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

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6.2 Signal Shaping 197

0.01 0.05 0.1 0.5 1 5 10

0.2

0.4

0.6

0.8

0.05 0.1 0.5 1 5 10

–10

|W| / W

–20 log W / W

ω t

–5

–15

–20

(a) (b)

Fig. 6.15 Absolute value of the transfer functions for the unipolar and the bipolar shaper as a

function of the dimensionless number

, assuming n = 4. The plot in (a) has a linear coordinate,

and the plot in (b) shows the two transfer functions in units of dB. The horizontal line indicates

the −3dB level. Both shapers attenuate frequencies higher than

≈1/t

. The transfer function of

the bipolar shaper shows a maximum at

√

n −1)/t

and monotonically decreases for lower

frequencies. At zero frequency the transfer function of the bipolar shaper is zero

The two transfer functions are depicted in Fig. 6.15. The unipolar shaper passes

all frequencies up to the 3-dB bandwidth limit of

= n



1/(n+1)

−1/t

, which

evaluates to

≈ 1.54/t

for n = 4. The transfer function of the bipolar shaper

has a peak at

√

n −1)/t

and lower and higher frequencies are attenuated.

For n = 4 the peak of the transfer function is located at

= 1/t

and the 3-dB

bandwidth limit is

≈ 1.81/t

, similar to the one from the unipolar shaper. For

our purposes it is sufﬁcient to assume that the bandwidth limits of a unipolar and

a bipolar shaper of the same peaking time t

are identical. For DC signals, which

means

= 0, the transfer function of the bipolar shaper is zero. The name ‘bipolar

shaper’ comes from the fact that the positive part of the signal is followed by a

negative part (‘undershoot’) of equal area, such that the time integral over the entire

signal is zero. This follows from the fact that H

bip

(0)=0, which can be seen by

using Eqs. (6.4d) and (6.4j):

lim

t→∞



−∞

bip



)dt



= lim

s→0

bip

(s)=H

bip

(0)=0. (6.39)

In fact, for any device with a transfer function having the property W(0)=0, the

integral of the delta response w(t) is zero, which means that the positive signal part

has an area equal to the negative signal part. The same is true for any output signal

(s)=W(s)V(s) of the device:

lim

t→∞



−∞



)dt



= lim

s→0

W(s)V (s)=W(0)V (0)=0. (6.40)

In physical terms, the zero in the transfer function at s = i

= 0 implies that DC

signals are fully attenuated. Only sinusoidal components with frequencies

> 0

pass the circuit, and because all individual sinusoidal components are bipolar the

entire signal is bipolar.

198 6 Electronics for Drift Chambers

A device with a transfer function W (s) having the property W(0)=0iscalledan

AC coupled device. If W(0) is non-zero it is called DC coupled. The drift tube circuit

discussed in Sect. 5.6.1 is an AC coupled system because the capacitor blocking the

high voltage U from the ampliﬁer also blocks the low-frequency components of the

signal.

In the following we discuss the processing of a wire chamber signal by the two

shapers discussed above. The currents ﬂowing into the input of the readout electron-

ics are typically not exactly equal to the current signals induced on the electrodes,

but they are already ‘shaped’ by the effects of the passive detector elements pre-

ceding the ampliﬁer together with the input impedance of the ampliﬁer. In order to

avoid the complications of this input circuit, which we discuss later, we consider for

the moment only the drift tube from Sect. 5.5, where the tube wall is set to potential

−U and the wire is grounded. We assume that the connection of the wire to ground

is performed through a current ampliﬁer of zero input impedance with transfer func-

tion W

uni

or W

bip

. The induced current signal is therefore equal to the current ﬂowing

into the ampliﬁer and is given by the form

i(t)=

ln(b/a)

1+t/t

0 < t/t

< t

max

−1. (6.41)

As the Laplace transform of this signal cannot be written in closed form we

calculate the output signal by numerical evaluation of the convolution of i(t) with

the ampliﬁer delta response. We assume the same sensitivity g and peaking time

for the unipolar and bipolar shaper. We must therefore perform the convolution

of i(t) with gh

uni

(t) or gh

bip

(t), where we insert

= t

/n for the unipolar and

= t

/(n −

√

n) for the bipolar shaper. We just present the integral for the unipolar

shaper, which reads as

v(t)=g



uni

(t −t



)i(t



)dt



→ (6.42)

v(t)

2ln(b/a)







−





−



−





1+(t



)



As an example we use a drift tube with a parameter b/a = 100, giving a value

of t

max

= 10

for the signal duration. For a t

value of 1 ns this evaluates to t

max

s. Figure 6.16a shows the ampliﬁer output signals for n = 4 and a value for

= 10. For a t

value of 1 ns this corresponds to an ampliﬁer peaking time of

10 ns. The bipolar signal returns to zero after a time of t/t

≈ 100, while the output

signal of the unipolar shaper shows a very long tail that will lead to signal pileup at

high counting rates.

For our assumed parameters the signal has a peak of v

/gq = 0.22, which means

that the ballistic deﬁcit is 78%. Figure 6.16b shows the ampliﬁer output signal for

increasing values of t

. The longer the peaking time the larger the signal and the

smaller the ballistic deﬁcit. For t

> t

max

the signal peak approaches v = gq and

the ballistic deﬁcit approaches zero. The pulse height is related to the signal charge

6.2 Signal Shaping 199

20 40 60 80 100 120 140

0.1

0.05

0.1

0.15

0.2

0.25

(a) (b)

50 100 150 200

0.1

0.2

0.3

0.4

0.5

0.6

v/gq v/gq

Fig. 6.16 (a) Output of a unipolar and bipolar ampliﬁer for a drift tube signal of form 1/(t + t

The ampliﬁer parameters are n = 4andt

= 10. The ballistic deﬁcit for this set of parameters

is 78%. (b) Ampliﬁer output for the unipolar shaper with n = 4 and various values of t

1,10,30, 150. The longer the peaking time the larger is the signal. For t

> t

max

= 10

, the peak

approaches a value of v

= gq

‘integrated’ by the ampliﬁer. Let us assume that we approximate the delta response

by a ‘box’ with a duration T

equal to the period that the delta response exceeds 90%

of its peak value. We can state that the width of the box is equal to the ‘ﬂat part’ of

the delta response, and for the unipolar and bipolar shaper discussed in this chapter

we have T

t

/2, as can be seen in Fig. 6.17a. We therefore assume a device with

a delta response h

box

(t)=1for0< t < T

and h

box

(t)=0 outside this interval. The

output signal of such a device is

v(t)=g



box

(t −t



)i(t



)dt



= g



t−T

i(t



)dt



(6.43)

and the peak of the signal is

= v(T

)=g



i(t



)dt



= gQ(T

2ln





. (6.44)

The signal peak is thus equal to the sensitivity g times the total charge induced

within the duration T

of the ﬂat top of the delta response. We call the time T

the

integration time of the ampliﬁer. Figure 6.17b shows the charge integrated during a

time T

= t

/2, together with the signal peak of the exact expression from Eq. (6.42).

We see that they are almost identical, which justiﬁes the above assumptions. In

order to have no ballistic deﬁcit, the ﬂat top of the delta response has to be longer

than the duration of the entire signal, being 10

in our case. Because a signal

duration of several microseconds is usually not compatible with rate requirements

we always use only a fraction of the induced charge. However, since the induced

charge increases just with the logarithm of the integration time, it takes only a time

of 200t

to integrate half the charge, corresponding to a ballistic deﬁcit of 50%. It

takes another 9800t

to induce the remaining 50% of the charge.

We can conclude that the peak of the ampliﬁer output signal is proportional to

the charge integrated during the duration of the ‘ﬂat top’ (>90% of the peak value)

200 6 Electronics for Drift Chambers

1 2 3 4

0.2

0.4

0.6

0.8

100 200 300 400 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

v / gq

(a) (b)

Fig. 6.17 (a) Approximation of the unipolar delta response by a box with a width corresponding

to the duration of the ﬂat top of the signal. (b) Output signal peak for the unipolar delta response

uni

(t) and the box delta response h

box

(t) with a width T

equal to the ﬂat top of h

uni

(t).Wesee

that interpreting T

as the ampliﬁer integration time is a very good approximation

of the ampliﬁer delta response. For wire chamber signals, where the induced charge

increases only with the logarithm of the elapsed time, this means that the ballistic

deﬁcit is 80% for integration times of ≈ 10–20t

and 50% for integration times of

≈ 100–200t

ns. In order to have a ballistic deﬁcit below 10% one has to integrate

for 10

–10

, corresponding to several tens of microseconds.

6.2.2 Signal Tail Cancellation

The long tails of the wire chamber signals result in undesired signal pileup even at

modestly high counting rates. There are several possibilities for removing this signal

tail. The simplest and most preferable is the use of the bipolar shaper shown in the

last section. For the example of t

= 1ns and t

= 10ns we saw that the width of the

positive signal part is ≈ 30ns followed by an undershoot of about 70 ns duration.

The baseline is ‘restored’ after a total duration of 100 ns.

There are, however, applications where such a bipolar pulse is undesirable and

we want to remove the signal tail without producing an undershoot. One reason

is the worsening of the signal-to-noise ratio for this shaping method, as pointed

out in Sect. 6.3. Another reason concerns the pulse width. We show in this section

that the ‘dead-time’ of 100 ns, mentioned in the example above, can be reduced to

about 40 ns with a unipolar shaping scheme. A third reason concerns crosstalk. We

have seen that the signal induced on the wire where the avalanche is taking place

has negative polarity and the signal induced on the neighbouring wire has positive

polarity. Applying a negative discriminator threshold to the wire signal registers

only the avalanche wire. If we read the wire signals with a bipolar shaping ampliﬁer

there is a negative undershoot of the cross-induced signals, which can also trigger

the discriminator, resulting in crosstalk.

The standard technique for realization of unipolar tail cancellation is the use

of one or several pole-zero ﬁlters [BOI 82]. As we saw in Sect. 6.1.3, a pole-zero

6.2 Signal Shaping 201

ﬁlter with time constants

and

transforms an exponential signal of the form

exp(−t/

) into an exponential signal with time constant

(Fig. 6.8). In order

to apply such a procedure to our wire chamber signal we ﬁrst approximate the

signal

i(t)=

1+t/t

= I

1+t/t

(6.45)

by a sum of exponentials with increasing time constants

= t

i(t)  I

∑

n=1

−

t/t

= I

∑

n=1

−t/

n+1

. (6.46)

By choosing the number of exponential terms N sufﬁciently large, the signal can be

approximated to any desired accuracy. The Laplace transform of the above expres-

sion is

I(s)=L [i(t)] = I

∑

n=1

s + 1/

. (6.47)

For illustration we use two exponentials:

i(t)=I



−t/

+ A

−t/



I(s)=I



s + 1/



(6.48)

and rewrite the expression as

I(s)=I

+ A

)

(s + 1/

)

(s + 1/

)

(s + 1/

)

+ A

. (6.49)

Applying a pole-zero ﬁlter with the time constants

and

to the signal, i.e.,

(s)=(s + 1/

)/(s + 1/

), results in the output signal

(s)=I(s)

s + 1/

= I

+ A

s + 1/

(t)=I

+ A

−t/

, (6.50)

which is a single exponential. This means that applying a pole-zero ﬁlter with

these specially chosen time constants to the sum of two exponential signals is

equivalent to ‘removing’ one of the exponential terms. The procedure is easily ex-

tended to the sum of several exponentials. Representing i(t) by the sum of three

exponentials we need two pole-zero ﬁlters to arrive at a single exponential, and

for N exponentials we need N −1 ﬁlters to remove all but the ﬁrst exponential

term.

In order to ﬁnd the constants A

and

that approximate i(t), we mini-

mize the area (charge) between i(t) and the sum of exponentials





1+t/t

−

∑

n=1

−

t/t



dt → min. (6.51)

202 6 Electronics for Drift Chambers

Fig. 6.18 The 1/(1 +t/t

)

signal together with the sum

of 1/2/3/4/5 exponentials that

approximate the signal to

within ±10%. Using three

exponentials the signal can be

approximated up to t = 100t

0.01 0.1 1 10 100 1000 10000

0.0001

0.001

0.01

0.1

t / t

For a given time interval T = kt

the number of exponentials N must be chosen

such that the signal is represented with the desired accuracy. Figure 6.18 shows

the chamber signal approximated by up to ﬁve terms. Using 1/2/3/4/5 exponen-

tial terms, we can represent the signal with 10% accuracy in a time interval of

3/20/100/350/1200t

As a numerical example we use three exponentials and two pole-zero ﬁlters:

1+t/t

 I



−

t/t

+ A

−

t/t

+ A

−

t/t



. (6.52)

The minimization procedure in Eq. (6.51) results in the parameters

= 0.62 A

= 0.29 A

= 0.052

= 0.92

= 5.6

= 57.

As before, we rewrite the expression with a common denominator

∑

n=1

s + 1/

+ bs + c

(s + 1/

)(s + 1/

)

(6.53)

a = A

+ A

b =

c =

Writing the polynomial in the numerator in the form as

+ bs + c = a(s + 1/

)

(s + 1/

), we get

I(s)  I

(s + 1/

)

(s + 1/

)

(s + 1/

)

(s + 1/

)

(s + 1/

)

. (6.54)

Inserting the ﬁt parameters we have

= 2.04t

and

= 26.9t

. By applying two

pole-zero ﬁlters with time constants

and

we transform the signal into a

single exponential

(s)=I(s)

(s+ 1/

)

(s+ 1/

)

(s + 1/

)

(s + 1/

)

= I

s + 1/

(6.55)

i(t)=I

+ A

−t/

= I

0.96e

−t/0.92t

. (6.56)

6.2 Signal Shaping 203

20 40 60 80 100

0.2

0.4

0.6

0.8

(a) (b)

200 400 600 800 1000

0.02

0.01

0.02

Fig. 6.19 (a) Wire chamber signal processed by a unipolar shaper of n = 4andt

= 10 together

with the same signal ﬁltered by of two pole-zero ﬁlters. The pulse width is reduced to a length of

T 40t

.(b) The same signal on a smaller vertical scale, showing a tail with a height corresponding

to about 1% of the chamber signal

Figure 6.19 shows the result of applying these two pole-zero ﬁlters to the wire cham-

ber signal of Eq. (6.45), which was processed by a unipolar shaper with n = 4 and

= 10. For this purpose we convoluted the signal v(t) from Eq. (6.42) with the

delta response of the pole-zero ﬁlter from Eq. (6.25):

(t)=



(t −t



)v(t



)dt



= v(t) −



−

−(t−t



v(t



)dt



. (6.57)

We see that the result is very satisfactory: the two pole-zero ﬁlters have considerably

reduced the signal length to about 40t

and only a small tail below 1% of the peak

value remains.

Improvement for Long Peaking Times: In the previous example, two of the

ﬁtted exponential time constants were rather small (0.92t

and 5.6t

) because they

had to represent the steep part of the original wire signal at very short times. But

for ampliﬁers with long peaking times t

 t

, the output pulse shape does not

depend on the exact form of the input at times < t

, but only on the signal charge

during this time. We can then achieve a closer approximation by ﬁrst convoluting

the exponential terms with the ampliﬁer response and then ﬁtting this signal to the

chamber signal processed by the ampliﬁer. The ﬁt can extend over times as long as

several hundred t

(t)=g



uni

(t −t



)

1+t



(t)=L

−1



uni

(s)I

∑

m=1

s + 1/



(6.58)

The parameters A

in the analytic expression of v

(t) are then determined such

that the ‘ampliﬁed’ signal v

(t) is represented as closely as possible. The analytic

representation of v

(t) can be expressed by making use of the relations

−1



uni

(s)

s + 1/



= L

−1



(n + st

)

n+1

(s + 1/

)



= Af(t,

)

204 6 Electronics for Drift Chambers

f (t,

)=t

n!e



n −



−(n+1)



−t/

−e

−nt/t

∑

m=0





n −







(6.59)

Thus for three exponentials the signal is expressed as

(t)=gI

f (t,

)+A

f (t,

)+A

f (t,

)). (6.60)

For a value of t

= 10, the signal v

(t) is approximated to within ±10% accuracy

up to 350t

by the parameters

= 0.60 A

= 0.10 A

= 0.016

= 2.48

= 18.8

= 191.

Using two pole-zero ﬁlters with these time constants, giving

= 9.37,

= 107.35

results in the output signal presented in Fig. 6.20. A small bump with a height of

0.5% of the peak value is followed by a remaining tail of only 0.25% relative height.

This is an improvement of a factor 2 compared to the previous scheme.

Although the unipolar-shaped pulse is shorter and avoids the undershoot, there

are several issues that have to be considered carefully when choosing this shaping

scheme. First, one has to evaluate the effects of the remaining signal tail, which lasts

up to 10

in our example. Although the tail is small, the statistical superposition

of many of them may result in signiﬁcant baseline ﬂuctuations. In addition, the pre-

cise cancellation of the signal tail requires a very good knowledge of the chamber

signal shape. Because the time constant t

depends on the ion mobility and applied

high voltage, the signal shape depends on the working point and the gas mixture

that is used. The optimized pole-zero ﬁlter constants are therefore in principle only

valid for a single well-deﬁned working point. Another word of caution concerns the

realization of the pole-zero ﬁlters. Resistors and capacitors have certain production

variations. In integrated circuits, the absolute values of R and C elements have typ-

ical errors of up to 10%, so the precise realization of the time constants is rather

difﬁcult. The tail cancellation effect is mostly deﬁned by the so-called pole/zero

ratio

, which itself is determined by the ratio of resistors. Although ratios of

20 40 60 80 100

0.2

0.4

0.6

0.8

200 400 600 800 1000

0.015

0.01

0.005

0.01

0.015

t / t

(a) (b)

Fig. 6.20 (a) Application of the two pole-zero ﬁlters to the wire chamber signal, where the pole-

zero time constants were determined after processing with the unipolar shaper. (b) The same signal

on a smaller vertical scale showing a tail of only 0.25% height

6.2 Signal Shaping 205

resistors on ICs are typically correct to about 1% it is still very tedious to ﬁne-tune

the time constants.

It is therefore advisable to use the bipolar scheme whenever possible owing to

its simplicity and insensitivity to working point variations and imprecisions of elec-

tronic elements. In the case where unipolar shaping is required for reasons outlined

in the introduction, it is usually essential to make the pole-zero time constants pro-

grammable in order to be able to compensate for process variations and adapt to

changing working points.

6.2.3 Signal Pileup, Baseline Shift, and Baseline Fluctuations

At high counting rates, when the time between signals is not long compared to

the signal pulse width, the individual signals pile up and cause a deterioration of

the signal quality. This can be calculated if one uses the fact that the individual

pulses come at random and therefore have an exponential distribution of the time

interval between any two consecutive signals. The probability P(t)dt for ﬁnding a

time interval t,t +dt is

P(t) dt =

−t/

dt =

−

dt, (6.61)

where

is the average pulse rate and

= 1/

is the average time between two

consecutive pulses. The effect of different pulse shapes at high counting rates is

demonstrated in the four parts of Fig. 6.21, where we see a rapid random sequence

of identical pulses treated by four different shapers. We use the remainder of this

section to quantify what we see. The rate

is taken to be

= 5 ×10

−3

, which

corresponds to 5 MHz when t

= 1ns.

For the following discussion it is useful to separate the signal into a ‘signal’

part of duration T and a ‘tail’ part (Fig. 6.22). The ﬁrst high-rate effect to discuss is

signal pileup, where the ‘signal’ part of two signals arrives within a time interval that

is smaller than the signal pulse width T. This pileup results in incorrect charge or

time measurement. In a particle tracking system, the particle trajectory is typically

measured by several readout channels, and the track reconstruction can detect the

channel with incorrect charge or time measurement. Consequently the measurement

from this channel is not used in the reconstruction, which results in an inefﬁciency

of this channel. The signal pileup probability, or inefﬁciency

, for a pulse width T

and a rate

is given by



−

dt = 1 −e

−

≈

for T 

. (6.62)

The rate requirement is therefore one key element for speciﬁcation of the maximum

pulse width T , which in turn speciﬁes the shaping strategy. The signal part of the

bipolar pulse in Fig. 6.16a has a duration of T ≈ 80t

while the signal part of the

206 6 Electronics for Drift Chambers

0 1000 2000 3000 4000 5000

–0.25

0.25

0.5

0.75

1.25

1.5

(a) (b)

0 1000 2000 3000 4000 5000

0.25

0.5

0.75

1.25

1.5

0 1000 2000 3000 4000 5000

0.25

0.5

0.75

1.25

1.5

t/t

0 1000 2000 3000 4000 5000

–0.25

0.25

0.5

0.75

1.25

1.5

t/t

–0.25

t/t

Fig. 6.21 A fast random sequence of chamber signals processed by four different shapers:

(a) unipolar shaper, (b) bipolar shaper, (c) unipolar shaper with two pole-zero ﬁlters, and (d) unipo-

lar shaper with two pole-zero ﬁlters and AC coupling. The rate dependence of the baseline for these

four shapers is treated in this subsection

unipolar pulse with two pole-zero ﬁlters in Fig. 6.20 has a length of T ≈ 40t

.This

causes an inefﬁciency of 40% for the bipolar shaper and 20% for the unipolar shaper

at the average pulse separation of 200t

The second high-rate effect is ‘tail’ pileup, where a signal is ‘sitting’ on the

superimposed tails of one or several preceding signals, causing a displacement of

the baseline. We would like to know the average value of the displacement (‘baseline

shift’) as well as its variance (‘baseline ﬂuctuation’). The baseline ﬂuctuations cause

a deterioration in the measurement accuracy rather than an inefﬁciency.

50 100 150 200 250

0.2

0.4

0.6

0.8

(a) (b)

20 40 60 80 100 120 140

0.2

0.4

0.6

0.8

1.2

1.4

Fig. 6.22 (a) Separation of the chamber pulse into a ‘signal’ part and a ‘tail’ part. (b) Signal pileup

leading to incorrect charge or time measurements. Removing these incorrect measurements leads

to inefﬁciency