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

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6.2 Signal Shaping 207

In the following we would like to compute the effect of the random superposition

of a number of tails from preceding signals. Looking at the wire chamber signal

processed by a unipolar shaper in Fig. 6.16, we see that the output signal shows a

long tail that approaches the baseline only slowly. The distinction between the signal

and the tail parts is of course somewhat arbitrary and we take the beginning of the

tail at a time t

min

when the signal level falls below 10% of the peak value, which

corresponds to t/t

= 70 for the assumed parameters. For t  t

, the convolution

integral in Eq. (6.42) is approximated by

v(t) ≈ gi(t)



∞

uni



)dt



≈

gqc

2ln(b/a)

min

< t < t

max

, (6.63)

where







∞

−y

dy =

n+1

. (6.64)

In order to calculate the average baseline and the baseline ﬂuctuation we use

Campbell’s theorem, which states the following: If signals of equal shape v(t) are

randomly superimposed at an average rate

, the average and the variance of the

resulting signal are given by

v =



∞

−∞

v(t)dt



∞

−∞

v(t)

dt. (6.65)

Inserting the expressions from Eq. (6.63) for the signal tail we ﬁnd an average base-

line

v and an r.m.s. baseline ﬂuctuation

v =

gqc

2ln(b/a)

max

min

and

≈

√

gqc

2ln(b/a)



min

. (6.66)

Normalizing these values to the peak v

of the signal v(t) given in Eq. (6.44), we

ﬁnd the values

ln(1+t

/2t

)

max

min

≈

√

ln(1+t

/2t

)



min

For the parameters of b/a = 100, n = 4, t

= 10, and t

max

=(b/a)

= 10

assuming that the beginning of the tail is at t

min

= 70, Fig. 6.23 shows the average

baseline shift and r.m.s. With a value of t

= 1ns and a rate

= 5MHz we ﬁnd a

baseline shift and r.m.s. corresponding to  15% and  5% of the average signal.

The baseline shift of 15% can be clearly made out in Fig. 6.21a. If the signal charge

q is not constant but is ﬂuctuating around an average value of

q with a variance of

, an extension of Campbell’s theorem tells us that the value of q in the expression

for

v must be replaced by q and the value of q in the expression for

must be

replaced by q

This example illustrates the undesirable effects of the long tail associated with

the simple unipolar shaper. We now set out to remove the signal tail as explained

208 6 Electronics for Drift Chambers

1 2 3 4 5

–5

unipolar

unipolar/2PZ

unipolar/2PZ/CR

1 2 3 4 5

unipolar

unipolar/2PZ+RC

ν t

–3

v/v

(a)

(b)

–10

ν t

–3

Fig. 6.23 Average baseline shift (a) and baseline ﬂuctuation (b) for various shapers, as a function

of the signal rate

in the previous section. Let us apply two pole-zero ﬁlters with time constants

and

, which remove the signal tail up to ≈ 3

. For times longer than 3

the

pole-zero ﬁlters are just scaling the signal without changing the shape, because the

transfer is then approximately constant,

+ 1/

≈

for



. (6.67)

The signal tail is therefore given by the expression from Eq. (6.63) scaled by this

factor:

v(t) ≈

gqc

2ln(b/a)

< t < t

max

. (6.68)

The expressions for the average and r.m.s of the baseline are equal those from

Eq. (6.66), scaled by

with t

min

replaced by 3

In order to express the baseline shift and ﬂuctuation as a fraction of the pulse

height we must ﬁrst ﬁnd the peak of the unipolar-shaped signal processed by two

pole-zero ﬁlters. We use the fact that the pole-zero ﬁlters remove all but the ﬁrst ex-

ponential, so the remaining signal is given by i(t)=Aexp(−t

) [Eq. (6.56)], where

we have deﬁned A = A

+ A

. The procedure for determining the pole-zero

parameters results in a time constant

which is much smaller than the ampliﬁer

peaking time t

, so we can assume that the entire signal charge is contained in a

delta signal and ﬁnd the ampliﬁer output peak v

= gQ = g



∞

i(t)dt =

. (6.69)

Thus, the average baseline shift and the r.m.s. ﬂuctuation are

max

≈

√



Figure 6.23 shows the results using the time constants that we determined in the

previous section. With t

= 1 ns at a rate of 5 MHz we ﬁnd a baseline shift of 3%

and a baseline ﬂuctuation of 0.8% of the average pulse height. In Fig. 6.21c we see

6.2 Signal Shaping 209

the sequence of input pulses for this shaping scenario, and the reduction of baseline

shift and ﬂuctuation is evident. These numbers can be still improved by using more

pole-zero ﬁlters with time constants

... The above formula can be directly

extended for this case. With this technique we can in principle reduce the baseline

shift and ﬂuctuation to any desired level, but it requires a very precise choice of ﬁlter

constants matched to the chamber input signal and the working point.

At this stage one might be tempted to AC couple the signal by adding a CR

ﬁlter with time constant

in order to remove the remaining signal tail. But

that introduces a new tail because any signal processed by a CR ﬁlter is strictly

bipolar. Let us nonetheless consider this possibility for a moment. We approximate

the unipolar signal by a box signal of duration T and amplitude v

,asshownin

Fig. 6.20. Sending this signal through a CR ﬁlter of time constant

we ﬁnd the

following output signal, sketched in Fig. 6.24:

v(t)

= e

−

t < T

v(t)

= −(1−e

−

t > T. (6.70)

The CR ﬁlter has created a new tail, an undershoot with an area equal to the area of

the positive signal part. The baseline shift and ﬂuctuation due to this tail are again

calculated with Campbell’s theorem, and assuming that

 T we ﬁnd

= −

ντ

(1−e

−

) ≈−



ντ

(1−e

−

) ≈ T



. (6.71)

The baseline shift is independent of the ﬁlter time constant and simply reﬂects the

area balance of the positive and negative signal parts. In case the average distance

between two pulses

is twice the pulse width

= 1/

= 2T we have v/v

= 0.5,

which means that the baseline has shifted by half of the amplitude. The baseline

ﬂuctuation decreases for a longer RC time constant

and can be reduced to any de-

sired level. The fact that the baseline shift cannot be removed and that we therefore

have a rate-dependent baseline makes this shaping scheme with a long RC time con-

stant undesirable for high-rate applications. An example of such an RC ﬁlter applied

to a sequence of chamber pulses is shown in Fig. 6.21d.

The simplest way of avoiding baseline ﬂuctuations is the bipolar shaper described

in the last section. The results of the previous paragraph also apply to this shaper

1 2 3 4 5

t/T

0.25

0.5

0.75

1.25

2 4 6 8 10

–0.5

–0.25

–0.5

–0.25

0.25

0.5

0.75

1.25

t/T

Fig. 6.24 Square signal processed by a CR ﬁlter. The CR ﬁlter introduces a signal tail with an area

equal to the positive signal part

210 6 Electronics for Drift Chambers

because the bipolar shaper is equal to a unipolar shaper with an additional CR ﬁl-

ter. But now the CR time constant is short—of the same order as the signal length

T—so this shaper creates a large, but short undershoot. The area balance is done

‘quickly’, and after a duration of two to three times t

the signal is back to the

baseline.

The tail of the output of the bipolar shaper is found by writing the output signal

in the Laplace domain and interpreting the s in the numerator as the derivative of

the unipolar output signal:

v(t)=L

−1

(

bip

(s)I(s)

)

= g

−1



bip

(s)I(s)



≈ gi



(t) lim

s→0

bip

(s)

(6.72)

v(t) ≈−gq

√

n+1

2ln(b/a)

. (6.73)

We see that the negative signal tail tends to zero as 1/t

, which guarantees a quick

return to the baseline, and the inﬂuence on the following pulses is eliminated. The

effect of this shaping scheme is shown in Fig. 6.21b. It can be seen that the pulse

width and therefore the inefﬁciency is increased, but the baseline is equal to zero

and the ﬂuctuations are eliminated.

Conclusion of this Section

There are two linear signal processing strategies by which we can avoid baseline

shift and baseline ﬂuctuations at high counting rates owing to the long tail of wire

chamber signals:

Unipolar Shaping: A unipolar shaper and several pole-zero ﬁlters matched to

the individual exponentials that represent the chamber signal are used. The advan-

tage is a ‘short’ unipolar response. There are, however, several disadvantages to this

approach. First of all, the entire electronics readout chain must be DC coupled. This

causes problems for the electronics design because the DC offset variations within

the electronics chain, which are due to component imperfections or temperature

variations, must be very carefully controlled. The DC coupling also makes the elec-

tronics sensitive to low-frequency noise, which is very undesirable. In addition, the

wire signals are intrinsically AC coupled owing to the capacitor that decouples the

wire high voltage from the ampliﬁer input, so a unipolar shaping strategy is simply

not realizable with a linear electronics circuit. In addition, the pole-zero parameters

must be carefully tuned to the signal tail constant t

, which depends on the chamber

geometry, the gas used, and even the operating voltage. Since the absolute values of

resistors and capacitors in most electronics technologies can vary by 10–20% with

respect to the nominal values, the needed precision of the achieved time constants

must be very carefully evaluated. A solution to this problem is the introduction of

programmable pole-zero time constants, but this, of course, adds another level of

complication.

6.2 Signal Shaping 211

Bipolar Shaping: A bipolar shaper, realized by a fast CR ‘differentiation’ with

a timescale of the same order as the peaking time, produces a large, but short, sig-

nal undershoot. The signal dead time is approximately doubled with respect to the

unipolar + pole-zero shaping scheme and, as we will see later, the signal-to-noise

ratio is slightly reduced. The simplicity of the realization and the reduced sensitivity

of the signal shape to the input signal parameters makes this scheme a very desirable

one.

Nonlinear Shaping: Another possibility for avoiding the signal tail, which was

not discussed here, is a nonlinear circuit for elimination of the signal tail. These

circuits are usually called ‘baseline restorers’ and are realized by different kinds of

nonlinear components such as, e.g., diodes. These nonlinear baseline restorers allow

one to avoid the area balance present in linear AC coupled systems, so it is possible

to realize unipolar shaping without undershoot even if the signal is AC coupled.

A widely used circuit for this purpose is the Robinson restorer, which is described

in detail in [NIC 73]. The important advantages of linear signal processing that were

praised in the fourth paragraph of the introduction to this chapter are then lost.

6.2.4 Input Circuit

Up to now we have considered a setup where the induced current signal is equal to

the current ﬂowing into the ampliﬁer. The detector capacitance, together with the

ﬁnite ampliﬁer input impedance, that shape the signal and the current ﬂowing into

the ampliﬁer is different from the induced one. In general, the ampliﬁer input current

is calculated from the induced currents by applying them to the network representing

the wire chamber and the other impedance elements connected to the electrodes

(Fig. 6.25). If we want to study effects of coupling among different electrodes and

readout channels, such as, e.g., crosstalk, we have to use the entire equivalent circuit

representing the chamber. For studying the readout signal on a single electrode we

x(t)

a) b)

ind

(t)

V(t)

ind

(t)

ind

(t)

Fig. 6.25 Equivalent input circuit relating the currents induced on the electrodes to the ampliﬁer

input current

212 6 Electronics for Drift Chambers

(s)

det

ind

(t) I

(t)

(s)C

det

ind

(t)

Fig. 6.26 Input circuit and equivalent block diagram for a typical cathode and a wire readout

channel

can neglect the impedances between the electrodes, which are typically large, and

just consider the electrode capacitance and ampliﬁer input impedance. Figure 6.26

shows the circuit for a typical cathode and wire readout channel. A cathode strip or

cathode pad is typically directly connected to the ampliﬁer input and the induced

current is divided between the input impedance R

and the detector capacitance

det

. As a result, the ampliﬁer input current is equal to

(s)=

1+ s

ind

(s)

= R

det

. (6.74)

This is equivalent to the transfer function of an RC integration stage, as indicated

in the equivalent block diagram in Fig. 6.27. The wire is also connected to the high

voltage through a load resistor R

and decoupled from the ampliﬁer with a capacitor

C. The ampliﬁer input current is therefore related to the induced current as follows:

(s) ≈

(1+ s

)

(1+ s

)

ind

(s)

= R

det

= R

C. (6.75)

This transfer function is equivalent to an RC ﬁlter with time constant

followed by

an CR ﬁlter with time constant

. This means that the wire chamber signal is AC

iind(t)

v(t)

k i(t)

v1(t)

iind(t)

v(t)

k i(t)

Rin

Cdet

Rin

Cdet

Input Circuit

n = 2 unipolar shaper

n = 2 bipolar shaper

double Pole/Zero filter

Fig. 6.27 Equivalent block diagrams for a cathode channel processed by a unipolar shaper with

n = 2 and a double pole-zero ﬁlter (top), and a wire channel processed by a bipolar shaper with

n = 2 (bottom)

6.3 Noise and Optimum Filters 213

coupled and the output signal of any linear signal processing chain connected to the

wire is strictly bipolar.

The current signal I

(s) is then processed by the readout electronics chain.

Figure 6.27 shows how the input circuit is represented in an equivalent block

diagram. The examples show a cathode signal processed by a unipolar shaper with

two pole-zero ﬁlters and a wire signal processed by a bipolar shaper. The ampliﬁer

output signals for the two scenarios are given by

V(s)=

1+ sR

det

(1+ s

)

n+1

s + 1/

ind

(s), (6.76)

V(s)=

1+ sR

det

(1+ sR

(1+ s

)

n+1

ind

(s). (6.77)

6.3 Noise and Optimum Filters

By this point we are familiar with various possible transfer functions and pulse

shapes for analyzing wire chamber signals so that for a given application we have a

choice for achieving the best performance. There is one more element which often

plays an essential role when designing optimal chamber electronics—the inevitable

noise present in all electronics elements as well as in the wire chamber itself, which

represents a background against which the signals have to be distinguished. In the

following sections we characterize noise as a general phenomenon, discuss some

relevant noise sources, and develop a method for studying the inﬂuence of the noise

source on the signal pulse shapes. In the end we want to ﬁnd the optimum transfer

function for a given application.

For optimum charge measurement in cathode pad and cathode strip chambers, the

noise may limit the measurement accuracy and we want to choose H(i

) such that

the signal-to-noise ratio is maximized. The choice of H(i

) for time measurement

is more complicated. The time resolution of a wire chamber is determined by the

chamber internal effects of primary ionization and electron dynamics together with

the electronics noise. In order to minimize the noise effect one has to maximize the

so-called ‘slope-to-noise’ ratio, which may sometimes pose conditions unfavourable

for exploiting the measurement accuracy of the chamber itself.

In principle the signal-to-noise ratio and slope to noise ratio can always be

improved by increasing the chamber gas gain. On the other hand, for reasons of

chamber ageing and rate limitations, one wants to work at the lowest possible gas

gain.

The ‘optimum ﬁlters’ that minimize the effect of noise are related in a unique way

to the input signal shape and the noise spectrum, but they are not always realizable

or practical. They serve as a guideline for choosing the parameters for a realistic

design and show how far it deviates from the theoretically optimal performance.

The optimum ﬁlter might result in a very long pulse which is not compatible with

rate requirements or it might increase the impact of the chamber internal effects.

214 6 Electronics for Drift Chambers

After reviewing some basic deﬁnitions we consider the two main noise sources:

passive detector elements, which are ‘irreducible’ noise sources, and ampliﬁer noise,

which is subject to speciﬁcation. Finally we discuss how to optimize the ampliﬁer

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

6.3.1 Noise Characterization

Noise Alone

The random statistical movement of the atomic constituents of matter is the

source of electrical noise on any signal line. It occurs as random ﬂuctuations of

currents and voltages. For a description of noise, statistical concepts are appropriate.

Any observation of noise happens during an observation time T . Let us assume

a random voltage signal n(t) in the example depicted in Fig. 6.28a, where T was

chosen to be 100 ns. The variance

of n(t) is a measure of the ﬂuctuation. If the

average of the noise signal

n is zero, the variance is given by

= n

−n



T/2

−T/2

n(t)

dt. (6.78)

The frequency content can be described by a discrete set of frequencies f

= k/2

with k = 1,2,....IfweletT go to inﬁnity, the frequency content of the noise

becomes continuous, and a noise power spectrum w( f ) may be deﬁned over the

continuous frequency variable f , so that the variance of the ﬂuctuation is given by



∞

w( f )df =



∞

. (6.79)

The noise power spectrum is the essential tool for the characterization of electronics

noise. It speciﬁes the contribution in the frequency interval df to the square of the

ﬂuctuating quantity. If n(t) is a voltage or a current on unit resistance, then n

(t) is

a power and w( f ) a frequency power density; hence the name.

20 40 60 80 100

0.01 0.02 0.05 0.1 0.2 0.5 1

–5

–10

–15

()

GHz

(a)

(b)

Fig. 6.28 (a) Random noise signal n(t).(b) Power spectral noise density w(f ) of the noise signal,

which characterizes the noise signal in the frequency domain

6.3 Noise and Optimum Filters 215

The noise power spectrum must obviously be related to the Fourier components

of n(t). The relationship is as follows: If the Fourier transformation is

n(t)=



∞

−∞

N(i

, (6.80)

then, using (6.78) and N

∗

(−i

)=N(

) and going to the limit,

becomes

lim

T→∞



|N(i

, (6.81)

which implies

w( f )= lim

T→∞

|N(i

. (6.82)

As an illustration, a typical power spectrum w( f ) has been deﬁned to be ﬂat up to a

cutoff frequency at 150 MHz, together with an example of a ﬂuctuating voltage n(t)

containing frequencies according to w( f ). Both quantities are depicted in Fig. 6.28.

The higher this cutoff the more ‘spiky’ the noise.

In the case where the noise power spectrum is constant for all frequencies we call

the noise ‘white’ because the noise signal contains all frequencies in equal amounts.

The power spectrum w( f ) is not only useful for calculation of the r.m.s. ﬂuctuation

of the noise signal, but it is also related to some other key parameters which charac-

terize a noise signal. As was shown by Rice [RIC 44] the average frequency of zero

crossings of n(t) is given by

zero

= 2



∞

w( f ) df



∞

w( f ) df

. (6.83)

The evaluation of Eq. (6.83) using the noise power spectrum in Fig. 6.28b gives

zero

= 0.31GHz and the noise signal in Fig. 6.28a counts 27 zero crossings in

100 ns, giving a rate of 0.27 GHz. For a noise spectrum that is constant up to a

sharp cutoff frequency f

this expression evaluates to f

zero

= 2/

√

3 f

= 1.155 f

The frequency of zero crossings of the noise signal is therefore approximately given

by the cutoff frequency of the power spectrum.

Noise in a Linear Network

Now we want to ﬁnd out how the spectral density of a noise signal is transformed

if it is sent through a linear network with transfer function H(iω). The noise sig-

nal N(iω) is transformed into N(iω)H(iω). Using Eq. (6.82), we see that the noise

power spectrum is transformed into

w( f ) → w( f )|H(i2

f )|

. (6.84)

216 6 Electronics for Drift Chambers

If we assume a noise signal with a constant power spectrum up to a given frequency,

i.e., w( f )=w

for f < f

, we ﬁnd a noise r.m.s. of

√

. If we send this

noise signal through a low-pass ﬁlter of unity gain with a sharp cutoff frequency

< f

, the noise r.m.s. becomes

√

, meaning that the r.m.s. is reduced

by a factor



/ f

. As shown later one can improve the signal-to-noise ratio by

ﬁlters that are passing only the frequencies contained in the signal and removing the

others.

Next we want to ﬁnd the noise power spectrum at a given circuit node of a linear

network in the case where there are several independent (uncorrelated) current and

voltage noise sources distributed throughout the network. Elementary circuit theory

tells us that for a set of voltage and current sources distributed in a linear network,

the voltage on a given circuit node can be calculated in the following way: The

contribution of each individual source is computed separately by ‘switching all other

sources off’. A source is ‘switched off’ by shorting it out if it is a voltage source

and by opening it if it is a current source. The ﬁnal voltage is then the sum of these

individual contributions. For the network in Fig. 6.29a, e.g., the voltage at node

(s) due to the voltage source V (s) and current source I(s) is

(s)=

(s)

(s)+Z

(s)

V(s)+

(s)Z

(s)

(s)+Z

(s)

I(s) . (6.85)

If these are noise sources with voltage noise power spectrum w

( f ) and current

noise power spectrum w

( f ), Eq. (6.84) tells us that the noise power spectrum

( f ) is given by

( f )=



(iω)

(iω)+Z

(iω)



( f )+



(iω)Z

(iω)

(iω)+Z

(iω)



( f ). (6.86)

In order to characterize the signal, noise, and impedance of a wire chamber electrode

connected to the input of an ampliﬁer, it is very useful to further simplify the picture

by using Thevenin’s and Norton’s theorems (Fig. 6.30):

A linear, active network which contains one or more voltage or current sources

can be replaced by a single impedance Z(s) with a series voltage source V

(s)

(Thevenin’s theorem), or by the same impedance with a parallel current source I

(s)

(Norton’s theorem).

The impedance Z(s) can be obtained by shorting out the voltage sources, open-

ing up all current sources, and ﬁnding the equivalent impedance of the network.

The Theveninn equivalent voltage V

(s) or Norton equivalent current I

(s) can be

(s)

(s)I(s) V

(s)

V(s)

a) b)

(s) w

(f)

(s)

Fig. 6.29 A voltage source and a current source embedded in a linear network