IEEE Ttanzactionz on NutwewA ScZence, Vot.NS-25, No. 1, Feb'ua/y 1978

OPTIMUM FILTERS WITH TIME WIDTH CONSTRAINTSFOR LIQUID ARGON TOTAL-ABSORPTION DETECTORS*

Emilio Gatti

Istituto di Fisica Politecnico MilanoMilano, Italy

and

Veljko Radeka

Brookhaven National LaboratoryUpton, New York 11973

Abstract

Optimum filter responses are found for triangularcurrent input pulses occurring in liquid argon ioniza-tion chambers used as total absorption detectors. Thefilters considered are subject to the following con-straints: finite width of the output pulse having aprescribed ratio to the width of the triangular inputcurrent pulse and zero area of a bipolar antisymmetricalpulse or of a three lobe pulse, as required for highevent rates. The feasibility of pulse shaping givingan output equal to, or shorter than, the input one isdemonstrated. It is shown that the signal-to-noiseratio remains constant for the chamber interelectrodegap which gives an input pulse width (i.e., electrondrif t time) greater than one third of the required out-put pulse width (i.e., resolving time).

Introduction

Electronic noise and electrode configuration inliquid argon total absorption detectors have beenstudied in Ref. 1. It has been shown that the lowerlimit for electronic noise is determined only by. thechamber volume, electron drift velocity, event re-solving time and the unity-gain frequency of the fieldeffect transistor. This limit is approximately thesame in the case of equal output waveforms for multiple-plate ionization chambers and large gap ionizationchambers where the electron drift is much longer thanthe event resolving time. In all cases it has beenassumed that the ionization chamber capacitance ismatched to the amplifier capacitance regardless of themethod used to achieve this condition. However, thebehavior of multiplate chambers in which the drift timeis approximately equal to the resolving time (i.e.,output pulse width) showed a large increase of noise,due to the assumption in the analysis of a weightingfunction of a fixed shape and the width equal to thedifference between the resolving time Xm and the elec-tron drift time td.

This filter is satisfactory for nearly ballisticexcitation (electron drift time less than one half ofthe resolving time). For the region of significantballistic deficit and especially when the electrondrift time is comparable with resolving time (outputresponse) one needs a general treatment based on thesearch of an optimum f ilter.

In this paper we investigate the behavior of theoptimum ENCS as a function of the ratio Cl of the drifttime td and of the finite output pulse width Xm forthe cases of unipolar output response wA(t), for zero

area three-lobe response wC(t) and for bipolar anti-symmetrical response wB(t).

First, we shall consider (Fig. 3) the case ofunipolar output wA(t) because it leads to a simplertreatment: we shall give thereafter the results forthree lobe pulse wC(t) and bipolar output pulse wB(t).In all cases, we shall see that ENCS has at least oneminimum for td/Xm between 0 and 1; however, ENCS in-creases slightly with respect to the minimum, movingto the right and, for td/xm = 1, ENCS is finite andpractically equal to the ENCS obtained with large gaps:td/Xm >> 1 (Figs. 4, 9, and 13).

Representation of Signals and Noise Power

Let us consider (Fig. 1) the triangular waveformof the input current:

p [l-(l-e td) 1 = Q 2 [1 l-eP Ptd s Ptd Ptd

(1)

where Qs is the pulse charge Iotd/2 = Qs.

We shall synthesize the responses of the optimumfilters hA(t), hB(t), hC(t) for the input current (1)by means of several steps: first let us shape the in-put pulse (1) by means of a filter D, into a 6-pulseQ 6(t). From Eq. (1) we obtain the transfer functionot filter D,

1 PtdD(p) = 2 1 - Ptd

1 - tdPtd

(2)

This transfer function can be implemented by the feed-back arrangement shown in Fig. 1.

The series noise of the preamplifierof as an input current noise of bilateralsity,

can be thoughtspectral den-

4CdC e2w2 /2 (3)

where en is the ms equivalent series noise voltage(i.e., untlateral density) of the field effect transis-tor (V/Ha ). The input noise current is a current

Research partially supported by the U. S. Departmentof Energy: Contract No. EY-76-C-02-0016 and by IstitutoNazionale di Fisica Nucleare.

0018-9499/78/0200-0676$00.75 (i 1978 IEEE676

formed by the superposi2ion of doublets in the Campbell'stheorem representation, and it produces at the outputof filter D a noise having the bilateral spectral den-sity given by,

2e2 6 4CC C n d

df d a 2 2(1 - coswtd) + w t2_ 2wtd sinwtd

(4)

As a second step, let us now look for a filter W to becascaded to D giving a 6 response of finite width km)expressed within the interval 0 --- Xm, by

co

w(t) = Ij F e m (n integer)n n

_ co

Its Laplace transform is given by,co

W(p) = (1 - e M)LinFp - irn

(5)

(6)

The noise at the filter output is, then,

transfer functions G(p), cascaded to L, as shown inFig. 3(b), in order to produce the optimum outputpulses WA(t), or wB(t), or wC(t).

The most general filter, having a 6-response oftotal width Xm, can be defined by a transfer function,

G(p) = (l-e ) / Gn - 2rrn-O

p - 1 A,1m(11)

But the transfer function G(p) excited by a ramp functiondoes not give a response confined to the width Xm with-out additional conditions. In fact, the output pulseis given, in the Laplace transform domain by,

0c

W(p) = -2 (1-e P\m) /n Gn 2np p -ix

co co

Q(l~~ePXF Gn -l l 7~ iGnQ (1-e-Pxm)E /1i

I

+ lD/inm)L2 ;2 2TT +-+1) 2T

2 2 xm - Xmn$o Xmn n7o

G ,1 + 0

p2 p

( 12)

1 Iw(iWI 2 dN2(w) dw

We can express lw(iw)12 as a double sum,

00 00

i2 ~ ~~~~~11|W(iw)I = 2 m FmFn(l-cosu)m) 2r 2nnLmcLn 2rrmW- 2Wnn-Cc -Cc

x

m

(7) It is apparent from this expression that, if w(t) hasto be zero outside of the interval 0 ---- Xm, we mustadd the conditions,

00

G GnG0 = 0 and /- n=0 .(13)

nio.(8)

The expression for W(p) becomes,

By using this expansion in Eq. (7) we note, upon in-spection of Eq. (4), that the individual terms aredivergent. This fact suggests to introduce an addi-tional low-pass filtering term in order to proceed withfilter synthesis.

We can design the output waveforms wA(t), wB(t),wC(t) indirectly, by means of their second derivatives,by defining first a filter L, as shown in Fig. 2. Thisis obtained simply by following the filter D by adouble integrator. In the same figure the 6-responseof the filter L is plotted:

=td e17 ek t>i~~=2 e d Lk kI (k- td±Kt -k) * (9)0

Filter L produces, as output waveform, the ramp func-tion Q t j(t) when fed by the input triangular pulse,Eq. (1;. At its output the bilateral noise density

W(p) = Q (l-e PXm) 7 Gn -( 1

nio m

1A

p-iTTn

xm0 (14)

Comparing Eq. (14) with Eq. (5), we can evaluate theterms of the Fourier representation of w(t) as func-tions of all Gn,

2Xxmn

F =o 4TT2 Z--CO

n¢o

Gn n ___ n2'- Fn = m nn n 4Tn2 n

(15)

In time domain, the output pulse can be written asfollows:

2 4w td

2(1-coswtd) + w td - 2wt sifwtj

cc

(10) w(t) s / 4TT2n2 (1

n/o m

- ei km FiL(t)-l(t-xm) . (16)

is present, due to the input noise current, Eq. (3).

The next step will be that of finding suitable

677

e2c Cn d a

2

We note that {w~t) 0} for t -.0 and t -W t th tWW t) = ° f 0

These properties of w(t) do not represent a loss ofgenerality because a discontinuity in w(t) or w'(t) att = 0 or t = Xm would imply (with the noise spectrumgiven by Eq. (4) at the input of W) an infinite noiseamplitude at the filter output. These continuity re-quirements are, therefore, compulsory for w(t).

The output noise will be given, recalling Eq. (10)and (11), by

2 e2CdCa r r u td 2(1-coswX)2 2Cd tN =2 J

2 22(1l-coswtd)+w td-2wtdsinwttdco 2

Ln n i(w-2TTn) JT

i(u-o km--i-~~

(17)

We note that Wnm(CL) is now finite for every integervalue of m and n. Let us now determine G(p) for specialuses of interest here which lead, respectively, to theoptimum output waveforms wA(t), wC(t), wB(t).

Optimum Unipolar Output Waveform WAM

Imposing symmetry about Xm/2 we have,

Gn = G_n .

Further, we note that the second condition from Eq. (13)is included in Eq. (23).

From Eq. (16), the maximum output, which occursfor L- 0.5 is,

mn

Q X2(Xm = smWA 2 4)T2

By inverting the order of sums and integral operations,we obtain,

co

In-CO

n$o

(24)

2 c co2 enCdCa \ 1'

N = 2r LL2T n$ojm

n7bo mio

co1Wf-cosukm

G.G (W 2TTn) (U 2TT)

2 4w~~~~d td2td dw

2(1-coswtd)+w t2-2wtdsinwtd

It is convenient to put the square of this expressionin the form,

22 4nQw( 2 )) =Qs m G_ m (l( _l)nXl(-l) . (25)

14-o -Co

n#o n~o

(18)

Introducing the following notation,

tdX" nX=U

m

we can write,

2 d32mn1COKN= 2 -n-nOGnGm .1 { (K-2rrn) (K-22Tm)

n#o m#o

a2K2 d}a2x22~~~~dx-C82)tC :ss

and finally,

e2CC 3 2cN2 =n d a m ,LGGwma2=d ZZn_c m nm

-CO -CO

(19)

From Eqs. (21) and (25), the signal-to-noise ratiosquared is,

Q 2x8ram ed1nr3XendCa

(wI( 2 j))N2

X,7- GnGm (- (1 X-(_1

-at_l(1 nOD -

-00 -CO

nio mio

~' G GW (at)/ n m nmLnLIn

-_ -CO

n¢o mio

(20)

(21)

0 (26)

and, calling, the ratio

n 00

Z nm ( -(1)nXl_(-l) )-O -Co

n#o m/ocO co

v7 G G W (a)n m rim

-CO -CO

n/o mio

2 (27)

where

co~= 2 2W a r c22 (1-C08x)runm(a) 2(l-coscta)4a x -2a0sinaK (-22Tn)(x-2TTnm)

_ (O( 22)

678

(23)

Gnn2 (- 1)

.

the equivalent noise charge becomes,

2. 21/2rr3/2ae c1/2c1/2ENC ndd as x1/24112

m

Introducing, with notations of Ref. 1 the volume Vthe ionization chamber, the drift velocity ve of thelectrons, and the dielectric constant s, we get,

d t2V2d e

and the equivalent noise charge can be written:

2. 21/2TT3/2ENC = 1/25 /

e c1/2 1/2v1/2n a

x3/2 -m e

where Qmn (ac) is given, according to Eqs. (22) and (34),by c

Q (a) =.4 C { 1-cos7 _

(28) mn a2 -_c (2_4TT2M2) ( 2_4n)

2( l-cosao)+a2 ,2_2ax.sintK

(29)

e C1/2 1/2v1/2Cn a

/Xm Ve

We maximize l as given by Eq. (27) by an optimumchoice of the G . It is immediately seen, by standardvariational metWods that, if

Qmn(a) is tabulated for a = 0.6 and a = 1 in Table I.In the same table the eigenvalues p(ct) for which thesystem of Eq. (35) has a nontrivial solution and thecorresponding eigenvectors Gm(a) are given. TheFm(a), according to Eq. (15), are also tabulated.

In Fig. 4, the coefficient = 15.75/9/2, to beused in Eq. (30)in order to calculate the ENCs, isplotted as a function of a. The asymptotic value 9.79for ax >> 1 can be compared with 8VW= 11.31 as given byEq. (17a) of Ref. 1. In Table I and Fig. 4, the numberof harmonics considered allows to estimate 4(a) with anaccuracy better of one per cent. 5(a,) shows a minimumfor ca = 0.6 and it is independent of a for a > 1. Fora -4 x, Eq. (22) gives: Qmn = .mn* 2TT, where 6mm is theKroeneker symbol; the eigenvalue 4 is 8/rr(]+l/81+1/623+-... ) = 2.584, and the corresponding eigenvector isGl=l, G3=1/9., G5=1/25. G7=1/49...G2=G4=G6=G8... = 0. This asymptotic filter (for a>>l)corresponds, as expected, to a g(t) of triangular shapeas shown in Fig. 5. The output pulse wA(t) is given by,

WA(t) = 12( t)Q-16(. ) for 0 . t . 1/2%mG-1 (- m1)nGW)avLm Gm n2m2 = Lmal nmwrm (31)

Mno

for every integer n, different from 0, the ratio (27)is stationary for variations of the GM.

Recalling that G = G,m, according to Eq. (23),the set of equations T31) can be written,

co

2)l-(-l)2_ (W m+ Wn)J Gm = 0 . (32)

Noting that the coefficient in brackets is invariantfor a change of n into -n, we can symmetrize Eq. (32)with respect to m and n and write,

co

[4(1-(-lm2n2(l-1 ,u(W+W +W +W G 0.4-M m2n2 m~in -mm m,-n -m,-n m

(33)We define Qmn as,

Q = W + W + W + W (34)mn mn -mn m,-n -m,-n

(37)

and is symmetrical about 2m as plotted in Fig. 5together with the overall weighting function hA(t) =,(t)* gA(t) of the whole filter, HA. In Figs. 6 and 7,gA(t), w (t) and weighting function of the whole filterHA are plotted for a = 1 and for a = 0.60.

In Fig. 8, the outputs of the whole filter (de-signed for a = 1) are plotted for triangular inputcurrent pulses of equal charge but shorter or longer bya factor 0.9 or 1.1 from the expected td = Xm. One maynote the sensitivity of the peak amplitude to the drifttime td, while the amplitude and width of the tails givean estimate of the sensitivity, as far as the tails areconcerned, to an input shape different from the expectedone. These data give, indirectly, an estimate of thesensitivity (for an ideal input pulse) to inaccuraciesin the response of the designed filter.

As expected, the peak amplitude sensitivity isquite high: 0.70 (sensitivity 1 is peculiar of a.x,while the sensitivity 0, for zero ballistic deficit,is obtained for a -. 0). For aL = 0.6, sensitivity is0.46.

Optimum Three Lobe Output Waveform wc(We have to add the condition of zero area for

wC(t) to the requirements of Eq. (13) and the symmetrycondition of Eq. (23). According to Eq. (15), thiscondition is,

(36)

Then the set of equations (31) becomes,

-,-n ~ n

[4 l)m2n2 - 'QmnJ =01n = 1,2 -- -

(35)

F =GM

=m F = 00 m2 Z/ m

1 1

mio

(38)

679

The maximum of (wc(t)) 2, occurring at t = 0.5, isaccording to Eq. (16), m

2 4 X a0

Cmm( Q2 r2 n

L io-comio n$o

GnGm m+-n- (-1)mmn (39)

n-i17 k) -lDn-( )((-l) _(-1)iLk i2k2

n4 2 2.Q n

QQi n Q G=- ( ik+ 72.:=k nn- T2 in i2 kn> Gk = 0

From Eqs. (21) and (39), the signal-to-noise ratiosquared is,

xM 2 2(2 Qsxm 1N2 8rr3Cc2 e2CdCa

The ENC5 is given by Eq.ratio,

nv nm mt nL w (-I)co co

nyio mio

C'GGWnniLnmnman rL/-CO -cc

n$o m/o

. (40)

(30), having defined as ,U the

xc co

7 C GnGm tmn11~~~(-1

4 = 4 1 1

IxLn/GnGmQnm1 1

(41)

The optimum waveform wC(t) is found again by maximizingEq. (41) by variations of the Gm, taking into account

the constraint, Eq. (38).

In this equation let us substitute Gn as a functionof all harmonics, Gr of lower order exploiting Eq. (38).We get,

n-12^ Gr

G = -nn r2

1

(42)

i = 1,2. n-i

Gn is given by Eq. (42), once the system of Eq. (44)has been solved by finding the eigenvalue k and thecorresponding eigenvector G1, G2, *..Gn-. 1

In Table II, the eigenvalues and the eigenvectorsare given for Ca = 0.36 and Cl = 1, and in Fig. 9 theC value, used in Eq. (30) in order to calculate the ENC,is plotted as a function of Ca. C shows a minimum forcL = 0.36 and it is independent of a for ca > 1.The number of harmonics considered allows to estimate: (cc) with an accuracy of better than one per cent. InFigs. 10 and 11, wC(t) and hC(t) = t(t)*g (t) are plot-ted for Cc = 1 and ox = 0.36. Figure 12 gives the outputresponses of the filter HC, designed for a = 1, forinput pulse width a = 0.9 and cL = 1.1. Sensitivitiesare 0.93 and 0.68, respectively, for filter designedfor Ca = 1 and Ca = 0.36.

Optimum Bipolar AntisymmetricalOutput Waveform w(t

We have to add the requirement of antisymmetrywith respect to %m/2, to the requirements of Eq. (13),which is expressed by

Gn = -Gn (45)

Equation (16) becomes, then,

co.2T ,

2T

wB(t) = Qs -n4TT2 (jeL mt -eie t)1 x 2

m

and by introducing

S = -iGn nand Eq. (41) becomes,

$ (46)

(47)

n-i n-iT- GiGkn(V F"

4 Li Gkk72 yl)n (l)(l)(l)1 1

4 =~n-lIn-1 ,- n4 - 2 n2 N

GiGkQik+ 2; Qnn - Qin 7 Qkn)1i Z1i k J1 1

the waveform B(t) can be written,

. (43)co

2Q T-w ( )=2S wwB(t) 4TT2~~ 2

L72 1m

Sn sin 2TTnn2 x

is maximum for the set of G1, G2, ... Gn- which is

a nontrivial solution of the Iollowing homogeneous

system of n-i equations,

where all Sn are real. By using Eq. (48), the maximum

of the output pulse squared can be written as:

One could synthesize optimum zero-area bipolar filters

without the requirement of antisymmetry and this would

lead to an output waveform similar to Deighton'sfilters.3 This development has not been dealt with

in this paper.

680

(44)

(48)

Q244QX co

2 4 s s= s mm 2rT 2Trw(t)j m nsin(-t sin(-nt )l 4sLm Ln 2 2 ( mtmax A. max16T 1 1 mmn m

(49)

where t has to be found.max

The noise squared, recalling Eqs. (10), (11), andthe definition of the Sn, can be written,

2 3 2coe2C C x3c2c_c_N2 n d a m Ln mnmn (50)

_co -oo

m/o n$o

If only n harmonics are considered, Eq. (55) can bewritten,

n n

4 Li j i2k2 ik

n n

Xi 2 SiSkRikI 1

Expressing the nth harmonic as a function of all thelower order ones by means of Eq. (13) (recallingEq. (47)), we have,

where Wnm is given by Eq. (22).can be extended only to positivesequently N becomes,

By Eq. (45), the sumsintegers, and con-

n-1-n

Sk

n nk k1

e2 CC X3a2coc2 n d a m 7N 2TT Lm Ln RmnSmSn11

where Rn is,

R = W + W -W -Wmn mn -m,-n -m,n m,-n

and Eq. (57) becomes,

(51) n-l n-1

4 L I S S ( « _«RiLk i (2k2 21 1

= n-l n-l_

1 1

,k,n + ik'in ikn2

2

Rk+

By means of Eq. (22), R n can be written

R~ (a) = 4 J{ (l-cosK0mnmn( )

0{ (%2_4rT2n2) (Q2-4TT2m2)

4TT2a2h2dx2(1-coscLx)+o2Kt2- 2cut sino

The signal-to-noise ratio can be written,

2 2LwB (t )] QS m

N e8 n3eCdCaa2

where 4 is given by,co co

4 ~~~217 2rr1 1 n2m sin( mnt aX)sin(-.-mtaxm mx Xm nmx

)n )S S RL 1m n m mn1 1

We introduce for brevity the notation,

sin(-2Tnt ) = 0Xm max n

In order to maximize 4, we introduce a tentativevalue for tmax/Xm and calculate the set of 5k accordingto Eq. (56). 1 is maximum when Eq. (59) is stationaryfor a variation of Sk. This happens when the Sk are a

nontrivial solution of the following system of n-1equations in S1) S2) Sn_l1

(53)n-Il

T, {t ffI ik i n

1-k i2k2 i2kn1

Rk n n

k2in ikn2

tt ik k R k-i Rkn+ nn Sk

(54)

(60)

i = 1,2 .... n-l

To solve this system of equations, the determinantis equated to zero and the eigenvalue 4 is found, as

well as the corresponding eigenvector S , S, Sn1'Sn. (5n is calculated by means of Eq. 58).) UsingEq. (48), the waveform is calculated, in particular the

(55) location tmax/Xm of the maximum, and so a new set ofis calculated and the whole process is iterated. Theprocess is rapidly convergent and few iterations are

generally sufficient.

In Table III the coefficients Rmn, the eigenvaluesand the eigenvectors are given for several values of a,

and in Fig. 13 the C value is plotted. The asymptoticvalue 30, for a > 1, can be compared with 36 as given by

(56) Eq. (17b) of Ref. 1 and the value 26.3, for ao t = 0.42,with the value 33.84 as given by Eq. (5) of Rev. 1 fora ,t= 0.33. For Table III and Fig. 14, the number of

681

(57)

(58)

(59)

2.33254, -016.92882,+002.79590,-02-1.05507,-Ol2.71334, -02-4.56121, -02

p(0. 6)=3. 21440

1.04945, -012. 79 590, -025.85671 ,+00

-2.00839,-012.68753, -02

-5.92365, -02

C(0.6)=8.78

TABLE I

Qnn (0.6)

-3.77540,-02-1.05507,-Ol-2.00839,-Ol6.44814,+002.03984, -01-9.67553,-03

= 1.00000,+00=-3.39315,-02= 7.77611,-02= 6.99204,-03= 2.32945,-02= 4.75132,-03

F0F1F2

F3F4F5F6

=-1.0016579= 1.00000.+00=-8.48289.-03= 8.64012.-03= 4.37003.-04= 9.31779.-04= 1.31981.-04

Qn (1)

-1.2 7346,-026. 27040,+00-1.27382,-02-1.27411,-02-1.27448,-02-1.27497,-02

A(l)= 2.58893

G = 1.00000,+00G2 = 2.35638,-03G = 1.13206,-OlG4 = 2.35732,-03G5 = 4.22635,-02G = 2.35892,-036

-1.27368, -02-1.27382,-026.27040 ,+00-1.27432,-02-1.27469, -02-1.27519,-02

C(1)= 9.79

F0F1F2F3F4F5r6

-1.27397,-02-1.27411, -02-1.27432,-026. 27040,+00-1.27499,-02-1.27 549,-02

=-1. 015070= 1.00000,+00= 5.89095,-04= 1.25785,-02= 1.47332,-04= 1.69054,-03= 6.55255,-05

TABLE II

5.99263, -014.67036 ,+003.07962,-Ol-1.65234,-Ol1.30693,-021.55421,-02

4.17201 ,-013.07962,-Ol6.95460,+00-5.03236, -01-7.90861,-02-3.86105, -02

-1.23327,-02-1. 65234, -01-5.03236,-Ol6.46971,+002. 50804,-011.24007,-Ol

.216338 C(0.36) = 33.86

G = 1.00000,+00G2 =-3.86311,+00G3 = 1.54022,-01G4 =-7.50064,-01G5 = 7.51727,-02G6 =-2.68650,-01

Qmn (1) (See Table I)

G = 1. 00000,+00GC =-3. 77856,+00G2 = 1.28780,-OlG =-9.51306,-OlG = 4.17458,-02G6 =-4. 27753,-Ol

p(l) = 0.158

C(1) = 39.62

5.02413 ,+002.33254,-Oi1.04945,-Ol

-3. 77540,-025.28985, -02

-2.17091,-02

5.28985,-022.71334,-022.68 753,-022.03984,-Ol6.28443 ,+00

-1.84490, -01

-2. 17091 ,-02-4. 56121, -02-5.92365,-02-9.67 553,-03-1.84490, -016. 13154,+00

GiG2G3G4G5G6

6.27041 ,+00-1.2 7346,-02-1.27368, -02-1.27397,-02-1.27435,-02-1.27484,-02

-1.27435, -02-1.27448, -02-1.27469,-02-1.2 7499,-026.27043 ,+00-1.27591,-02

-1.27484,-02-1.27497,-02-1.27519,-02-1.27549,-02-1.27 591,-026.27038 ,+00

7.11176,+005.99263, -014.17201, -01-1.23327,-028.63438,-026.55755,-02

p(O.36) =0

8,63438,-021.30693,-02

-7.90861,-022. 50804,-Ol5. 63745,+002.02628,-02

6.55755,-021.55421,-02

-3.86105, -021.24007,-Ol2.02628,-026.80214,+00

F=F2=F4=3F5=45F=6

1.00000 ,+00-9.65779,-Ol1.71136,-02

-4. 68790, -023.00691,-037.46251,-03

F=1F2=F =3F =4F5=5F=6

1.00000 ,+00-9.44640, -011.43089, -02

-5.94567, -021.66983,-03

-1.18820,-02

682

TABLE III

Rmn (0.42)

2.19 536 ,+005.68257,-017.82415,+004.40555, -014.17703 ,-Ol2.75691,-Ol

tmax = 0.308xm

1.62968 ,+004.87026, -014.40555, -015.98256 ,+004. 19020,-Ol1.87853,-Ol

p(0.42) = 0.360

C(0.42) = 26.25

1.31248 ,+004.37666,-Ol4.17703 ,-014.19020, -016.78266 ,+00-8.73150,-02

F1F2F3

F5F6

1.07449 ,+003.32657,-Ol2.75691,-Ol1.87853,-Ol

-8.73150,-026.35910,+00

= 1= -3.3207,-Ol= -5.6864,-02= -1.4981,-02= -1.1265,-02= -8.1690,-03

R (l)

-2.17829,-03-2.12921,-036.28053,+00

-3.27514,-03-4.06991,-03-5.01348,-03

-2.05836,-03-2.45632,-03-3.27 514,-036. 27890,+00-5.40231,-03-6 .73195,-03

-2.11441,-03-2.91194,-03-4. 06991,-03-5.40231,-036.276273,+00-8.67043,-03

-2.28483,-03-3.48519 ,-03-5.01348,-03-6.73195,-03-8.67043,-036.27230,+00

F1 1

F2 = -2.9601,-01F3 = -7.7151,-02F4 = -1.4403,-02F5 = -1.2676,-02F6 = -9.2529,-03

harmonics considered allows to estimate C(a) with an

accuracy better than one per cent. In Figs. 14 and 15,wB(t) and hB(t) = gB(t)*(t) are plotted respectivelyfor a = 1, ca = 0.42.

Figure 16 gives the output responses of thefilter HB designed for aL = 1, for input pulse widtha = 0.9 and a = 1.1. Sensitivities are 0.85 and 0.57,respectively, for aL = 1 and a = 0.42.

Conclusions

It has been shown that the signal-to-noise ratioremains constant for the chamber interelectrode gap

which gives an input pulse width (i.e., electron drifttime) greater than about one third of the requiredoutput pulse width (i.e., resolving time). Sensitivityto pulse shape, which could arise from changes in drifttimes, increases from 0 to 1, passing from L = 0 toa = x. Sensitivity is given for ao tand = 1 forfilters HA, HB, Hc. The ratio of tRe tail amplitudeto the peak amplitude is roughly equal, for a = 1, tothe relative deviation of actual drift time from thedesign's nominal one. The triangular input currentassumed is well representative of the current pulsedelivered by all types of liquid argon total absorptiondetectors (either sampling or nonsampling) and, there-fore, the results are useful for filter design in allthese cases.

Filter D can be easily implemented according toFig. 1. In order to avoid instabilities and overload

due to pileup (before the final differentiator), an

exponentially decaying pulse is produced instead of a

current step by suitably lowering the feedback loopgain. The final differentiator is substituted by a

zero pole cancellation differentiator. Thus an ex-

ponentially decaying waveform with a time constantmuch shorter than the width of the input triangularpulse is obtained at the output.

This unit inserted between a conventional chargepreamplifier and a conventional gaussian shaping ampli-fier allows us to approximate easily the calculatedfilters.

References

1. V. Radeka; High Resolution Liquid Argon TotalAbsorption Detectors: Electronic Noise andElectrode Configuration. IEEE Trans. Nucl. Sci.,NS-24, no. 1, 293-298 (1977).

2. V. Radeka and N. Karlovac; Time Variant Filtersfor High Rate Pulse Amplitude Spectrometry Semi-conductor Nuclear-Particle Detectors and Circuits.Proceedings of the Conference of Gatlinburg.Publication 1593. National Academy of Sciences,1969.

3. M. 0. Deighton; Minimum-Noise Filter with GoodLow-Frequency Rejection. Proceedings IspraNuclear Electronics Symposium, May 1969, Euratom.

683

1.24924,+013.58717,+002. 19 536 ,+001.62968 ,+001.31248,+001.07449,+00

S3S4S5S6

3.58717 ,+006.37697,+005.68257,-014.87026,-Ol4.37666,-Ol3.32657,-Ol

1.00000,+00-1.32828 ,+00-5.11778,-Ol-2.39709,-Ol-2.81635,-Ol-2.94086,-Ol

6.27844,+00-2.68998,-03-2.17829,-03-2.05836,-03-2.11441,-03-2.28483,-03

-2.68998,-036.28113,+00-2.12921,-03-2.45632,-03-2.91194, -03-3.48519 ,-03

1S =

3S3 =45=6

1.00000,+00-1.18406,+00-6.94363, -01-2.30453,-Ol-3.16921,-01-3.33105,-Ol

u(l) = 0.275tmax 0.30Xm

C(1) = 30.03

QJIQ(t)

1iFig. 1. Filter D for transformation of the inputcurrent triangular pulse into a 6(t) pulse.

Fig. 2. Filter L obtained by cascading a doubleintegrator to filter D. 6-response offilter L.

0 0.5 1.0 1.5 a

Fig. 4. Parameter C proportional tto ENCS (to be used

in Eq. (30)) versus a =

d(ratio drift time

kmtd and output pulse with km) for the uni-polar output pulses wA(t). Fig. 3. Filter H obtained by cascading to D or L a

suitable "optimum" filter W or G in order

to obtain waveforms of the type wA(t),wB(t), wC(t) with the lowest equivalentnoise charge ENCS.

Fig. 5. Input pulse i(t), output pulse wA(t), re-

sponse gA(t), and overall 6-response hA(t)(weighting function) for a - .

Fig. 6. Input pulse i(t), output pulse w (t), re-

sponse hA(t) (weighting functiont for = l

684

filter 0

A11

10

9

0-

i(t)

(t)

Fig. 10. Input pulse i(t), output pulse wC(t), re-sponse hC(t) (weighting function) forQ = 1.

Fig. 7. Input pulse i(t), output pulse w (t), re-sponse hA(t) (weighting function forCl = 0.6 (a value for which minimum ENCs isobtained).

a -1

A

Fig. 8. Output pulses wA(t) for input pulses havingwidth td, shorter (0. 9 Xm), equal, andlonger (1.1 km), than the nominal value forthe filter HA designed for a = 1, that istd = X . Sensitivity of mismatch referredto peak amplitude and tail amplitude can beobserved.

CC

45

40

35

a =0.36

hc(t)

im

Fig. 11. Input pulse i(t), output pulse w (t), re-sponse hC(t) (weighting function* fora = 0.36 (oa-value for which, at constantXm, minimum ENCs is obtained).

C

1.U

Parameter C proportional to ENCs (to be usedin Eq. (30)) versus aL = td (ratio of drift

xmtime td and output pulse width Xm) for three-lobe-zero-area output pulses wC(t).

Fig. 12. Output pulses wC(t) for input pulses havingwidth td, shorter (0.9 Xm), equal, andlarger (1. 1 Xm), than the nominal value forthe filter HC designed for a = 1, that istd = x . Sensitivity of mismatch referredto peatZ amplitude and tail amplitude can beobserved.

685

aFig. 9.

-

n r 4 n " , -0- -4e

U.5

a

35S

30

U U.5 1.0 1.5 a

Fig. 13. Parameter C proportional to ENCS (to be used

in Eq. (30)) versus a = Ld (ratio of driftm

time td and output pulse width Xm) for thetwo lobes antisymmetrical pulses wB(t).

a -1

a

Fig. 16. Output pulses wB(t) for input pulses havingwidth td, shorter (0.9 Xm), equal, andlarger (1.1 Xm), than the nominal value forthe filter HB designed for a = 1, that istd = Xm. Sensitivity of mismatch referredto peak amplitude and tail amplitude can beobserved.

a =1

Fig. 14. Input pulse i(t), output pulse wB(t), 6-response hB(t) (weighting function) fora = 1.

t) la = 0.42

(t)

Fig. 15. Input pulse i(t), output pulse wB(t), 6-response h (t) (weighting function) fora = 0.42 (a value for which, at constantXm, minimum ENCS is obtained).

686

L;) I. r