i a n t filter was an R C high-pass filter with 2.2 R C equal to 100 nsec. A timing error of 6.81 nsec fwhm and 12.6 nsec fw(0.1)m was o b t a i n e d for this case. The time-variant filter with an R C high-pass filter before the gate and an R C low-pass filter after the gate with 2.2 R C equal to 20 nsec provided the best results of those filters examined. A
t i m i n g error of 5.63 nsec fwhm and 10.58 nsec fw(O. 1)m was measured.
References l) T. D. Douglass, Ph. D. Dissertation (,Tile University of Tennessee, Knoxville, Tennessee, 1968). e) J. C. Balland, Theses (Universit6 de Lyon, 1967). :r) F. V. Kuo, Network analysis and synthesis t Wiley, New York, 1965). ~) C. H. Nowlin and T. V. Blalock, Signal-to-noise ratios of gated filters for nuclear pulse amplifiers, presented at Semiconductor Nuch'ar Particle Detectors and Circuits Conf. (Gatlinburg, Tennessee, May, 1967). ~) D. G. Lampard, IRE Trans. Circuit Theory CT-2, no. 1 ~,1955) 49.
INSTRUMENTS
AND METHODS
66 (i968)
181-192; © NORTH-HOLLAND
PUBLISHING
CO.
THE A P P L I C A T I O N OF TIME-VARIANT FILTERS TO TIME A N A L Y S I S T. D. DOUGLASS and C. W. WILLIAMS
ORTEC Inc., Oak RMge, Tennessee, U.S.A. and J. F. PIERCE The University of Tennessee, Knoxville, Tennessee, U.S.A. Received 26 August 1968 Timing with semiconductor nuclear radiation detectors, particularly Ge(Li) detectors, is increasing in importance in the field of nuclear research. In determining the time of occurrence of a nuclear event from such a detector, the present limitation by current timing methods is charge collection variations in the detector. Tile ultimate limitation of timing unczrtainty, however, m any semiconductor detector is given by the ratio of noise to signal slope at the point of discrimination. For a minimum wdue of timing error, tl~erefore, a low noise level and fast rise time signal are necessary. Lowest noise and fastest rise time are contradictory requirements; and therefore, an optimum filtt:r and a minimum timing error can be determined. The optimum filter is normally non-realizable; however, the minimum tinting error associated with the optimum filter provides a good basis for comparison of realizable filters.
For a given noise spectrum and signal spectrum, an examination of the effect of certain realizable filters onAtiming error is presented. The filters included are RC low-pass, RC high-pass, and the combination of RC low-pass, RC high-pass. Both the time-invariant and time-variant cases were examined, and theoretical data is compared for each case indicated to demonstrate the advantages of the time-variant filter for timing. The advantages of the time-variant filter are lower discriminator levels, allowing the reduction of the effect of charge collection variations on timing error, and lower noise/slope error at low discriminator levels. A time measurement on 511 keV gamma rays using a 20cm 3 coaxial GelLi)detector and time-variant filtering on the preamplifier output signal resulted in a timing error spectrum with fwhm equal to 5.63nsec and fw(.0.1)m equal to 10.58 nsec.
1. Introduction
2. The signal and noise from a charge-sensitive preamplifier
Recently, much emphasis has been placed on the m e a s u r e m e n t of time associated with signals from the charge-sensitive preamplifier used with Ge(.Li) detectors. The accuracy of time m e a s u r e m e n t is limited by charge collection variations in the detector and noise from tile detector and preamplifier. The use of filters on the preamplifier signal affects both of these limitations. The effect of charge collection variations is very dep e n d e n t o n the p a r t i c u l a r detector. In addition, charge collection variations affect the time m e a s u r e m e n t spect r u m in a n o n - G a u s s i a n m a n n e r p r o d u c i n g a very irregular spectrum. Thus, a theoretical description of the effect of filters o n charge collection variation~ for a general detector would be very difficult, if not impossible, to accomplish. The effect of filters o n the error in time m e a s u r e m e n t due to noise is less difficult to analyze. In addition, since the error due to noise is the present ultimate l i m i t a t i o n for time m e a s u r e m e n t of signals from a charge-sensitive preamplifier, an analysis of the effect of filters on this error should provide a s t a n d a r d for c o m p a r i s o n . Also, the e x a m i n a t i o n of particular filters may lead to conclusions a b o u t the possible a p p l i c a t i o n of these filters to reduce errors due to charge collection variations. JANUARY 1969
The simplified equivalent circuit of the semiconductor detector, charge-sensitive preamplifier system is shown in fig. 1 ~}. A l t h o u g h this analysis is for the charge-sensitive preamplifier, the results are valid for any integrating preamplifier. The a s s u m p t i o n of a single pole for the Laplace t r a n s t o r m of the active portion response is made. Thus,
G(.s) = - K / ( T , s +
I),
(l)
where K is the voltage gain, T~ is the time c o n s t a n t associated with the impulse response of G(s), a n d s is the Laplace t r a n s f o r m variable. F o u r circuit equations can be written to find the i n p u t to o u t p u t transfer function. These equations are
and
Id(S)= It(s)-Io(0,
(2)
If(s) = l E e ( s ) - E o ( O ] s G ,
(3)
Ic(s) = - E~(s) s( C + Ca),
(4)
Eo(s ) = G(s)E~(s).
(5)
Solving eqs. (1)-(5) simultaneously, the transfer function is
e o ( s ) / l d ( s ) = -- K {s(C + Cd + Cf + K C f ) ( T o s + 1)}-1 , (6) 181
T.D. DOUGLASS et al.
182
sure of the charge collection time. The total area of id(t) is equal to Q as required. The preamplifier output signal is found from the current signal and the input to output transfer function. The Laplace transform of id(t ) is
)[ cfs Nb(O~) = Nd(tO) + Ni((.O)
Ec(S) Id(S)
If(s)
'S' 1 s(c~d/
I El(S)
%L
[
No( i)
la(s ) = Q/(T~s + 1).
Substituting eq. (11)into eq. (8), solving for Eo(s), and defining A equal to -Q/C•,
Eo(s) = A{s(Tos+ I)(Tcs+ 1)}-' Fig. 1. A simplified equivalent circuit of the semiconductor detector, charge-sensitive preamplifier system. where
To = T a ( C . - [ - C d - i - C f ) / ( C ' - b C , l q - C f ' - l - K C f ) .
(7)
If K is assumed to be very large, eqs. (6) and (7) simplify to
Eo(s)/I,l(s) = -- {sCf(Tos + 1)} -~,
(8)
To = Ta(C + Ca + Cf)/(KCf).
(9)
ia(t) = ( Q / T c ) e x p ( - t/T¢),
(10)
where Q is the total charge collected and Tc is a mea-
(12)
The determination of the noise power spectral density at the output requires that the proper transfer function be found for each noise source. For Nb(co), the power transfer function of voltage-out to current-in at the noise source is I Gb(co)l2 -- {C2co2(T02co2+1)} - '
(13)
At the output the effect of Nb(co) is
Nob(co) = N~(co)/ {C~co~(Tbco ~ + l)}.
and The detector current signal, ia(t), is a function of several variables. These variables include the energy of the absorbed radiation, the location of the charge created in the detector, the electric field distribution in the detector, the hole and electron mobility, and the detector geometry. For a particular detector and monoenergetic radiation all of these variables are constant except for the location of the charge created in the detector. A variation in the charge location is caused by a difference in the point radiation enters the detector, by a variation in the range of the radiation, and by the possibility of complete absorption of the radiation from a multiple interaction process of absorption as opposed to a single interaction. A variation in the charge location results in different charge collection properties and, therefore, different detector current signal shapes. The total charge collected and the total area of the current signal, however, are equal for monoenergetic radiation. The determination of the average detector current signal shape for a single detector is very difficult. Moreover, establishing a signal shape that can be used in a theoretical analysis for a general detector is formidable without making some compromise. A first approximation to the average current signal shape of a general detector is an exponential, or
(11)
(14)
The power transfer function for a voltage source at Na(co) is I Ga(co)[2 = (Ca-l- C + Cf) 2 / {Cf2(To2co2 + 1)}.
(15)
Thus, the effect of Na(co) at the output is Noa((D) = [(C d -1- C -[- Cf) 2 / {C2(T02co2 + l)}]X,(co).
(l 6)
Nb(e)) and Na(co) are both white noise sources; therefore, the total output noise power spectral density is
No(co) = [b / {co2(r2co 2 + 1)}3 + [a/(T2co 2 + l)], (17) where a and b are constants given by
b = Nb(CO)/C 2,
(18)
a = {(Cd+C+CO2/C~}N,(CO).
(19)
and The two terms of No(c0) are known to be equal in the range of 105-106 rad/sec for current detector-preamplifier systems. In time measurement using Ge(Li) detectors, the measurement is made at a time much less than the charge collection time. Thus, only a knowledge of the high frequency properties of the signal and noise are necessary. Nob(co) is practically negligible as compared to No,(co) in the frequency range above 10 7 rad/sec. Therefore, in this study, the output noise power spectral density that will be used is
No(co ) = a/(T~co 2 + 1).
(20)
3. The optimum filter and time measurement
From geometrical considerations, the timing error,
TIME-VARIANT
aT, due to noise is seen to be related to the noise and signal by
ar = e./[de(t)/dt],
183
FILTERS
G m Eo w o)eJOtdo) -~
(21)
where e. is the noise level and de(t)/dt is the signal slope at the time of discrimination1). The relationship of eq. (21) is true only if the slope and noise do not change appreciably over the period in time of a T. For an arbitrary signal and noise, an optimum filter for minimizing a T in eq. (21) can be found. For a signal and noise spectrum of Eo(o)) and No(o)) into a filter with an arbitrary transfer function, G(o)), the signal slope and mean square noise voltage out are
I G o)
o) do)_-<
-oo
<
(IEo(o))12/go(o))}o2do),
(30)
--o9
and substituting eq. (25) into eq. (30),
~__< (2~) -~
{le, o(o))[2/No(o))}o)2dm.
(31)
~-oo
The maximum value for p~ is, therefore,
Pr2 (Max) = ( 2 0 - '
-oo
{IEo(o))12 / No(o)))o)2 do); (32)
and the maximum value is obtained only when
del(t)/dt = (2zr) -1
Eo(o))G(o))jo)eJ~"do),
(22)
and 2 enl
= (27t)-'
No(o))lG(o))12do).
(23)
f ~ --00
u(o)) = kv*(co),
where k is some constant and v*(o)) is the complex conjugate of v(o)). By substitution of eqs. (26) and (27) into eq. (33) and solving for G(o)),
G(o)) = {kE*(o))/No(o))}(-jo))e -j°~'.
The mean square error in time measurement is then a~ = 27z
//[ f]
(24)
Maximizing 1/a~- is equivalent to minimizing a~-. Thus, by defining Pr = a t - 1,
p-Sr=If~oEo(o))G(o))jo)eJ'°'do)l:' / No(o))lG(o))12do).
(25)
By defining
u(o)) : G(o))N~(o))
(26)
v(o)) = {Eo(o))/ No(o))}jo)e j°'
(27)
and
and using the Schwarz inequality,
f:(,()d o) vo)
co < --o5
luo) (), 2do)
s
Ivo) ( , 12do, (28)
the expression, 2
i.i
o) o o9 ¢oeJ'°t do)
<=
I -vr:
o)
o) o) -o0
{lEo O) 2/No
(1)
[f
{IEo(o))12/No(o))}o)2do) -o~
1-1
(35)
A similar analysis for the optimum filter for the measurement of time was performed by Balland2). Eqs. (34) and (35) relate the optimum filter and the minimum error in time measurement for an arbitrary signal and noise and, also, for an arbitrary measurement time. The optimum error in time measurement is independent of the time the measurement is made; however, the optimum filter is a function of this time. Thus if the noise and signal spectra are known, the optimum error in time measurement is known; and if, in addition, the measurement time is chosen, the optimum filter can be described. To simplify the description of the optimum filter, the filter can be broken down into two parts. The first filter, the whitening filter, changes the input noise spectrum into white noise. The necessary transfer function is [ G,(o))[ = [W/No(o))] ~, (36)
coZdo),
(29) is obtained. Rearranging eq. (29),
a~ (opt.) = 2~
where W is a constant. The signal and noise spectrum at the whitening filter output are
o(
< --
(34)
Eq. (34) is a representation of the optimum filter for timing, and the minimum error in the measurement of time is the inverse of eq. (32),
No(o))[ G(o)) 21do)
2re
(33)
Ew(o) ) : E o ( O ) ) G I ( O ) )
(37)
Nw(o)) = W.
(38)
and
T.D. DOUGLASS et al.
184
The second filter operates on the signal and noise f r o m the whitening filter in order that the transfer function of both filters is equivalent to the o p t i m u m filter described by eq. (34). The transfer function of the second filter or matched filter is G2((D
kE*(oo)(-j~o)e -j'°'',
) =
(39)
where t a is the m e a s u r e m e n t time. Analysis of Gz(~o) reveals that the impulse response of the matched filter is the mirror image of the signal slope from the whitening filter, delayed by the m e a s u r e m e n t time, t~. The advantage of using two filters to describe the optim u m filter is that the first filter, the whitening filter is a function of the input noise alone and the second filter, the matched filter, is a function of the signal from the whitening filter alone. Using eq. (12) and (20), an o p t i m u m filter and minim u m error in time m e a s u r e m e n t for the signal and noise from a charge-sensitive preamplifier can be found. If a whitening filter with a transfer function, Gl(co ) = jooT0 + 1,
ew(t)
(42)
The impulse response of G2(oJ) is p r o b a b l y the most revealing description of the filter. As described previously, the impulse response of G2(o)) is the mirror image of the signal slope from the whitening filter delayed by the measurement time, 11. In fig. 2, the procedure for determinmg the impulse response is outlined for the signal and noise from a charge-sensitive preamplifier. F o r a realizable filter, the impulse response must be zero for t less than zero3). Thus, the o p t i m u m filter represented by the impulse response in fig. 2 is not physically realizable. The m i n i m u m error in the measurement of time is found by the substitution of eqs. (12) and (20) into eq (35). Performing these substitutions and simplifying,
A2/(2aa
o-~.(opt.)=
((~2T2+1) --,7,
a 2, (opt.) = 2aTc/A 2.
]
ld{~)
. (43)
c~r (opt.) = (2aTc)~/A.
(45)
Since the o p t i m u m filter is not physically realizable, I I
= Signal
(44)
The root mean square (rms) value for the m i n i m u m error in time m e a s u r e m e n t is
(41)
The matched filter portion of the o p t i m u m filter has a (a)
G2(w ) = k[A / { - j e o ( - j o , r c + 1)}] ( - j o , ) e-J'"n.
Evaluating the integral,
is applied, the noise and signal f r o m the whitening filter are N,.(~o) = a (40) and Ew(o) ) = A/{jco(jcoT~+ 1)}.
transfer function described by
dew (t} dt
=
S ig[lal
sl~pe
A 'Pc
e
'
c
i - e -t/Tc)
/
IL t]
f
D = time
dew{-t )dt
of m e a s u r e m e n t
F -l [- }tUEw* (~)1
t
tI
(d
,r Itl
A -(1 I t ) I ' "t/-'
Tc
/ I I -t: 1
t 1
Fig. 2. An outline for the determination of the impulse response of the optimum filter, a. Signal from the noise whitening filter; b. Signal slope from the noise whitening filter; c. Signal slope mirror image; d. Optimum filter impulse response.
TIME-VARIANT
the minimum error represented by eq. (45) is impossible to attain. Although neither the optimum filter nor the minimum error in time measurement are realizable in this case, useful information is still available from a knowledge of their values. For example, the minimum error is a good standard for the comparison of realizable filters. In addition, the impulse response of the physically non-realizable optimum filter may provide insight into the design of a filter that is realizable. 4. Time-invariant and time-variant RC filters
A group of filters that have particular usefulness in nuclear instrumentation are the resistance-capacitance filters. These filters in the simplest forms consist of RC low-pass, RC high-pass, and the combination of RC low-pass, RC high-pass circuits. The application of these filters in reducing the error in time measurement is the object of this section. The most straightforward use of these filters is as a time-invariant, linear, passive filter. The procedure for determining the error in time measurement will be the -2-~-±2 and same for each filter. The rms noise voltage, {en} the signal slope, del(t)/dt, at the filter output for each filter will be found. From the values for e n2 and del(t)/dt, the rms error in time measurement is 2 • a,r = (e,,)~ / [de,( t) / dt].
(46)
To perform an evaluation of each filter, a r normalized to the optimum value is plotted as a function of the measurement time, q, and the filter time constant, T~, for each of the filters examined. In order to do this, particular values of To and T~ must be chosen; or the variables, t~ and Tt, may be normalized in some manner. The most general approach is to normalize the variables. The normalization used in-this analysis is to define T o equal to T~ and to normalize the filter time constant, T~, and the measurement time, q, to To or T~. For a filter transfer function of G(s) and the noise from the charge-sensitive preamplifier into the filter, the rms noise voltage and the signal slope at the filter output are
-[
2 ~ (e,,)~= ( 2 ~ z )1
and
f
{a/(T2~j)2+l)}]G(~o)]Zd~o
de~(t)/dt = L-I[AG(s)/(Ts+ 1)z],
7
(47) (48)
where T i s equal to T o and T~ and L - I implies the inverse Laplace transform. Substitution of eqs. (47) and (48) into eq. (46) gives the expression for the rms error in time measurement as a function of the particular time-invariant filter and the measurement time,
FILTERS
185
Time-variant filters have received considerable attention in the improvement of pulse amplitude measurement recently. For several reasons, time-variant filters appear to possibly have application in time measurement. For example, the response of a time-variant filter may more closely approximate the optimum filter response than that of the time-invariant filter. The noise/slope error is decreased in this case. In addition, the steady-state noise level of a time-invariant filter which limits the minimum discriminator level is not a limitation using time-variant filters. The error in time measurement due to pulse shape variations can, therefore, be reduced in this case. Using the RC lowpass, RC high-pass, and the combination of RC lowpass, RC high-pass filters and a linear gate, certain time-variant filters are examined. A filter whose impulse response varies as a function of the time the impulse is applied can be classified as a time-variant filter. The time-variancy may be due to a variation in the element values of the filter as a function of time, a variation in the gain of the filter as a function of time, or a combination of the two variations. A special family of time-variant filters exists in which the filter is composed of a linear gate or switch followed by a time-invariant linear network. Nowlin and Blalock 4) have examined this family of time-variant filters for the general case of the gate opening and closing at arbitrary times with white, 0)2, and 1/co2 noise at the gate input. A simplification to this family of time-variant filters is for the gate to open on the arrival of the signal and to close after the measurement time. The transfer function of this filter is
G2(s),
to <=t <=t , ,
Gg(s) =
(49) t0,
t+
where G2(s) is the transfer function of the time-invariant filter following the gate, t o is the time of arrival of the signal, and ti'- is some time after the measurement is made. Since the gate is open to the signal for t o < t < ti i , the effect of filtering on the signal is time-invariant. Thus, the signal slope for some G(s) is given by eq. (48). The effect of the filter represented by eq. (49) on noise, however, is time-variant. Lampard 5) gives equations adequate for the special case of noise given by eq. (20) that describe the output noise as a function of time and filter. The expression for the rms value of noise at the output of the time-variant filter represented by eq. (49) is {~,2(t)}~ =
If, f, 0
0
g2(u)g2(v)qS~,(u- v)dudv
7
, (49a)
186
T.D. DOUGLASS et al.
where (g~t) is mean square transient noise, g2(t) is the impulse response of the filter after the gate, q~a(z) is the autocorrelation function of noise at the input to the gate, and u and v are dummy variables. The autocorrelation function's relationship to the preamplifier output noise power spectral density is 4~a(z) = F-'[No(cO) I G,(co) 12],
(50)
where F-1 is the inverse Fourier transform and Gl(cn) is the transfer function of the filter between the preamplifier and the gate. From eqs. (49) and (50) the transient noise of a time-variant filter can be found for particular filters before and after the gate. With this value for noise and the slope given by eq. (48), the error in time measurement can be determined for the timevariant filter• RC low-passfilter: T h e transfer function of the timeinvariant R C lowpass filter is G(s) = ( T , s + 1 ) - ' ,
(51)
where T 1 is equal to R C . T h e rms, steady-state, output noise voltage and output signal slope are (e2) ~ = [½a I(T, + T)] ~
(52)
and
noise is { ~ ( t ) } ~ = { ( a / T ) ( T ~ / T 2 - 1) I [-~(Ti/T+ I) × x(1-e-Z'l'r')+e-t/re
-'/T'--I]} ~
By defining X = T , / T and Y = t/T, substituting eqs. (53) and (58) into eq. (46), and normalizing to the optimum value aL={(1--X) 2[(X+l)(1-e-zr/x)+2e-re-r/x-2]1} • {2(X 2 - 1): [X e - fix _ ( _ X + Y - X Y ) e - r] }- ,,
O(s) = T l s / ( T , s + 1).
(60)
The rms, steady-state output noise voltage and output signal slope are (e2)~ = [ ( ½ a / T ) ( T , / T ) / ( T , / T + I ) ]
(61)
½,
and d e l ( t ) / d t = ( A / T ) ( T / T 1 - 1)-2"
•
{_ (TIT,)e-'/~,+r(rlr,)--{1--(r, lr)}tlrde -"T} (62)
- {e-'/T(1 - t i T ) l ( 1 - T i l T ) } +
-T,/Ty}].
(53)
By defining X = T , / T
and Y = t / T and substituting eqs. (52) and (53)into eq. (46),
<,~ = A-I{½aT/(1 + X)}&[{X /(1 - X y } e - ' ' < (54)
The timing error for the optimum filter is a T (opt.) = (2aT)¢/A.
½(1+ x )
-+ [ { x / ( 1 - x y } e
- { e - r(1 - r ) / O
-x)}
(55)
-1
(63)
{eV2(t)} ~- = (a/T) ~ [½(1 - T i l T ) -1 {e -2tiT' - 1} -
+(TilT+l)-'
{e-tlr'e-tlr-1}+½]
+t.
(64)
~. = {(1 - X) z [(1 + X) (e- 2fix _ 1) -
-xy)
e - r ] - i,
(56) where by definition ar = a t ~ a t (opt.)
•[-e-rlX+(l-Y+Xr)e-r]
The normalized transient timing error is
-r/~-
+ ((1 - 2 x ) / ( i
a . = ~ { X l(1 + X ) } + [ ( 1 - X ) Z l X ] •
- {1 - ( T i l T ) Z } - ' {e -'/r' e-'/T-- I}+
By normalizing eq. (54) to the optimum, ~ =
The normalized timing error with X and Y as defined previously is
For the R C high-pass filter after a linear gate, the rms, transient, output noise is
- { e - r(1 - r ) / ( 1 - x ) } +
+ {(1 - 2 x ) / ( 1 - x ) 2 } e - r] - ,.
•
(59)
where a L is the normalized timing error of the timevariant R C low-pass filter. In fig. 3, crL and 6 L are shown as a function of Y for several values of X. RC high-pass filter." The transfer function of the timeinvariant R C high-pass filter is
dei( t) / dt = ( A / T ) [ {(T1/T ) e -tIT'/(1 - T , / T ) 2 } -
+ ((1-2T,/T)e-'/~/(1
(58)
(57)
for the time-invariant R C low-pass filter. With the R C low-pass filter after a linear gate as described previously, a time-variant filter is produced. The signal slope is given by eq. (53) since the filter is time-invariant to the signal. The rms, transient, output
- 2X(e- rlx e- y - 1) + (1 - X2)] :~} • • {2x(a - x z ?
[ - e - r/x + 0 -- r + X r ) e - Y]) - ' . (65)
In fig. 4, an and 5n are shown as a function of' Y for several values of X. RC low-pass, RC high-?ass filter: The transfer function of the time-invariant R C low-pass, R C high-pass filter is G(s) = T 2 s i { ( T l s + l ) ( T a s + l ) } , (66)
TIME-VARIANT
where T~ is the R C time constant of the low-pass network and T 2 is the R C time constant of the high-pass network. Two special cases for the relationship of T1 to ~ were examined to demonstrate the effect of this filter. The first case is with equal time constants for T 1 and 72. The rms, steady-state, output noise voltage and output signal slope for T2 equal to T 1 are
187
FILTERS
The normalized timing error is O'HL= [(X-{-0.1) -1 -(X-{-|)-13½ / ( 2 ( 9 9 ) {[--~(1 - X ) / ( X -
+ ~y(1 + 8 X ) / ( X - 1)23 e- r _ - x[(10x-
(eZ~)~ = ( a / T ) ' [ ¼ ( T a / T ) / ( T , / T + 1)23~,
(67)
and de l( t) / dt = ( A / T ) { ( T I l T ) l ( 1 - T~ /T)2 } • • [e -tIT {1 - ( t / T ) + 2 / ( T / T 1 - 1)} +
+e-tiT'{1 - ( t / T ~ ) - 2 / ( l
(68)
-TILT)}].
The normalized timing error is + Y)]-I. (69)
gate with 7"2 equal to T 1, the rms, transient, output (O,2(t)}~ = (a/T) t { [ ¼ ( T J T ) / ( T , / T + 1)2] + + e- 2t/r, [ _ ¼(T~ IT) I(Y, IT - 1) 2 +
+½(t/T)/(T1/T- 1) 2 - ½ ( t 2 / T 2 ) / ( T 1 / T e - t / r , [ ( 1 + T / T , ) -2 ( T , / T - I )
-(t/T)(T,/T-1)-Z(T,/T+I)-']}
- 1)] + -2 -
(70)
~.
The normalized transient timing error is 2x
x e - 2r/x [ _ X + 2 Y - ( 2 Y 2 / X 2) ( X - 1)3 + +4e-r e-r/x[x2-
Y ( X + I)]} ~x
x ( ( 2 x / 2 ) X ( X + l) 2 [ e - r ( l
[ 8 . 1 ( l - 10X)] -1 e- l°r}.
(74)
For the R C high-pass filter with T2 equal to 0.1 T b e fore the linear gate and the RC low-pass filter with T~ variable after the linear gate, the rms, transient, output noise is
- Y+X+XY)+
+e-r/x(-1-r/x-x+r)]}-l.
(71)
In fig. 5, aLn and a L . are shown as a function of Yfor several values of X. The second special case of the RC low-pass, RC highpass filter examined was for T 2 to equal 0.1 T a n d for T 1 to vary. The rms, steady-state, output noise voltage and output signal slope are (e2.)~ = (a/99T) ~ [½(TI/T + O. 1)-1 _ ½(T1/T + 1)- 1]¢ (72)
and d e l ( t ) / d t = ( A / T ) {[-~-(1 - - t / r ) / ( T 1 / r 1)+ +~T(1 + ST,/T) / ( T 1 / T - 1 ) 2] e -t/r -
-(T,/T)(IOT,/T- 1)-' ( T , / r
+ e-1 o,/r e-t/v' _ 1] - ~ [ ( T , / T ) 2 - 1] - ' x x [½(T~/T + 1)(1 - - e - 2 t / T ' ) +
+ e -tIT e-'Iv' - 1]} ~.
(75)
The expression for O'nL is found by dividing eq. (75) by eq. (73) and normalizing to the optimum value given in eq. (55). In fig. 6, aHL and a . L are shown as a function of Y for several values of Jr.
noise is
3 {X(X-1)2+(X+I)
1)23 - ' e - r / x -
x [ ½ ( I O T 1 / T + 1)(1 - e - 2,/r' ) +
For the R C low-pass, R C high-pass filter after a linear
~LH = ( l - - X )
-
l)(X-
{~2(t)}¢ = ( a / T ) " { 9 - ~ [ ( I O T 1 / T ) 2 + 13 -1 x
crLn = [X ~ / {(2x/2) (1 + X)}] [(1 - X ) 3 / X ] • '[e-r(1- Y + X + XY)+dr/x(-1-Y/X-X
+e-,/r
1)+
-
x e -'/r' -[8.1(I -IOTI/T)]-'e-,o,/r}.
×
(73)
5. Conclusions In this study, an examination of the effect of filtering on the measurement of time has been performed. The optimum filter and the minimum noise/slope error were determined for both an arbitrary signal and noise and for the signal and noise from a charge-sensitive preamplifier. The error in the measurement of time using several R C filters, both time-invariant and time-variant, was calculated; and this error was compared to the error using the optimum filter. Experimental verification of the results obtained theoretically was performed for the signal, noise, and filters described. An evaluation of the filters examined reveals that for the measurement of time of signals from Ge(Li) detectors, time-variant filters have two distinct advantages over time-invariant filters. Lower discriminator levels or earlier measurement times is the first advantage. The lower discriminator levels are possible because the noise level from the gate is very low in comparison to the noise level from any of the time-invariant filters. This first advantage allows the reduction of the effect of charge collection variations• The second advantage is the lower noise/slope error at low discriminator levels for most of the time-variant filters. This advantage decreases the discriminator level at which the noise/slope error becomes dominant•
T . D . DOUGLASS et al.
188 o-~ ,oo
~,, 1 [ I k l
-
~,>
°,,
t
NIX kllkL\
el\
0o
. . . . . . .
k x L&k \
,~kxl I \ xi
"~o
__
e r
_
'°
1 I Ill
tlAII
\t
Y
\
f
- 1//I
\ ~
\
i li
//II1,1
(~)
\~ XI\ !! \\',,k~\~",L'K~, lX,l.'kl'i,, N.q.\ll,
'~
I l
\
li
I I _
-
\
I
.~X_~X_J~\
\
IX
NIX
lj
l\
i
k
I
= 2
.
3 Y
4
.
. ~
.
\
'
\\1
YltlX J,
. o
•
2
8
3
4
5
e 91o
t/T
=
\
I I
I I
] %,
] 1
11/)
,%
%,
IV li
ir//t /;
(b)
// I // I //
\\%. \\'%.
1 1
/ t/)'Yl
:://
)4
J/, V
.01
02
03
04
.05
06
oo
1
~
3 Y=
4
5
6
.7
8
~) 1
2
3
I
I [ I/I
,)
5
6
7
0
~ 10
t/T
FJg. 3. T h e effect o r RC low-pass filters on the m e a s u r e m e n t o r time. a. Timcdnvariant
filters; b. T h ] ] e - v a r i a n t filters.
TIME-VARIANT
189
FILTERS
cT.
(a
2
.3 Y=
4
5
~
7
e
9
1
.4
5
.e
7
e
9
1
t/T
(b
01
02
03
o4
os
oe
oe
1
.2
3 Y~.
2
3
4
s
6
7
e
s 10
t/T
Fig. 4. T h e effect of R C high-pass filters on the m e a s u r e m e n t of time. a. Time-invariant filters; b. Time-variant filters.
190
D O U G L A S S et al.
T.D.
C~L H 90
I\ Io~
X=TI/T=O~
I
lick 2d
\
O2
\\
eo
\
\
\
\
\ Ca)
\
\
\
\ \\ \
\
\
\ r~
\
,,
\ \
X
\
\
\
\ \
\
\
~
\
\
\\'
\ \1
\
\
I
\\
\
~
I
,,'~
",, \ \ , , / ' , \
/
/
/
X
3
y =
/ ,/I
/
,~N< .2
/ '/
,,.
4
15
6
7 8 91
2
3
4
5
6
7 8910
.4
.5
.e
.7 .8 .g 1
2
3
4
5
~
7
t/T
(bl
.O1
,02
,03
.04
.05 ,0~
.0~
.1
.2
.3
Y=
t/T
Fig. 5. The effect of R C low-pass, R C high-pass filters on the measurement of time for Tz = T2. a. Time-invariant filters; b. Time-variant filters.
El g 10
TIME-VARIANT
191
FILTERS
CTHL
(a)
ol
.o2
03
04
05
.os
08
1
.2
3 Y=
4
5
e
4
5
e
7 8
9
1
t/T
~L
Cb
01
02
03
04
os
.o(s
08
1
2
3
y=
7
8
G) 1
2
3
4
5
o
7
t//T
Fig. 6. T h e effect o f R C high-pass, R C low-pass filters on the m e a s u r e m e n t o f time for 1"2 a. T i m e - i n v a r i a n t filters; b. Time-variant filters.
=
0.l T.
B 9 10
192
T.D. DOUGLASS et al.
The use of the filters described previously in a Ge(Li) detector, preamplifier system resulted generally in much i m p r o v e m e n t in c o m p a r i s o n with the time measurem e n t on the unfiltered preamplifier signal. The detector was a 20 cm 3 true coaxial detector detecting 511 keV g a m m a rays. The best t i m e - i n v a r i a n t filter was an R C high-pass filter with 2.2 R C equal to 100 nsec. A timing error of 6.81 nsec fwhm and 12.6 nsec fw(0.1)m was o b t a i n e d for this case. The time-variant filter with an R C high-pass filter before the gate and an R C low-pass filter after the gate with 2.2 R C equal to 20 nsec provided the best results of those filters examined. A
t i m i n g error of 5.63 nsec fwhm and 10.58 nsec fw(O. 1)m was measured.
References l) T. D. Douglass, Ph. D. Dissertation (,Tile University of Tennessee, Knoxville, Tennessee, 1968). e) J. C. Balland, Theses (Universit6 de Lyon, 1967). :r) F. V. Kuo, Network analysis and synthesis t Wiley, New York, 1965). ~) C. H. Nowlin and T. V. Blalock, Signal-to-noise ratios of gated filters for nuclear pulse amplifiers, presented at Semiconductor Nuch'ar Particle Detectors and Circuits Conf. (Gatlinburg, Tennessee, May, 1967). ~) D. G. Lampard, IRE Trans. Circuit Theory CT-2, no. 1 ~,1955) 49.