Journal of Modern Physics, 2013, 4, 42-47
doi:10.4236/jmp.2013.45B009 Published Online May 2013 (http://www.scirp.org/journal/jmp)
A Comparative Study of Amplitude and Timing
Estimation in Experimental Particle Physics using
Monte Carlo Simulation
Hongda Xu1, Datao Gong2, Yun Chiu1
1Departmentof Electrical Engineering, University of Texas at Dallas, Richardson, TX, USA
2Departmentof Physics, Southern Methodist University, Dallas, TX, USA
Email: dgong@mail.smu.edu
Received 2013
ABSTRACT
Optimal detection of liquid ionization calorimeter signal in experimental particle physics is considered. A few linear
and nonlinear approaches for amplitude and arrival time estimation based on the χ2 function are compared in simulation
considering the noise sample correlation introduced by the analog pulse shaper. The estimation bias of the first-order
approximation, a.k.a linear optimal filtering, is studied and contrasted to those of the second-order as well as the ex-
haustive search. A gradient-descent technique is presented as an alternative to the exhaustive search with significantly
reduced search time and computation complexity. Results from various pulse shapers including the CR-RC2, CR-RC3,
and CR2-RC2 are also compared.
Keywords: Liquid Ionization Calorimeter; Detection; Optimal Filtering; Amplitude and Timing Estimation; χ2
Function; CRm-RCn pulse Shaper; Linear Optimal Filtering; Exhaustive Search; Gradient Descent; Monte
Carlo
1. Introduction
In particle physics experiment, it is common to measure
the small charge signal collected by the detector in pres-
ence of electronic noise. A collection of data acquisition
and signal processing techniques are well developed to
optimize the singal-to-noise ratio (SNR) in such systems
[1]. Due to the recent development of optical transmis-
sion techniques, a trend in detector readout system design
is to continuously digitize the detector signal and con-
tinuously transmit the data out of the front-end to the
control room, in which complex signal processing can be
imposed to improve the overall detector system per-
formance [2]. This paper reviews several of such signal
processing techniques for liquid ionization calorimeters
and compares their performance in simulation. A novel
accurate, fast, and low-cost gradient-descent technique is
also introduced in the paper.
2. Liquid Ionization Calorimeter
Liquid ionization calorimeter is an energy-measurement
detector widely deployed in many particle physics ex-
periments [3-6]. Since the liquid gap between the elec-
trode plates is narrow, e.g., about 2 mm in ATLAS liquid
argon calorimeter, the ionization triggered by the
electromagnetic shower is instantaneous and the process
is followed by a drift of the electrons towards the anode
plate. Thus, the detector output signal is modeled as a tri-
angular current pulse. Liquid ionization calorimeter usu-
ally exhibit so long drift time, dependent on the drift ve-
locity and the gap size [7]. For example, the drift time in
AT- LAS liquid argon calorimeter is about 450 ns, much
longer than the bunch crossing time which is 25 ns. To
avoid long dead time and to reduce noise in the meas-
urement, a CRm-RCn pulse shaper – a chain of integrator
(RC) and differentiator (CR) circuits – is always employed
in the analog front-end before digitalization [8]. The gen-
eral transfer function of a CRm-RCn shaper is given as
follows.



1
m
s
mn
s
s
Hs
s
(1)
where τs is the time constant of the shaper. An intuitive
way of understanding the functionality of the shaper is
that the integrator limits the input bandwidth and slows
down the rising edge of the current pulse for analog-
to-digital conversion (ADC) while the differentiator re-
stores the baseline quickly by removing the long tail of
the pulse to reduce the possibility of signal pileup. Care-
fully choosing the time constant τs gives a smooth shaper
Copyright © 2013 SciRes. JMP
H. D. XU ET AL. 43
output waveform with minimal pedestal recovering time,
which can largely relax the ADC sample rate while re-
taining sufficient samples for post-processing.
Figure 1 sketches the shaped waveform as well as the
triangular current pulse from the detector. The parame-
ters used for modeling waveforms in Figure 1 are ex-
tracted from [9] for ATLAS liquid argon calorimeter, e.g.,
a CR-RC2 shaper with τs set to 13 ns, and the output
peaking time is approximately 50 ns.
3. Detection and Signal Processing
While detection is a general signal-processing topic that
has been well studied, what we are most interested in par-
ticle physics experiments is how to precisely measure the
amplitude and timing information of the sampled wave-
form of the detector output – the amplitude A represents
the energy of the incident particle shower and the timing
signifies the arrival time τ of the particle thus our ability
to correlate signals and events in time.
The general approach of determining amplitude and
timing information from limited number of samples can
be derived from the theory of optimal filtering [1,10].
The technique in the core constitutes a search algorithm
to project the samples to the known signal space of the
detector front-end including the preamplifier, the shaper,
and the analog-to-digital converter to maximize the SNR
of the detected signal. Two leading noises are typically
considered in the optimization process, the electronic
noise and the pileup noise. At their origin, the dominant
electronic noises, e.g., thermal noise and shot noise, are
white. Temporal correlation however is introduced by the
detector front-end processing, particularly the pulse
shaper, and needs to be considered in the optimization
procedure. Unlike the stochastic deteriorating effect of
-100 0100 200 300 400 500 600
-0.2
0
0.2
0.4
0.6
0.8
1
time [ns]
norm a lized amplitude
det ect or pulse
sha per o utp ut
samples
proce ssed sa m pl es
Figure 1. The triangular current pulse produced by a liquid
ionization calorimeter and the output waveform of the
analog pulse shaper. The five samples (at the sample rate of
40 MS/s) for post-processing are highlighted.
the electronic noise, the signal pileup by nature is deter-
ministic and should be treated as inter-event interference
(IEI). In particle detectors operating at high luminosity
levels, many events are produced at each bunch crossing.
The densely packed calorimeter cells and the long tail of
the detector current pulse tend to aggravate the IEI prob-
lem, leading to an equivalent model termed pileup noise
to highlight the statistical property of the effect rather
than its deterministic physical origin.
Many works have been reported for detetctor signal
processing and some are cited as follows. A linear opti-
mal filtering approach was reported in [10], in which the
amplitude and arrival time of the incoming pulse are es-
timated by the weighted sum of a few relevant samples
of the ADC output. The noise autocorrelation matrix is
utilized in this technique to improve the estimation accu-
racy. With the continuous data output in the upgraded
ATLAS liquid argon calorimeter, a Wiener filter ap-
proach was reported to reduce the pileup noise [11].
4. Monte Carlo System Model
A Monte Carlo simulation platform for modeling the ana-
log front-end of liquid inonization calorimeter is con-
structed in MATLAB® / SIMULINK®. The model takes
into account the detector noise and the input-referred
front-end electronic noise as well as the shaper frequency
response. Other design parameters are extracted from
ATLAS calorimeter system [9] – the ionization current
for EMB (electromagnetic barrel) is approximately
3μA/GeV, the typical value for charge drift time is 450ns,
a bipolar CR-RC2 shaper with a time constant τs of 13 ns,
and a sample rate of 40 MS/s which yields five signal
samples to be used for estimation. The peaking time at
the output of the shaper is close to 50 ns due to the con-
tribution from the shaper and the RC delay of the pream-
plifier. The CR-RC2 shaper is a good compromise be-
tween the number of filtering stages, power dissipation,
and performance. In our model, the ADC quantization
noise is not included.
In the simulation, the front-end electronic noise is as-
sumed to be Gaussian distributed with a standard devia-
tion of 10% of the peak value of the current pulse. De-
tector calibration is assumed and the ideal output pulse
shape is known a priori.
5. Amplitude and Timing Estimation
In line with the differing physical origins of the elec-
tronic noise, pileup noise and their effects on the detec-
tion process, we will concentrate on the amplitude and
timing estimation for liquid ionization calorimeters con-
sidering electronic noise only in the rest of this paper.
Pileup noise will be reported in a future study.
Copyright © 2013 SciRes. JMP
H. D. XU ET AL.
44
5.1. χ2 Exhaustive Search
Considering the correlation between the noise samples
introduced by the shaper, the χ2 function can be defined
as follows [10]:





2,iiijj j
ij
ASAgtVSAgt

 
(2)
whereVij is the weight matrix for the measured samples.
V is the inverse of the noise autocorrelation matrix R
with Rij = <ni·nj> and ni is the noise sample.
The χ2 function defines a non-negative quadratic error
surface as a function of A and τ between the noisy
samples Si and the known pulse shape g(ti) as sketched in
Figure 2. A straightforward approach to determine the
best estimate for A andτis to perform an exhaustive
search on the error surface. Albeit not computationally
efficient, the exhaustive search result establishes a
baseline for the estimation approaches covered in the
subsequent sections.
The Monte Carlo simulation results of the χ2 exhaus-
tive search are shown in Figure 3 and Figure 4 The per-
formance of the method is limited by the finite step size
employed by the search algorithm. No obvious trend for
the estimation error is observed.
5.2. Least-Square Exhaustive Search
The derivation of the weight matrix V (or the noise
autocorrelation matrix R) requires precise knowledge of
the impulse response of the detector front-end. The com-
putation of the χ2 function in Eq. (2) also dictates N2 mul-
tiplications for N samples when the off-diagonal entries
of V are nonzero. In practice, the magnitude of the
off-diagonal entries of R can be small relative to the
main diagonal entries. In such cases, the V matrix can be
well approximated by the identity matrix. Thus, Eq. (2)
reduces to
Figure 2. The quadratic error surface of the χ2function in
terms of A and τ.
4.1 4.114.12 4.13
x 10
-7
0
2
4
6
8
10
12
14
16
18
20
A
est
2.8 2.933.1 3.2
0
5
10
15
20
25
30
est
[ns]
Figure 3. Histogram of 100 Monte Carlo runs for amplitude
(left) and arrival time (right) estimation using the χ2 exhaustive
search method. The standard deviation of the detector noise
is set to 10% the peak value of the detector current pulse.
The sample period T = 25 ns, A0 = 4.1134e-7, and τ0 = 3 ns.
-10 010
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1x 10
-9
del ay [ns]
<A
est
- A
0
>
-10 010
-0.1
-0 .08
-0 .06
-0 .04
-0 .02
0
0.02
0.04
0.06
0.08
0.1
del ay [ ns]
<
est
-
0
> [ns]
2
-ES
LS-ES
2
-ES
LS-ES
Figure 4. The χ2 versus the least-square search results of
1000 Monte Carlo runs: the mean estimation error and
standard deviation for amplitude Aest (left) and arrival time
τest (right). τ0 = [T/2, T/2]. Other simulation parameters are
identical to those of Figure 3.


2
2,ii
i
ASAgt


(3)
This is identical to the least-square metrics to fit N
samples to the known pulse shape g (t).
In our Monte Carlo simulation, the above argument is
confirmed with a CR-RC2 pulse shaper. Again, an ex-
haustive search is employed to determine the optimal fit
of the five samples to g (t). The estimation errors for A
and τ are plotted in Figure 4 and overlaid with the χ2
exhaustive search results – the two results are nearly
identical.
Copyright © 2013 SciRes. JMP
H. D. XU ET AL. 45
5.3. χ2 with First-Order Taylor Expansion
Taylor expansion can be performed on g(t) in the vicinity
of τ = 0 to reduce the computation complexity of the χ2
function, i.e.,
'
iii
A
gtAgtAgt

  (4)
Where g'(t) is the first-order derivative of g(t). Thus,

2
121 2
'
iiiijj j
ij
Sg gVS gg
 
 
'
j
(5)
where α1 = A and α2 = Aτ.
Compared to Eq. (2), Eq. (5) defines a first-order ap-
proximated quadratic error surface in terms of A and τ,
which can be used to perform a search or to directly de-
rive a closed-form analytical solution to the problem. The
latter has been done in [10] and results are quoted as fol-
lows


12435
2153
1
1
QQ QQ
QQ QQ

 
4
(6)
where
1
2
3
4
5
2
12 3
''
'
'
iij j
ij
iijj
ij
iijj
ij
iij j
ij
iijj
ij
QgVg
QgVg
QgVg
QgVS
QgVS
QQ Q
 
(7)
5.4. Linear Optimal Filtering
A linear optimal filtering technique was proposed in [10]
to minimize the computing effort involved in determina-
tion of the amplitude and arrival time information. The
formulation of the optimal filter is quoted as follows
ii
i
ii
i
A
uaS
A
vbS


(8)
The coefficients of ai and bi are given as



aVg Vg'
bVgVg'
(9)
where λ = Q2/Δ, κ = Q3/Δ, μ = Q3/Δ, ρ = Q1/Δ, and Q
and Δ are defined in Equation (7).
The advantage of this technique is that the filter tap
values are pre-calculated.Thus, the computation can be
performed on the fly when data samples arrive, suitable
for continuously operated detectors such as the proposed
upgrade for ATLAS. It is also useful in resource-con-
strainted implementation, e.g., FPGA or DSP, or latency-
sensitive applications.
It can be shown that linear optimal filtering is equiva-
lent to the χ2 method of first-order approximation [10].
The simulation results of both for A and τ are illustrated
in Figure 5. It is interesting to note that the estimation
error exhibits a quadratic dependence on τ as predicted
by Eqs. (4) and (5) fortruncating the second- and higher-
order terms in the Taylor expansion.
5.5. χ2 with Second-Order Taylor Expansion
The first-order Taylor expansion of the χ2 function leads
to a rather large estimation error or bias when τ is large
4.9% for amplitude and 12% for arrival time when τ
reaches ±T/2 in Figure 5,. One way to mitigate the large
error is to iterate the series expansion and Equation (5)
by re- calculating the g' and Q or, in the linear optimal
filtering case, re-derive the filter tap values ai and bi and
iterate Equation (8). Simulation results are shown in Fig-
ure 6 for linear optimal filtering with two iterations. The
computing over- head in either case is significant. Another
solution is to resort to a second-order Taylor expansion,
 
2
1
'
2
iii
'
'
i
A
gtAgtAg tAgt

  (10)
Where g''(t) is the second-order derivative of g(t).
-10 010
-2.5
-2
-1.5
-1
-0.5
0
0.5
1x 10
-8
del ay [n s]
<A
es t
- A
0
>
-10 010
-1.5
-1
-0.5
0
0.5
1
1.5
2
del ay [n s]
<
es t
-
0
> [ns]
2
-E S
2
-ES-2O
2
-ES-1O
LOF
2
-E S
2
-ES-2O
2
-ES-1O
LOF
Figure 5. The simulation results of 100 Monte Carlo runs
for the first-order χ2 exhaustive search, linear optimal
filtering, and second-order χ2 exhaustive search: the mean
estimation error and standard deviation for amplitude Aest
(left) and arrival time τest (right). Simulation parameters
are identical to those of Figure 4.
Copyright © 2013 SciRes. JMP
H. D. XU ET AL.
46
-10 010
-2.5
-2
-1.5
-1
-0.5
0
0.5
1x 10
-8
del ay [ ns ]
<A
est
- A
0
>
-10 010
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
del ay [ ns ]
<
est
-
0
> [ns]
1st iter ation
2nd it er at ion
1st iter at ion
2nd it er at ion
Figure 6. The simulation results of 100 Monte Carlo runs
for the linear optimal filtering case with two interations.
Simulation parameters are identical to those of Figure 4.
The error surface of the second-order approximation
can be similarly defined as of Equation (5). However, the
nonlinearity of the second-order term excludes the possi-
bility of a closed-form analytical solution. Thus, an ex-
haustive search is performed instead and the simulation
results are shown in Figure 5 as well. We note that the
estimation error in this case exhibits a cubic dependence
on τ as the truncation error in Equation (10) is dominated
by the third order.
5.6. Gradient Descent Approach
As simulation results evidenced so far, the exhaustive
search approach produces the best estimation accuracy at
the cost of high computation complexity. To improve the
efficiency of the search, a gradient-descent approach is
devised. As illustrated in Figure 2, the bowl-shaped error
surface of the χ2 function exhibits a global minimum.
Starting from a random initial point on the surface, a
search direction can be derived by comparing the value
of the function at the current position to those offset by a
step size away in both A and τ direction. The current po-
sition is then advanced in the direction that minimizes the
function value. The process is itereated until convergence at
the bottom of the surface. Simulations for the gradient-
descent approach were performed with four random
starting point and the resulting search paths are plotted in
Figure 7. Comparing to the exhaustive search method in
which the whole error surface is evaluated, the gradient-
descent approach significantly reduces the computation
involved. Table 1 compares the typical required number
of iterations for the two cases.
6. Other Shapers
The amplitude and timing estimation techniques covered
in Sec. 4 are also tested with CR-RC3 and CR2-RC2 pulse
shapers. For the same 40-MHz sample rate, the peak of
g(t) falls in the middle between two samples in contrast
to the case of the CR-RC2 shaper in which the peak is
very close to a sample point. Figure 8 and Figure 9 plot
Figure 7. Simulated gradient-descent search paths for four
random starting point on the error surface of the χ2function.
The x-axes are delay (in ns) and the y-axes are amplitude.
Table 1. Effciency comparison between gradient-descent
and exhaustive search methods.
Method Gradient Descent Exhaustive Search
Iterations 680* 603201
Computations
per
iteration
90 multiplications
72 additions
3 comparisons
30 multiplications
24 additions
1 comparison
*Averaged over different delay times.
-100 0100 200 300 400 500600
-0.2
0
0.2
0.4
0.6
0.8
1
time [ns]
norm alized am plitude
-10 010
-3
-2
-1
0
x 10
-8
del ay [ns]
<A
est
- A
0
>
2
-ES
2
-ES -2O
2
-ES -1O
LOF
-10 010
-1.5
-1
-0.5
0
0.5
1
del ay [ ns ]
<
est
-
0
> [ns]
2
-ES
2
-ES -2O
2
-ES -1O
LOF
detec t or puls e
shaper output
samples
proce ssed s am ples
Figure 8. The simulation results of 100 Monte Carlo runs
for CR-RC3 shaper. Simulation setupis identical to that of
Figure 4. Standard dev. bars are not shown for clarity.
Copyright © 2013 SciRes. JMP
H. D. XU ET AL.
Copyright © 2013 SciRes. JMP
47
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-100 0100 200 300 400 500 600
-0.5
0
0.5
1
time [ns]
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x 10
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0
>
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-ES
2
-ES-2 O
2
-ES-1 O
LOF
detect o r pulse
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samples
processed sampl es
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