1 Beijing, January 2008 Luciano Musa Signal Processing for TPCs in High Energy Physics ( Part I)...

1 Beijing, January 2008 Luciano Musa

Signal Processing for TPCs in High Energy Physics

(Part I)

Beijing, 9-10 January 2008

Luciano Musa - CERN

Outline

• Introduction to signal processing in HEP

• Detector signal processing model

• Electronic signal processing

• Preamplifier and Shaper

• Analogue to Digital Conversion

• Digital signal processing


Signal Processing in High Energy Physics

Introduction

Signal Processing is a way of converting an obscure signal into useful information

Signal processing includes signal formation due to a particle passage within a detector, signal amplification, signal shaping (filtering) and readout

The basic goal is to extract the desired and pertinent information from the obscuring factors (e.g. noise, pile-up)

The two quantities of greatest importance to be extracted from detector signals are:

• amplitude: energy, nature of the particle, localization

• time of occurrence: localization, nature of the particle



Means of Detection

Each detection method has to extract some energy from the particle to be detected

Nearly all detection methods (Cerenkov and Transition Radiation Detector being an exception) make use of ionization or excitation

Charged particles: ionization and excitation is produced directly by the interaction of the particle electromagnetic field with the electrons of the detection medium

A typical particle energy (today’s experiments) is of the order of few 100MeV

to GeV, while the energy loss can be below the MeV level. This is an example of

a nondestructive method for detection of charged particles.

All neutral particles must first undergo some process that transfers all or part

of their energy to charged particles. The detection method is destructive



Detection of Ionization (1/2)

In most ionization detectors the total ionization charge is collected using

an externally applied electrical field

Sometimes an amplification process by avalanche formation in a high

electrical field is used. Examples of detectors are:

a) Proportional chamber (MWPC, GEM, Megas);

b) Time Projection Chamber

c) Liquid-argon chamber;

d) Semiconductor detector.

All of them provide a certain amount of charge onto an output electrode

The electrode represents a certain capacitance

For signal-processing point of view these detectors are capacitive sources,

i.e. their output impedance is dominated by the capacitance



Detection of Ionization (2/2)

This common feature of all detectors for particle physics allows a rather unified

approach to signal processing

Despite of common features among various detectors used in high-energy physics, great differences exist among them

• The typical charge at the detector output can differ by six orders of

magnitude

• The output capacitances can differ by the same factor

• Signal dynamics

• Pulse repetition rate


Signal Induced by a Moving Charge

d

x

E

Vb

i

QA,el = -qV’A(P)

V’A

= -qx

d

QA,ion = qV’A(P)

V’A

= qx

d

Anode (A)

Cathode (C)

A constant induced current flows in the external circuit

i = dQA,el

dt= - q

ddxdt

Parallel Plate Ion Chamber

Applying

Green’s

Theorem

Example I

P(x)


- - -

- - -

- - -

- - -

-

+ +

+ +

+ +

+ +

+ +

+

anode

cathode

Avalancheregion

(amplification)

Charged particle

electron – ionpair

E ≠ 0

primary ionization

Ioncloude

Electroncloude

gas

Signal Induced by a Moving Charge

Cylindrical Proportional Chamber

i(t) = i0 / (1+t/t0)

Example II


Cd CiQ · s(t)

A

noiseless

preamplifier

Detector Signal Processing Model

The detector is modeled as a current source, delivering a current pulse

with time profile s(t) and charge Q, proportional to the energy released,

across the parallel combination of the detector capacitance Cd and the

preamplifier input capacitance Ci.

signal

processor

i2W=b

e2W=a

parallel

white noise

series

white noise

e2f=c/|f|

i2f=d·f

series

1/f noise

parallel

f noise

noise power spectral density

Ax[a+b/(Cd+Ci)2]

A(Q/(Cd+Ci))If s(t) = t)


Electronic Signal Processing

F(f) U(f)

F(f)

fU(f)

f

h(f)

f

Noise floorf0

f0

f0

Improved Signal/Noise Ratio

Example of signal filtering - the figure shows a “typical” case of noise filtering

In particle physics, the detector signals have very often a very large frequency spectrum

The filter (shaper) provides a limitation in bandwidth, and the output signal shape is different with respect to the input signal shape.

Signal Processor



F(f) U(f)

f(t)

fu(t)

f

h(f)

f

Noise floor

f0

f0

Improved Signal/Noise Ratio

The output signal shape is determined, for each application, by the following parameters:

• Input signal shape (characteristic of detector)• Filter (amplifier-shaper) characteristic

The output signal shape is chosen such to satisfy the application requirements:

• Time measurement• Amplitude measurement• Pile-up reduction• Optimized Signal-to-noise ratio


Cd+Ci i(t)

A

noiseless

signal

processor

T(s)

n(

f(t)

System Transfer function AT(s) / (s (Cd+Ci))

System Impulse response function vs(t) = ʆ-1AT(s) / (s (Cd+Ci))

Output root mean square noise [vN2]1/2

Optimum Processor maximize

{Q A/(Cd+Ci) MAX ʆ-1(I(s) T(s) / s) } / [vN2]1/2

OPTIMUM PROCESSOR

Signal Processing for Charge Measurement



Example RC=0.5s=j

R

C VoutVin

VinRXc

XcVout

CjfCjXc

1

2

1

VinRCj

Vout

1

1Low-pass (RC) filter

1 2 3 4 5

0.5

1

1.5

2

Integrator s-transfer function

H(s) = 1/(1+RCs)Impulse rsponse function

RCtet /

RC

1)(h

Log-Log scalet

f0.01 0.05 0.1 0.5 1 5 10

0.05

0.1

0.2

0.5

1

1 2 3 4 5

0.2

0.4

0.6

0.8

1

Step function response|h(s)|



R

C

VoutVin

VinRXc

RVout

CjfCjXc

1

2

1

VinRCj

RCjVout

1

High-pass (CR) filter

1 2 3 4 5

-2

-1.5

-1

-0.5

0.5

1

Differentiator time function

RCtett /

RC

1)()(h

H(s) = RCs/(1+RCs)

0.01 0.05 0.1 0.5 1 5 10

0.05

0.1

0.2

0.5

1

|H(s)|

Example RC=0.5s=j

1 2 3 4 5

0.2

0.4

0.6

0.8

1

Step function response



R

C

Vout

Vin2RCjωC(1

RCjωVout

R

CVin

1

Combining one low-pass (RC) and one high-pass (CR) filter :

Example RC=0.5s=j

CR-RC time functiont/RCet/RC)(1h(t)

1 2 3 4 5

-0.2

0.2

0.4

0.6

0.8

1

0.01 0.05 0.1 0.5 1 5 100.015

0.02

0.03

0.05

0.07

0.1

0.15

0.2

CR-RC s-transfer function

h(s) = RCs/(1+RCs)2

Log-Log scale f

|h(s)|

2 4 6 8 10 12 14

0.025

0.05

0.075

0.1

0.125

0.15

0.175




R

C

Vout

VinnRCjωC(1

RCjωVout

R

CVin

1

Combining N low-pass (RC) and one high-pass (CR) filter :

N times

Example RC=0.5, n=5s=j

CR-RC4 time functiont/RC3ett/RC).(4h(t)

2 4 6 8 10

-0.005

-0.0025

0.0025

0.005

0.0075

0.01

CR-RC4 s-transfer function

H(s) = RCs/(1+RCs)5

0.001 0.0050.01 0.05 0.1 0.5 1

0.0001

0.0002

0.0005

0.001

0.002

0.005

0.01

0.02

Log-Log scalef

|H(s)|

2 4 6 8 10

0.002

0.004

0.006

0.008

0.01

0.012



Preamplifier - Shaper

Preamplifier Shaper

(t) (t)

I O

What are the functions of the preamplifier and the shaper (in an ideal world) ?

• Preamplifier - An ideal integrator : it detects an input charge burst Q (t). The output is a voltage step Q/Cf•u-1(t). It has a large signal gain such that the noise of the subsequent stage (shaper) is negligible.

• Shaper - A filter with : characteristics fixed to give a predefined output signal shape, and rejection of (input) noise components outside of the useful output signal band.

u(t)



Cd

Qi

Cf

Vi V0

Zi=

Active Integrator (“charge-sensitive amplifier”)

Inverting Voltage Amplifier

dVo / dVi = -A vo = - A vi

Input Impedance = (no signal current flows into amplifier input)

Voltage difference across Cf: vf = (A+1)vi

Charge deposited on Cf: Qf = Cfvf = Cf(A+1)vi

Qi=Qf (since Zi = )

Effective Input capacitance 1)(Af

Ci

vi

Q

iC

GainfC1

fC1

1AA

iCA

iviCiAv

idQodV

QA (A>>1)



Qi is the charge flowing into the preamplifier …. but some charge remains on Cdet

What fraction of the signal charge is measured?

1

iCdetC

1

1

detC

iC

sQ

sQiC

iQ

detQ

iviC

sQiQ

(if Ci >> Cdet)

A=103

Cf = 1pFCi = 1nF Cdet = 10pF Qi / Qs = 0.99

Cd

Qi

Cf

Vi V0

Zi=

Active Integrator (“charge-sensitive amplifier”)

Inverting Voltage Amplifier

dVo / dVi = -A vo = - A vi

Input Impedance = (no signal current flows into amplifier input)



Preamplifier Shaper

CR-RC shaperIdeal Integrator

(t)

I O

(t)

1 2 3 4 5

-0.2

0.2

0.4

0.6

0.8

1

1 2 3 4 5

0.2

0.4

0.6

0.8

1

0.01 0.05 0.1 0.5 1 5 100.015

0.02

0.03

0.05

0.07

0.1

0.15

0.2

0.2 0.5 1 2 5 100.1

0.2

0.5

1

2

5

t

f

t

f

TRANSFER FUNCTION

s1

2s)CR(1sCR

H(s)

2 4 6 8 10 12 14

0.025

0.05

0.075

0.1

0.125

0.15

0.175

Output signal for an “ideal” input charge

t

2s)CR(1

CRO(s)

CR

t

eCR

to(t)


5 10 15 20 25 30 35

0.02

0.04

0.06

0.08

0.1

2 4 6 8 10

-0.005

-0.0025

0.0025

0.005

0.0075

0.01


Preamplifier Shaper

CR-RC4 shaperIdeal Integrator

(t)

I O

(t)

1 2 3 4 5

0.2

0.4

0.6

0.8

1

0.2 0.5 1 2 5 100.1

0.2

0.5

1

2

5

t

f

t

f

TRANSFER FUNCTION

s1

5s)CR(1sCR

H(s)

Output signal for an “ideal” input charge

t

5s)CR(1

CRO(s)

CR

t4

eCR

4to(t)

0.001 0.0050.01 0.05 0.1 0.5 1

0.0001

0.0002

0.0005

0.001

0.002

0.005

0.01

0.02

f



Vout

Cf

n IntegratorsDiff

Semi-Gaussian Shaper

Cd

Basic scheme of a Preamplifier-Shaper structure

T0 T0 T0

Vout(s) = Q/sCf . [sT0/1+ sT0].[A/(1+ sT0)]n Vout(t) = [QAn nn /Cf n!].[t/T0s]n.e-nt/Ts

Peaking time Ts = nT0

Vout (normalized to 1 ) vs. n Vout peak vs. n2 3 4 5 6 7

0.2

0.4

0.6

0.8

1

Vout(peak) = QAn nn /Cf n!en


5 10 15 20

0.01

0.02

0.03


Preamplifier Shaper

(t) z(t)

I O

u(t)Non Ideal Integrator CR_RC shaper

s)1

T(11

2s)CR(1sCR

H(s)

T1= 10 R C

2(RC) 81t]9RC10t/9e RC [RCt/RC-eo(t)

2s)CR(1s)CR10(1sCRO(s)



Preamplifier Shaper

(t) z(t)

I O

u(t)Non Ideal Integrator CR_RC shaper pz cancellation

s)1

T(11

2s)CR(1

s1

T1

H(s)

RC

t

eRC

to(t)

2s)CR(1CRO(s)

2 4 6 8 10 12 14

0.025

0.05

0.075

0.1

0.125

0.15

0.175Pole-zero cancellation



Basic scheme of a Preamplifier-Shaper structure with p-z cancellation

Vout

CfN IntegratorsDiff

Semi-Gaussian Shaper

Cd Tp T0

Rp

T0

n

0Ts1

A

0Ts1

pTs1

fC)

fTs(1Q(s)outV

By adjusting Tp such that Tp = Tf, we obtain the same shape as with a perfect integrator

Ts = nT0

sT

nt

esTt

n!f

CnnnAQ(t)outV


Analogue vs. Digital Filters

Advantages of Digital Filters

Finite-duration impulse responses are achievable Time-varying filters (even self-adaptive) are realized without

any special component by simply programming a different set of numbers in the filter

Certain realization problems, such as negative element values, and practical problems, such as inconveniently large components at low frequencies, do not arise

Programmability Greater accuracy is achieved No sensitive to environmental conditions (e.g. temperature,

supply voltages, etc.)

Disadvantages of Digital Filters

Power Limited by A/D speed and resolution

Real-time Digital Filters


Analogue to Digital Converter

Analog

Input

D0

D3

D2

D1DigitalOutput

Example - 4 bits A/D ConverterExample - 4 bits A/D Converter

t

analoginput

SamplingClock

4-bit

reconstructedsignal

0000

1111

Output Code

t

0100 1 0111…

1

1

1

16 possible output codes

ADC

Full scale amplitude

LSB=Full scale/2N

62.5 mV for 1V/4bits


signal

clock

jitter

Time and Amplitude Resolution

Limiting Factors in the Time and Amplitude resolution

• Time: Aperture time and Clock Jitter

• Amplitude: Noise floor



24

6810

1412

16182022

010K 100K 1M 10M 100M 1G 10G 100G

Heisenberg1Kohm thermal

1ps jitter

SamplingSample/s

ENOB Accuracy – Speed



The Uniform Sampling Theorem


Introduced by Shannon in 1948 (original idea by Nyquist in 1928)

It establishes the theoretical maximum sampling interval for complete signal reconstruction

The theorem holds (rigorously) only for physically unrealizable band-limited signals

Band-limited signals are a good approximation of many signal encountered in practice


0 cc

|F()|

tfωF

0ωF cωω

cπf2cω

Fourier Spectrum of a band-limited function f(t)

Theorem

f(t) is uniquely determined by its values at uniform time intervals that are 1/2fc

seconds apart

cω

nπf

cf2

nfnf

nπtcω

nπtcωsinn

nnftf

2fc

Nyquist frequency


Cardinal function

p()

(low pass filter)


The z-Transform

The Laplace transform reduces constant-coefficient linear

differential equations to linear algebraic equations Continuous-Time

Domain

Discrete-Time

Domain

The z-transform reduces constant-coefficient linear

difference equations to linear algebraic equations

Four tools for the analysis, synthesis, and understanding of linear time-

invariant electronic systems, either continuous or discrete time:

• Fourier transform

• Laplace transform

• Hilbert transform

• z-transform


The z-Transform

The ideal sampler circuit

T

f(t)

F(s)

fs(t)Fs(s)

f(t) is a real-life causal signal f(t) = 0 for t < 0

0 T 2T 3T 4T 5T

. . .

f(t)

fs(t)

t

We consider f(t) as modulating d(t)

d(t) is a train of periodic impulse

functions of area T

0nnTtδnTfTd(t)f(t)(t)sf


The z-Transform

The sampling process regarded as a modulation process

0nnTtδnTfTd(t)f(t)(t)sf

0nnTsenTf(s)sF

LAPLACE TRANSFORM

Unfortunately Fs(s) contains exponentials, hence is not algebraic in s.

sTez lnzT1sor

0nnznTfzFtsfZ

Z-transform


Mapping the s-Plane into the z-Plane

sTez

Re

Im

z-plane

0 1Re

Im

s-plane

0

j

jωσs

Tωjez

Sampled time functions that exponentially

decrease with increasing time

Sampled time functions that exponentially

increase with increasing time

Oscillating sampled time functions

Z=1, constant or increasing functions

depending on the multiplicity of the pole


Frequency-Domain Characteristics

0

|F(j)|

0

|Fs(j)|

ss

n

tsjnωetf

T

1(t)sf

0n)sjnωF(s

T

1(s)sF

EFFECT OF SAMPLING

Function not band-limited

Aliasing or Foldover


The z-Transform

0

|F(j)|

0

|Fs(j)|

ss

cc

cc ss

0

|Fs(j)|

sscc ss ss

Band-limited function

s 2c

s< 2c


Difference Equation

antialias

filterA&D DSPS&HANALOG

WORLDD/A reconstr.

filter

ANALOG

WORLD

Key element of a sampled-data system

Difference Equation

The operation of the filter is described by a difference equation that relates

u(nT) as a function of the present input sample f(nT) and any number of past

input and output samples

m

1iiTnTu

iK

q

0iiTnTf

iLnTuRecursion

formula

input samples f(nT) output samples u(nT)DIGITAL FILTER

Readout & Recording


Difference Equation

Example of a Difference Equation Computation

u(nT) = 3f(nT) – 2f(nT – T) + 6f(nT - 2T) + 2f(nT – 3T) – u(nT-T)

First-order difference equation (m=1, q=3)

0

1

2

3

4

5

1 2 3 4 t/T

Input function

0

10

20

30

40

50

12

34 t/T

Ouput function

-10

-20

5

6

7

…

fs(t) us(t)


The recursiveness of the Difference Equation suggest for its implementation:

a) program a General Purpose Computer

b) program a Digital Signal Processor

c) Hardwired Digital Filter

Digital Filter Simulation of a First-Order Difference Equation

f(nT)0

LTnTu1

KnTu

Recursion

formula

L0 + T

K1

f(nT)

u(nT)+

L0

T

Adder

Multiplier

Delay (Register)

Difference Equation


Digital filter network values are easily obtained, often by inspection

m

1i

iTnTuiK

q

0i

iTnTfiLnTu

Difference equationDifference equation

m

1i

iziK1

q

0i

iziL

F(z)U(z)H(z)

System FunctionSystem Function

Canonic FormL0

-K1

L1

L2

Lq

-K2

-Kq

-Km

+ +Z-1 Z-1Z-1Z-1fs(t) us(t)

Feed-forward path

Feed-back path

m delays

Digital Networks

IIR FILTER


Digital Networks

Digital filter network values are easily obtained, often by inspection

m

1i

iTnTuiK

q

0i

iTnTfiLnTu

Difference equationDifference equation

m

1i

iziK1

q

0i

iziL

F(z)U(z)H(z)

System FunctionSystem Function

Non-recursive

L0

L1

L2

Lq+ +Z-1 Z-1Z-1fs(t) us(t)

q delays

If all Ki’s are 0

FIR FILTER

1 Beijing, January 2008 Luciano Musa Signal Processing for TPCs in High Energy Physics ( Part I)...

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