Signal Processing for TPCs in High Energy Physics ( Part I) Beijing, 9-10 January 2008
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Transcript of Signal Processing for TPCs in High Energy Physics ( Part I) Beijing, 9-10 January 2008
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
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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
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Signal Processing in High Energy Physics
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
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Signal Processing in High Energy Physics
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
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Signal Processing in High Energy Physics
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
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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)
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- - -
- - -
- - -
- - -
-
+ +
+ +
+ +
+ +
+ +
+
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
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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)
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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
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Electronic Signal Processing
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
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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
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Electronic Signal Processing
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)|
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Electronic Signal Processing
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
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Electronic Signal Processing
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
Step function response
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Electronic Signal Processing
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
Step function response
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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)
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Preamplifier - Shaper
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)
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Preamplifier - Shaper
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)
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Preamplifier - Shaper
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)
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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
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
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Preamplifier - Shaper
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
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5 10 15 20
0.01
0.02
0.03
Preamplifier - Shaper
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)
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Preamplifier - Shaper
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
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Preamplifier - Shaper
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
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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
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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
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signal
clock
jitter
Time and Amplitude Resolution
Limiting Factors in the Time and Amplitude resolution
• Time: Aperture time and Clock Jitter
• Amplitude: Noise floor
Analogue to Digital Converter
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24
6810
1412
16182022
010K 100K 1M 10M 100M 1G 10G 100G
Heisenberg1Kohm thermal
1ps jitter
SamplingSample/s
ENOB Accuracy – Speed
Analogue to Digital Converter
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The Uniform Sampling Theorem
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
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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
The Uniform Sampling Theorem
Cardinal function
p()
(low pass filter)
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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