Blum W., Riegler W., Rolandi L. Particle Detection with Drift Chambers

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6.3 Noise and Optimum Filters 227

20 40 60 80 100

500

1000

1500

2000

2500

3000

3500

4000

ENC (electrons)

20 40 60 80 100

(ns)

500

1000

1500

2000

2500

3000

3500

4000

ENC (electrons)

(ns)

(a)

(b)

Fig. 6.39 Equivalent noise charge (ENC) as a function of the peaking time, calculated from

Eq. (6.115). The three noise components are plotted separately, using typical values for the dif-

ferent noise sources: parallel noise (R

= 10k

) curves 1, series noise (R

= 150

)curves2,1/ f

noise curves 3. The resulting total noise curves S exhibit a shallow minimum. (a) unipolar shaping,

(b) bipolar shaping, both with n = 4

The three components of the ENC for the cathode channel are due to the parallel

white noise, series white noise, and the 1/ f noise of the ampliﬁer. The noise r.m.s.

for the bipolar and the unipolar shaper as a function of ampliﬁer peaking time is

shown in Fig. 6.39. The three components of the noise exhibit a very distinct de-

pendence on the ampliﬁer peaking time. The parallel noise is proportional to

√

whereas the series noise is proportional to 1/

√

. The contribution from the 1/ f

noise is independent of the ampliﬁer peaking time. The noise

shows a minimum

at the peaking time t

opt

given by

opt

= C



ENC

opt

= 8kTC



+ BC

. (6.117)

The time constant

= C



is the so-called noise corner time constant,so

the optimum peaking time is of the same order as

. In the following we neglect

the small 1/ f component of the noise. For estimation of the order of magnitude for

opt

and ENC

opt

we take an example where R

= 10k

and R

= 100

at room

temperature T = 295 K, which results in

opt

[ns]=C

[pF]

ENC

opt

= 350



[pF](K

)

1/4

electrons. (6.118)

Assuming a detector capacitance of C

= 20–50 pF and that K

and K

are of the

order of unity, we ﬁnd optimum peaking times in the range from 20–50 ns and ENC

values between 1500 and 1800 electrons. We will see later that the minimum possi-

ble ENC for any kind of shaping function is given by ENC

min

= 4kTC



,so

we can rewrite the expression for the ENC as

ENC

)=ENC

min





opt



. (6.119)

228 6 Electronics for Drift Chambers

Fig. 6.40 The factor by

which the equivalent noise

charge ENC of a unipolar and

a bipolar shaper with the

same peaking time differs

from the minimum obtainable

ENC, represented as a

continuous function of n,the

number of stages. Upper

curve: bipolar shaper; lower

curve: unipolar shaper

2 4 6 8 10

1.1

1.2

1.3

1.4

1.5

(4KpKs)^1/4

The factor (4K

)

1/4

tells us the amount by which the ENC for a given ampliﬁer

transfer function differs from the theoretical minimum. Figure 6.40 shows this value

for the unipolar and the bipolar shaper.

It is interesting to investigate how the parameters K

and K

are related to the

actual shape of the electronics delta response. With the help of Parseval’s theorem

we can perform the calculation of K

and K

in the time domain



∞

|H(i



∞

h(t)

dt, (6.120)



∞

H(i



∞



(t)

dt. (6.121)

The parallel noise is given by the time integral over h(t), which means that it in-

creases as the length of the electronic delta response, and so the peaking time

increases. The series noise is given by the integral over the derivative of h(t), and

because a shorter delta response has steeper rising and falling edges, the series noise

decreases for increasing peaking time.

Up to now we have examined how to minimize the variance

of the ampliﬁer

output noise, which is equivalent to maximizing the signal-to-noise ratio when the

chamber signal is a delta pulse I(t)=Q

(t).

However, our wire signal is different from a delta signal and shows the 1/(t +t

)

time dependence. As discussed in Sect. 6.2.1 the peak of the ampliﬁer output signal

is approximately given by the charge integrated during the duration T

≈t

/2ofthe

‘ﬂat top’ of the delta response, i.e.,

) ≈ gQ(t

) ≈

2ln(b/a)





. (6.122)

If we want to optimize the signal-to-noise ratio we must ﬁnd the best peaking time

by maximizing the expression





)

ENC

)



2ln(b/a)



ENC

min



(1+t

/2t

)

opt

6.3 Noise and Optimum Filters 229

Fig. 6.41 Optimum peaking

time t

Wire

opt

for a wire chamber

signal as a function of the

time constant t

, normalized

to t

opt

which is the optimum

peaking time for a delta input

signal

0.5 1 1.5 2

2.5

7.5

12.5

17.5

opt

/ t

opt

wire

/ t

opt

where t

opt

gives the optimum signal-to-noise ratio for a delta input signal. The max-

imum of the expression can only be found numerically, and Fig. 6.41 shows the

optimum peaking time as a function of the signal tail t

For t

values that are much shorter than t

opt

the wire chamber signal closely

resembles a delta pulse and the optimum peaking time for the wire readout is equal

opt

. In the case where t

is similar to or longer than t

opt

, the optimum peaking time

Wire

opt

is longer because integrating more of the induced charge is still improving the

signal-to-noise ratio.

Terminated Transmission Line

If the signal propagation time along an electrode is of the same order as the signal

pulse width as illustrated in Fig. 6.42, the electrode must be terminated in order to

avoid multiple signal reﬂections. A transmission line with characteristic impedance

can be terminated with a resistor R

= Z

. In this case, Z(s) consists of a sim-

ple resistor R

, which represents all the electrical characteristics of a terminated

transmission line. The chamber model parameters are given by

Z(s)=R

I(s)=I

ind

(s) w

( f )=

4kT

. (6.123)

While the load resistor R

for the wire channel is usually dimensioned such that

it is a negligible source of noise compared to the ampliﬁer noise, the termination

(f)

(s)

Z(s)

(s)

gH(s)

ind

(s)

w(f)I(s)

Fig. 6.42 Transmission line channel connected to to an ampliﬁer and the universal model of the

readout channel

230 6 Electronics for Drift Chambers

resistor R

of a transmission line is uniquely deﬁned by the chamber geom-

etry and has typical values of 50–500

. Therefore it is usually the termina-

tion resistor noise that dominates over the ampliﬁer noise for a transmission line

readout.

The equivalent input noise is

w( f )=

4kT

+ i

4kT





. (6.124)

For this discussion we neglect the 1/ f noise by setting B = 0, but because the order

of magnitude of the input impedance is similar to the chamber impedance we cannot

neglect Z

as was done for the cathode channel. We therefore have

(s)=

+ R

I(s)=

I(s) w

( f )=

w( f ). (6.125)

The output noise for the unipolar and the bipolar shaper is therefore



∞

( f )=g

4kT





. (6.126)

First we see that the noise r.m.s. is monotonically increasing with the peaking time

, so the smallest noise r.m.s. is achieved for the shortest peaking time. To ﬁnd the

ENC we send, as before, a delta signal Q

(t) into the ampliﬁer, which gives the

output pulse height v

= g

Q, so the equivalent noise charge is

ENC

4kT





= c

. (6.127)

For the cathode channel discussed in the previous section, the noise is entirely de-

ﬁned by the ampliﬁer noise i

and e

and therefore depends only on the ampliﬁer

design. For the terminated transmission line channel, the noise of the termination

resistor represents an irreducible noise ﬂoor and the ampliﬁer noise is negligible

when R

 R

and R

 R

As before, we are interested in optimizing the signal-to-noise ratio for the wire

chamber signal:





)

ENC

)



2ln



(1+t

/2t

)

→ max.

The numerical evaluation shows that the expression has a maximum at t

≈ 8t

If the input signal were a delta current pulse the signal-to-noise ratio would con-

tinuously improve by reducing the ampliﬁer peaking time. Since the wire chamber

signal has a tail and the induced charge only increases with the logarithm of time,

there is an optimum ampliﬁer peaking time, that maximizes the signal-to-noise

ratio.

6.3 Noise and Optimum Filters 231

(f)

(s)

Z(s)

(s)

gH(s) gH(s)

ind

(s)

w(f)I(s)

Fig. 6.43 Wire connected to to an ampliﬁer and the universal model of the readout channel

Wire Channel

The case shown in Fig. 6.43 contains more elements than there were for the cathode

and the transmission line channels. For the wire channel we have the following

electrical parameters:

Z(s)=

1+ sR

I(s)=

1+ sR

(C +C

)

ind

(s)

( f )=

4kT

(

1+(

(C +C

))

≈

4kT

for C C

f 

The loading resistor R

is a source of thermal noise which is ‘ﬁltered’ by the detector

capacitance C

and the decoupling capacitor C. The expression for w

( f ) represents

the white noise of the resistor ﬁltered by a high-pass ﬁlter with time constant

C. With the typical numbers of R

= 1M

and C = 1nF we have

= 1ms.

Therefore, only frequencies of f < 160 Hz are attenuated, which is negligible for the

typical ampliﬁer bandwidth of a few MHz, and we can neglect the effect of C and

. We assume again that the ampliﬁer input resistance is negligible with respect

to the chamber impedance and we also neglect the 1/ f noise term. The equivalent

input noise is then

w( f )=

4kT





+4kTR

. (6.128)

This expression has the same form as the one for the cathode channel and the results

from there apply to this case as well.

We have seen that the unipolar shaper gives a better signal-to-noise ratio than the

bipolar shaper, but the question arises as to whether one can improve the situation

still more by yet another kind of electronics transfer function. This problem can be

solved in a remarkably general and elegant way, and it is possible ﬁnd the maximum

obtainable signal-to-noise ratio from the ampliﬁer input signal and the spectral noise

density at the ampliﬁer input.

6.3.4 A Universal Limit on the Signal-to-Noise Ratio

In the last sections we found that we get different ENC values for different ampli-

ﬁer transfer functions, and that for a speciﬁc shape of a delta response there is an

232 6 Electronics for Drift Chambers

optimum peaking time. We found that the unipolar shaper results in a smaller ENC

than the bipolar shaper, and the question arises as to whether we can decrease the

noise level by yet another transfer function. Or going even further we can ask, what

is the maximum obtainable signal-to-noise ratio?

The reason the signal-to-noise ratio can be improved by speciﬁc ﬁlters is the fact

the signal and the noise have different frequency spectra. So it is possible to improve

the signal-to-noise ratio by speciﬁc ﬁlters. The mathematical problem that we have

to solve can be formulated as shown in what follows.

A signal F(i

) with a superimposed noise having a power spectrum w(

) is

processed by an ampliﬁer with transfer function H(i

). Which transfer function

H(i

) maximizes the signal-to-noise ratio?

Using Eq. 6.3 and 6.84 we can write the ampliﬁer output signal g(t) and the

variance of the noise at the ampliﬁer output as

g(t)=



∞

−∞

F(i

)H(i

ωσ



∞

)|H(i

. (6.129)

The output signal g(t) has a maximum at some time t

when the signal-to-noise

ratio S/N at the ampliﬁer output is







g(t

)







∞

−∞

F(i

)H(i





∞

−∞

)|H(i

. (6.130)

This expression has an upper limit given by the Schwarz inequality, which gives the

following relation for two complex-valued functions

(x) and

(x):





∗

(x)

(x)dx



≤



(x)|



(x)|

dx, (6.131)

where the equal sign applies if

(x)=c

(x) (c

=const). If we insert

(

∗

)



)

−i

(



)H(i

) (6.132)

and extend the integration limits to inﬁnity, the inequality reads as





∞

−∞

F(i

)H(i



≤



∞

−∞

|F(i

)



∞

−∞

)|H(i

, (6.133)

and therefore





≤



∞

−∞

|F(i

)



∞

|F(i

)

. (6.134)

This result is remarkable in two respects. On the one hand, the solution to the prob-

lem is surprisingly simple, while, on the other hand, the result has far-reaching

consequences: Whichever transfer function H(i

) we choose, the (S/N) will always

6.3 Noise and Optimum Filters 233

be lower than the value indicated above, which can be calculated in a straightforward

fashion from the signal and the noise spectrum at the ampliﬁer input.

Now we would like to know the optimum transfer function H(i

) and the output

signal G(i

). From the above relations we ﬁnd that the equal sign applies if

∗

)



)

−i

= c



)H(i

) → H(i

∗

)

−i

. (6.135)

We can omit the factor exp(−i

) because it corresponds simply to a time delay

according to Eq. (6.4e). This transfer function is unique up to a constant, and

it provides the maximum signal-to-noise ratio. The ampliﬁer output signal is then

proportional to

G(i

)=F(i

)H(i

|F(i

)

g(t)=F

−1

[G(i

)]. (6.136)

We note that this signal g(t) is symmetric around t = 0 because G(i

) is real.

As an example we consider a wire chamber cathode pad channel, where we found

a noise power spectrum of w(

)=4kT/R

+ 4kTR

= a

+ b

. Assum-

ing that the detector signal is a delta current pulse, we ﬁnd the following optimum

signal-to-noise ratio

f (t)=Q

(t) F(i

)=Q





≤



∞

+ b

(6.137)

The optimum transfer function is

H(i

+ b

h(t)=F

−1

[H(i

)] = c

−|t|/

. (6.138)

The time constant

is called the noise corner time constant. The delta response of

the optimum ﬁlter h(t) has a peak at t = 0 and is symmetric around t = 0 from where

it decays exponentially. This delta response is called the ‘inﬁnite cusp’ function. As

we saw earlier, a unipolar shaper with n = 6 results in a signal-to-noise ratio that is

just 15% worse than the optimum one that would be achieved by using the inﬁnite

cusp delta response.

The current signals in a wire chamber are not delta currents but have the hy-

perbolic 1/(t + t

) form. If we use several pole-zero ﬁlters to remove the signal

tail such that only a single exponential is left, the chamber signal is given by

f (t)=A

exp(−t/

) and the maximum achievable signal-to-noise ratio is smaller

than the one described above. By writing the signal amplitude in terms of the signal

charge Q =



f (t)dt = A

,wehave

f (t)=

−t/

F(i

+ 1/





≤

ab(1+

)

(6.139)

234 6 Electronics for Drift Chambers

In a previous section we found that it is very useful to approximate the cham-

ber signal by a sum of exponentials, which can be done for every possible signal

shape to any desired accuracy. By evaluating the above expressions for an in-

put signal of the form f (t)=

∑

n=1

exp(−t/

) we can ﬁnd the maximum

achievable signal-to-noise ratio for an arbitrary signal shape. We again write the

amplitudes in terms of the charge ‘contained’ in each exponential Q

= A

.The

result is





≤

∑

n=1

∑

m=1



(

)



. (6.140)

The expression can be used to investigate by what factor the noise for a realistic

ampliﬁer transfer function deviates from the theoretical minimum.

6.4 Electronics for Charge Measurement

In the previous sections we discussed the processing of a wire chamber signal for

several typical electrode conﬁgurations, including the effects of electronic noise. Up

to now the signal induced on the electrode was assumed to have the form 1/(t + t

which is characteristic for any electrode conﬁguration as long as the avalanche elec-

trons are moving in the coaxial region of the wire.

However, the signal shape applies only for a single electron that arrives at the

wire and creates a single avalanche. It would also apply for a localized cloud of

electrons arriving within a time shorter than t

. But the time spread of the incoming

electrons is generally much larger than t

, so the total signal is a superposition of

single-electron signals i

(t)=I

/(1+t/t

ind

(t)=

∑

n=1

(t −t

), (6.141)

with arrival times t

and avalanche sizes G

. The output of the front-end electronics

is then also a linear superposition of the responses to the single electron pulses. The

question arises as to how we choose the transfer function H(i

) in order to arrive

at the best signal-to-noise ratio for this situation. Clearly, the electron arrival times

vary from event to event and the electronics can only be adjusted for an average

situation.

We ﬁrst investigate the case of a TPC, where the arrival of the electrons at the

sense wires assumes a Gaussian form owing to the diffusion along the drift path. If

we deﬁne an arrival time distribution n(t) such that n(t)dt is the average number of

electrons that arrive between t and t + dt,wehave

n(t)dt =

√

πσ

−(t−T)

dt, (6.142)

6.5 Electronics for Time Measurement 235

where

characterizes the electron diffusion along the drift path. Each arriving elec-

tron will induce a signal of the form Gi

(t), where G is the gas gain, and the discrete

sum of time-delayed signals of Eq. (6.141) becomes a convolution of n(t) with i

(t):

i(t)=G



∞

−∞

(t −t



)n(t



)dt



. (6.143)

We realize that from a formal point of view, the arrival time distribution n(t) en-

ters like a linear signal processing device in an electronics chain. Using the Fourier

transform of n(t) we see that this becomes even more evident because we have

I(i

)=GI

)N(i

). (6.144)

We can therefore apply all the results from the previous sections for optimizing the

electronic transfer function in order to achieve the best signal-to-noise ratio.

The optimum achievable signal-to-noise ratio for the average signal is given by







∞

−∞

)N(i

)

. (6.145)

If we assume that for reasons of high rate we remove the signal tail with a series

of pole-zero ﬁlters such that the single electron signal assumes an exponential form

with a time constant

which is negligible with respect to the arrival time distribu-

tion, we can assume I

(t)=Q

(t). If we assume in addition that the superimposed

noise is white, the optimum transfer function according to Eq. (6.135) is

H(i

∗

)

, (6.146)

a Gaussian with the same width as the arrival time distribution.

With a wire chamber geometry shown in Chap. 7 the arrival time distribution will

be different from track to track, depending on the incident angle, the path length

variation, and diffusion. The distribution n(t) is given by the function f

(t, y) in

Eq. (7.7) and results in a width given by Eqs. (7.8) and (7.9). The optimizing process

can then be conducted as before. In general the optimum ampliﬁer peaking time is

such that the width of the delta response is approximately equal to the electron

arrival time distribution.

The actual measurement of the charge can be done in different ways. One either

records the peak of the signal by a so-called ‘peak sensing ADC’, or one samples

the signal at regular intervals and reconstructs the signal peak by ﬁtting the known

response to the sample points.

6.5 Electronics for Time Measurement

In this section we investigate which front-end electronics should be used in order

to extract the best time information from the chamber signal. In drift chambers, the

time information is used to determine the track position by measuring the arrival

236 6 Electronics for Drift Chambers

time of the ionization electrons at the sense wires. In so-called trigger chambers, the

time information is used to determine the time when a charged particle is passing

the detector. For the choice of the transfer function H(i

) which optimizes the time

measurement accuracy we have to consider several effects: the electronic noise,

pulse-height ﬂuctuations, and the arrival time distribution of the electrons.

The ﬂuctuations of the electron arrival times are discussed in detail in Sect. 7.4.

For long electron drift distances the diffusion effect dominates and the average elec-

tron arrival time gives the best time measurement. For short drift distances, the

primary ionization ﬂuctuations together with effects such as track inclination and

path length variations dominate and the ﬁrst arriving electrons show the smallest

arrival time ﬂuctuations. Mathematically, the optimum time measurement would be

achieved by recording the arrival time of each individual electron and performing

statistical analyses of the electron arrival time spectrum for each event. Clearly this

is not practical. With realistic front-end electronics we can approximately realize

some effective weighting of the arrival times and we can choose whether the time

information should be given by the ﬁrst electrons or some effective average of sev-

eral arriving electrons.

There are several ways of extracting the time information from the signal. We can

record the signal shape by sampling the signal at regular intervals and ﬁnd the time,

e.g., by ﬁtting the known average pulse shape to the signal as shown in Fig. 6.44a.

This requires of course the recording of multiple data points per signal, and it is only

useful in detectors where the signal shape does not vary signiﬁcantly from event to

event. In the following we discuss how to directly extract a single time value from

the signal by applying a discriminator, as shown in Fig. 6.44b. A discriminator is

a device that creates a square ‘logic’ output pulse when the input signal crosses a

certain applied threshold. The edge of the logic pulse can then be measured by a

suitable device, typically called a time to digital converter (TDC). For the discus-

sion of the individual contributions to the time resolution we need the following

expressions: the current signal i

(t) induced by a single ion moving from the wire

5 10 15 20 25 30

100

5 10 15 20 25 30

0.2

0.6

1.4

Time (ns)

Threshold (mV)

5 10 15 20 25 30

100

Signal (mV)

Time (ns)

(a) (b)

Fig. 6.44 (a) Sampling of the signal in regular time intervals for extraction of time time informa-

tion. (b) Application of a discriminator to the signal where the time information is determined by

the threshold crossing time