An Optimization of 16-Point Discrete Cosine Transform Implemented into a FPGA as a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower Experiments

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An Optimization of 16-Point Discrete Cosine
Transform Implemented into a FPGA as a Design
for a Spectral First Level Surface Detector Trigger
in Extensive Air Shower Experiments
Zbigniew Szadkowski
University of Łód´ z
Department of Physics and Applied Informatics,
Faculty of High Energy Astrophysics, Łód´ z
Poland
1. Introduction
The Pierre Auger Observatory is a ground based detector located in Malargue (Argentina)
(Auger South) at 1400 m above the sea level and dedicated to the detection of ultra
high-energy cosmic rays with energies above 1018eV with unprecedented statistical and
systematical accuracy. The main goal of cosmic rays investigation in this energy range is to
determine the origin and nature of particles produced at these enormous energies as well as
their energy spectrum. These cosmic particles carry information complementary to neutrinos
and photons and even gravitational waves. They also provide an extremely energetic stream
for the study of particle interactions at energies orders of magnitude above energies reached
at terrestrial accelerators (Abraham J. et al., 2004).
The flux of cosmic rays above 1019eV is extraordinarily low:
event per square-kilometer per century. Only detectors of exceptional size, thousands of
square-kilometers, may acquire a significant number of events. The nature of the primary
particles must be inferred from properties of the associated extensive air showers (EAS).
The Pierre Auger Observatory consists of a surface detectors (SD) array spread over 3000
km2for measuring the charged particles of EAS and their lateral density profile of muon
and electromagnetic components in the shower front at ground, and of 24 wide-angle
Schmidt telescopes installed at 4 locations at the boundary of the ground array measuring
the fluorescence light associated with the evolution of air showers:
subsequent deterioration during a development.
cross-calibrations between different experimental techniques, controlling and reducing the
systematic uncertainties.
Very inclined showers are different from the ordinary vertical ones. At large zenith angles
the slant atmospheric depth to ground level is enough to absorb the part of the shower that
follows from the standard cascading interactions, both of electromagnetic and hadronic type.
Only penetrating particles such as muons and neutrinos can traverse the atmosphere at large
zenith angles to reach the ground or to induce secondary showers deep in the atmosphere and
close to an air shower detector.
on the order of one
the growth and
Such a "hybrid" measurements allow
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The ability to analyze inclined showers with zenith angles larger than 60◦induced by
neutrinos or photons essentially increases the acceptance of the surface array and opens a
part of the sky that was previously inaccessible to the detector. These showers provide a
new tool for ultra high energy cosmic rays interpretation because they are probing muons
of significantly higher energies than vertical showers. Spectral triggers offering a pattern
recognition in a frequency domain may improve a standard detection technique based on
the signal coincidences from many PMT channels above some thresholds in the time domain.
The "old" muon shower fronts have only a small longitudinal extension, which is leading
to short detector signals also in time. To identify these showers at the presence of "young"
showers with a large electromagnetic component one may need a very good spectral
sensitivity to the fast muon component in the trigger.
The main advantage of the spectral trigger is the scaling feature.
coefficients depends only on the shape of signals, not on their amplitudes. Triggers sensitive
on the shape of FADC traces may detect events with expected characteristics i.e. the fast
attenuated, very short peaks related to the muonic, flat fronts coming from very inclined
showers. Independence of the amplitude is especially promising for the Auger North, where
due to a single PMT in the surface detectors the coincidence technique cannot be used. In
order to keep reasonable trigger rate for the 1st level trigger (ca. 100 Hz), the threshold for the
1st trigger should be much higher than for example in the Pierre Auger Observatory, where
3-fold coincidences attenuated a noise.
The set of the DCT
Fig. 1. Position of triggered surface detectors on the Auger array for the very inclined shower
(θ = 83.5◦) nr 1155555. Muons triggered only few surface detectors, although they crossed
several hundred detectors. A distance between opposite detectors is 54 km.
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An Optimization of 16-Point Discrete Cosine Transform Implemented into a FPGA as a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower
Experiments3
2. Triggers
Two different triggers are currently implemented at the 1st level. The first is a single-bin
trigger generated as 3-fold coincidence of the 3 PMTs at a threshold equivalent to 1.75 vertical
emitted muons. The estimated current for a Vertical Equivalent Muon (IVEM) is the reference
unit for the calibration of FADC traces signals and corresponds to ca.
This trigger has a rate of about 100 Hz. It is used mainly to detect fast signals, which
correspond also to the muonic component generated by horizontal showers. The single bin
trigger is generated when the input signal is above the fixed thresholds calculated in the
micro-controller during the calibration process. It is the simplest trigger useful for high-level
signals. The second trigger is the Time over Threshold (ToT) trigger that requires at least 13
time bins above a threshold of 0.2 IVEM. A pre-trigger ("fired" time bin) is generated if in
a sliding time window of 120 × 25 ns length a coincidence of any two channels appears.
This trigger has a relatively low rate of about 1.6 Hz, which is the expected rate for two
muons crossing the Auger surface detector. It is designed mainly for selecting small but
spread-in-time signals, typical for high energy distant EAS or for low energy showers, while
ignoring the single muon background (Abraham J. et al., 2010).
Cherenkov light generated by very inclined showers crossing the Auger surface detector can
reach the PMT directly without reflections on Tyvec liners. Especially for "old" showers the
muonic front is very flat. This together corresponds to very short direct light pulse falling on
the PMT and in consequence very short rise time of the PMT response. For vertical or weakly
inclined showers, where the geometry does not allow reaching the Cherenkov light directly
on the PMT, the light pulse is collected from many reflections on the tank walls. Additionally,
the shower developed for not so high slant depth are relatively thick. These give a signal from
a PMT as spread in time and relatively slow increasing.
Hadron induced showers with dominant muon component give an early peak with a typical
rise time mostly from 1 to 2 time bins (by 40 MHz sampling) and decay time of the order
of 80 ns (Aglietta et al., 2005). The estimation of the rise time for the front on the base of
one or two time bins is rather rough. The rise time calculated as for two time bins may be
overestimated due to a low sampling rate and an error in a quantization in time. Higher
time resolution would be favorable. The expected shape of FADC traces suggests to use a
spectral trigger, instead of a pure threshold analysis in order to recognize the shape of the
FADCtracescharacteristicforthetracesofveryinclinedshowers. Themonitoringoftheshape
would include both the analysis of the rising edge and the exponentially attenuated tail. A
very short rise time together with a relatively fast attenuated tail could be a signature of very
inclined showers. We observe numerous very inclined showers crossing the full array but
which "fire" only few surface detectors (Fig. 1). For that showers much more detectors should
have been hit. Muonic front probably produces PMT signals not high enough to generate
3-fold coincidences, some of signals are below of thresholds (see Fig. 2). This may be a reason
of "gaps" in the array of activated surface detectors.
50 ADC-counts.
3. Discrete Fourier Transform vs. Discrete Cosine Transform
There are several variants of the DCT with slightly modified definitions. The DCT-I is exactly
equivalent (up to an overall scale factor of 2), to a DFT of 2N - 2 real numbers with even
symmetry. The most commonly used form of the Discrete Cosine Transform is DCT-II.
¯Xk= αk
N−1
∑
n=0
xncos
?π
N(n +1
2)k
?
(1)
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a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower Experiments

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240
FADC traces from "Ramon"
PMT1
PMT2
PMT3
PMT3
1.93 VEM
1.75 VEM
160
C counts
ADC
80
0
240 244 248 252256 260
time bin
240
FADC traces from "Christian"
PMT1
PMT2
PMT3
PMT3
1.75 VEM
1.64 VEM
160
C counts
ADC
80
0
240244 248252256260
time bin
240
FADC traces from "Juancho"
PMT1
PMT2
PMT3PMT3
1.75 VEM
1.68 VEM
160
counts
ADC
80
0
240244 248252256 260
time bin
Fig. 2. FADC traces (in ADC-counts) of a horizontal shower (no. 01145055: θ = 83.3◦)
registered in three detectors: Ramon, Christian and Juancho, respectively, and shown for the
range of (240 - 265) time bins. Only the signal in the Ramon detector (1.93 IVEM) is above the
standard threshold of 1.75 IVEM. Signals in Cristian (1.64 IVEM) and Juancho (1.68 IVEM)
detectors are below the standard thresholds and they are detected by chance (compare a
registration efficiency for a similar event shown in Fig. 1). For all very inclined showers the
rising edge corresponds to one or two time bins.
where α0=
The DCT-III form is sometimes simply referred to as "the inverse DCT" (IDCT). A variant
of the DCT-IV, where data from different transforms are overlapped, is called the Modified
Discrete Cosine Transform (MDCT). The DCT is a Fourier-related transform similar to the
DFT, but using only real numbers. DCT are equivalent to DFT of roughly twice the length,
operating on real data with even symmetry (since the Fourier transform of a real and even
function is real and even), where in some variants the input and/or output data are shifted by
half a sample. The DCT-II and DCT-IV are considered as the alternative approach to the FFT.
In fact, the FFT routine can be supplied in an interleaving mode, even samples treated as real
data, odd samples as imaginary data. A trigger based on Discrete Fourier Transform (DFT)
(Radix-2 FFT) (Szadkowski, 2006) has already been implemented in the 3rd generation of the
Front FEB based on Cyclone™ Altera®chip (Szadkowski, 2005b). However, for real signal xn
1
√Nand αk=
2
√Nfor k ≥ 1.
¯X N
2+k=
N−1
∑
n=0
xne−j2π
N(N
2+k)n=
N−1
∑
n=0
xn(−1)n?
ej2π
Nkn?∗=¯X∗
N
2−k
(2)
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An Optimization of 16-Point Discrete Cosine Transform Implemented into a FPGA as a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower
Experiments5
and
symmetric, the imaginary part is asymmetric. The useful information is contained only in 1st
?
changing from zero to
2
N
?
N
2
th?
?
spectral line of¯Xk, k = 0,1,...,N-1 is lying on a symmetry axis: the real part is
N
2+ 1spectral lines for k = 0,1,...,N/2 corresponding to frequencies fk= k · f0= k
fsmpl
withgrid.
1
NΔt,
fsmpl
3.1 Pedestal independence
The analog section of the FEB has been designed to have a pedestal of ca. 10 % of the full
FADC range in order to investigate undershoots. However, the pedestal is relatively sensitive
on the temperature. Daily variation of the pedestal may reach 5 ADC-counts. The trigger
pedestal-independent is very welcome. Let us consider signal with a constant pedestal:
¯Xk(ped) =
N−1
∑
n=0
(xn+ ped)F(k,n) =¯Xk+ ped
N−1
∑
n=0
F(k,n) =¯Xk+ ped ×W
(3)
F(k,n) = cos
?kπ
N(n +1
2)
?
(4)
Due to symmetry and parity of the cosine, we get for odd and even indices respectively:
W = 2
N
2−1
∑
n=0
?
cos
?kπ
2
?
cos
?π
N
?
n +1
2
?
k −kπ
2
??
=
⎧
⎪
⎪
⎩
⎨
0, k − odd
2
N
2−1
∑
n=0
F(k,n) , k − even
(5)
By a recursion, repeating (5) we get finallyN
2= 2 and k = 0,N
?π
2. For k =N
2
1
∑
n=0
cos
2
?
n +1
2
??
= 0(6)
In a consequence for k > 0 the DCT coefficients are independent of the pedestal.
3.2 Scaling
The DCT algorithm has a significant advantage in comparison to the FFT one. The structure of
DCT coefficients is much simpler for interpretation and for a trigger implementation than the
structure of the FFT real and imaginary coefficients (compare 4th of the FFT data vs. 2nd row
for the DCT coefficients in Fig. 3). For the exponentially attenuated signals from the PMTs
higher DCT coefficients (scaled to the 1st harmonics)
ξk=
¯Xk
¯X1
(7)
are almost negligible, while both real and imaginary parts of the FFT (scaled to the module of
the 1st harmonics) give relatively significant contributions and are not relevant for triggering.
When a peak appears in the pure attenuated signal (last column in Fig. 3) the structure of the
DCT dramatically changes and trigger condition immediately expires, while modules of FFT
components almost do not change. The structure of FFT harmonics for the last graph in Fig. 3
would be more suitable for a trigger (almost negligible imaginary part for higher harmonics
and also relatively low real harmonics), however it corresponds just to situation, when the
383
An Optimization of 16-Point Discrete Cosine Transform Implemented into a FPGA as
a Design for a Spectral First Level Surface Detector Trigger in Extensive Air Shower Experiments