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A spectral 1st level FPGA trigger for detection of very inclined showers based on a 16-point discrete cosine transform for the Pierre Auger Observatory

Z. Szadkowski

Department of Physics and Applied Informatics, University of Ło´dz´, Pomorska 151, 90-236 Ło´dz´, Poland

a r t i c l e i n f o

Article history:

Received 22 October 2008 Received in revised form 1 February 2009 Accepted 23 March 2009 Available online 22 April 2009 Keywords:

Pierre Auger Observatory FPGA

Triggers

Discrete cosine transform Digital signal processing

a b s t r a c t

The paper describes a new spectral trigger based on the 16-point discrete cosine transform (DCT) algorithm that was implemented into an FPGA. The DCT trigger allows recognition of FADC traces with a very short rise time and fast exponential attenuation related to a narrow, flat muon component of very inclined extensive air showers generated by hadrons and starting their development early in the atmosphere. The discrete cosine transform, based on only real coefficients in the frequency domain, provides much more sensitive trigger conditions and a simpler interpretation in comparison to a discrete Fourier transform (DFT) that is based on complex coefficients or their absolute values. It also offers a scaling feature. The ratio of the DCT coefficients to the 1st harmonics depends only on the shape of signals, not on their amplitudes. However, an implementation of the DCT into an FPGA requires more resources than DFT even based on an FFT algorithm.

1. Introduction

The Pierre Auger Observatory is a ground based detector located in Malargu¨e (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[1]. 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.

The ﬂux of cosmic rays above 10¹⁹eV is extraordinarily low: on the order of one per square-kilometer per year. 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 an array of surface detectors (SD) spread over 3000 km² for 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 four locations at the

boundary of the ground array measuring the ﬂuorescence light associated with the evolution of air showers: the growth and subsequent deterioration during a development. Such ‘‘hybrid’’

measurements allow cross-calibrations between different experi- mental techniques, controlling and reducing the systematic uncertainties.

Inclined showers are different from the ordinary vertical showers, because they must be due to penetrating particles.

At large zenith angles the slant atmospheric depth to ground level is enough to absorb the early 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 (Fig. 1).

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. However, detection of these showers requires new sophisticated techniques, such as spectral triggers, which is an interest of the present paper.

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 we need a very good spectral sensitivity to the fast muon component in the trigger.

Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/nima

Nuclear Instruments and Methods in Physics Research A

doi:10.1016/j.nima.2009.03.255

Tel.: +48 42 635 56 59; fax: +48 42 635 56 17.

E-mail address:zszadkow@kfd2.phys.uni.lodz.pl

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Presently, triggering systems for inclined showers at the Southern Pierre Auger Observatory in Argentina are working well.

The goal of this paper is to examine new ways of optimizing the spectral sensitivity of such triggers. We will study this in the context of the data acquisition and triggering schemes of the Auger Observatory, but the methods we develop may also be used in other experiments.

The main advantage of the spectral trigger is the scaling feature. The set of the DCT 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 veryinclined showers.

Independence of the amplitude is especially promising for the Auger North, where qdue 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

Shower fronts

Shower core hard muons

1000g/cm² 3000g/cm²

EM shower Auger surface

detector array

Fig. 1. Development of showers generated early in the atmosphere. Only a dominating muonic front reaches the surface detectors of Auger.

PMT1PMT2 PMT3 1.93 VEM 1.75 VEM

0 80 160 240

PMT1 PMT2 PMT3 1.75 VEM 1.64 VEM

0 80 160 240

0 80 160

240 PMT1

PMT2 PMT3 1.75 VEM 1.68 VEM 240 244 248 252 256 260 time bin

240 244 248 252 256 260 time bin

Fig. 2. FADC traces (in ADC-counts) of a horizontal shower (no. 01145 055:y¼83:3) 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 tank (1.93 IVEM) is above the standard threshold of 1.75 IVEM. Signals in Cristian (1.64 IVEM) and Juancho (1.68 IVEM) tanks are below the standard thresholds and they are detected by chance (compare a registration efﬁciency for a similar event shown in Fig. 5). For all very inclined showers the rising edge corresponds to one or two time bins.

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in the Auger South, where 3-fold coincidences attenuated a noise (Fig. 2).

2. Triggers

The surface detector of the Pierre Auger Observatory is a 1600 water Cherenkov tank array on a triangular 1.5 km grid. Each tank contains 12 m³of super-pure water. Ultra-relativistic particles in air showers passing through the water generate the Cherenkov light which is detected by three 9-in. photo-multiplier tubes (PMTs) placed at the top of each tank. Signals are extracted both from the anode and the last dynode of each PMT, amplified to achieve 15-bit dynamic range, filtered by a five pole anti-aliasing filters with the 3 dB cut-off at 20 MHz and finally digitized at 40 MHz by a 10-bit FADC[2].

Two different triggers are currently implemented at the 1st level [3–5]. The ﬁrst is a single-bin trigger generated as 3-fold coincidence of the three 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[6]and corresponds to ca. 50 ADC-counts. 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 0.2IVEM. 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 tank. 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.

Cherenkov light generated by very inclined showers crossing the Auger tank can reach the PMT directly without reflections on Tyvec^s liners (Fig. 3). 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 inducedshowers with dominant muon component give an early peak with a typical rise time mostly from 1 to 2 time bins (by 40 MHz sampling) (Fig. 4) and decay time of the order of 80 ns [7](see alsoFig. 7).Fig. 4shows that for very inclined showers (a range 60290) the front of edges is much sharper than for less inclined calculated from pedestal level to the maximal value. 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 is strongly recommended. The expected shape of FADC traces (see inclined shower inFig. 2) suggests to use a spectral trigger, instead of a pure threshold analysis in order to recognize the shape of the FADC traces characteristic for the traces of very inclined showers.

Cherenkov light

PMT

PMT μ

μ

Fig. 3. A geometry of Cherenkov light paths for horizontal and vertical muons crossing the Auger tank.

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The monitoring of the shape 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 (see Fig. 5). For that showers much more tanks should have been hit. Muonic front produces PMT signals not high enough to generate 3-fold coincidences, some of signals are below of thresholds. This is a reason of ‘‘gaps’’

in the array of activated tanks.

A next-generation Auger Observatory, ‘‘Auger North’’, is currently being designed. It is intended to be located in Colorado, USA, and be larger in size than the Southern Observatory in Mendoza, Argentina. In order to improve the time resolution of measured events and consequently the pointing accuracy of the surface detector, the speciﬁcation of the surface electronics assumes 100 MHz sampling of the analog signals [8]. For this

reason SD electronics for the Auger North requires a new front-end.

The 100 MHz sampling chosen for the Auger North is a compromise between a requirement of improvements (better time resolution) and a hardware limitation (i.e. keeping power consumption on the level appropriate for a solar panel supply, an adjustment of a trigger rate and a size of transmitted data for the bandwidth of the microwave link).

The Auger North design assumes to use a single PMT in each surface detector. The coincidence technique for a noise suppression cannot be used any longer. In order to keep a reasonable trigger rate and not to saturate the communication channel, the trigger threshold has to be signiﬁcantly increased. But, this yet more reject under-threshold signals. The newspectral approach allows a recognition of a shape of signals independently of their amplitudes. However, this requires a new hardware trigger to assure registered events also for off-line analysis.

3. Improvements

The Pierre Auger Observatory has been designed to study cosmic rays mostly above 10^18:5eV. For the Auger North Observatory, a possible new front-end based on the Cyclone III^TM FPGA and with 100 MHz sampling frequency [9] is being studied.

A spectral trigger based on discrete Fourier transform (DFT) (Radix-2 FFT) [11] has been already implemented in the 3rd generation of the front end board (FEB) based on Cyclone^TMAltera^s chip [5]. It was tested for several weeks in the test tank simultaneously with the standard threshold and ToT triggers. However, tuning of boundaries for DFT coefﬁcients looked not unequivocally converging procedure relating to the trigger rate. Automatically tuned boundaries allowed also a registration of events with rather not expected shape. This was a motivation to look for an other approach with simpler interpretation of parameters. A comparison of DFT and DCT structures suggested simpler interpretation of the DCT coefﬁcients more suitable for triggers (compare graphs inFig. 6).

The DFT for a real signal xn

¯XN=2þk¼X^N1

n¼0

xne^jð2^p^{=NÞ N=2þk}ð Þⁿ

¼X^N1

n¼0

xnð1Þⁿ½e^jð2^p^=NÞkn¼ ¯X_N=2k (1)

and N=2th spectral line of ¯Xk, k ¼ 0; 1; . . . ; N 1 is lying on a symmetry axis: the real part is symmetric, the imaginary part is asymmetric. The useful information is contained only in 1st N=2 þ 1 spectral lines for k ¼ 0; 1; . . . ; N=2 corresponding to frequencies fk¼k f0¼kð1=NDtÞ, changing from zero to fsmpl=2 with fsmpl=N grid.

The new spectral trigger based on the discrete cosine transform can be implemented only in this new FPGA. The previous generations of FPGA did not contain enough resources for an implementation of the advanced DCT algorithm.

The new trigger allows:

Good trigger efﬁciency on very inclined showers.

An opportunity to built more sophisticated triggers in one Auger North design where each tank will be equipped with only one PMT. Noise reduction by 3-fold coincidences of the three installed PMT in the Auger South is then no longer possible. New and more elaborate noise suppression methods techniques may be required.

10000

1000

100

1 2 3 4 5 6 7 8 time bins

events number

0-30deg 30-60deg 60-90deg

Fig. 4. Distributions of rise times for all reconstructable showers from October 2008 grouped in ranges 0230, 30260and 60290calculated from a pedestal to the peak.

Fig. 5. Position of triggered tanks on the Auger array of event no. 01155 555.

Muons triggered only few tanks although they should have crossed partially 10 or more tanks. A distance between opposite tanks is greater than 54 km.

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4. Discrete cosine transform vs. discrete Fourier transform

There are several variants of the DCT with slightly modified definitions [10]. The most commonly used form of the discrete cosine transform (DCT-II) defined as follows:

¯Xk¼

a

k

X

N1

n¼0

xncos

p

N n þ1 2

k

(2)

where

a

0¼1= ffiffiffiffi pN

and

a

k¼2= ffiffiffiffi pN

for kX1, has been selected for our considerations as the alternative approach to the FFT. Due to a

direct sample in the output data this variant is independent of a pedestal.

The DCT is a Fourier-related transform similar to the DFT, but using only real numbers. DCTs are equivalent to DFTs 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).

DCT for real signal xn gives independent spectral coefﬁcients for k ¼ 0; 1; . . . ; N 1, changing f_k also from zero to f_smpl=2, but with f_smpl=2N grid. DCT vs. DFT gives twice better resolution.

0 20 40 60 80 100 ADC counts

time (ns) ADC samples: x₀= x₁= 0 ADC

counts

time (ns) ADC samples: x₀= 0 ADC

counts

time (ns) ADC samples: x₀= max ADC

counts

time (ns) ADC samples : a new small

incoming pulse

-70 -35 0 35 70 105

ξ_k (%) ξ_k (%) ξ_k (%)

index (k) DCT -II

index(k) DCT -II

0 20 40 60 80 100

%

index (k)

index (k)

%

index (k)

%

index (k)

-100 -50 0 50

%

index (k)

Re Im

0 20 40 60 80 100

-60 -30 0 30 60 90

0 20 40 60 80 100

-80 -40 0 40

%

index (k) Re Im

0 20 40 60 80 100

-50 -25 0 25 50

%

index (k) Re Im

0 20 40 60 80 100

-20 0 20 40 60 80 100

0 20 40 60 80 100

-50 -25 0 25 50 75 100

%

index (k) Re Im

Shape_A Shape_B Shape_C Shape_D

0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250

1 3 5 7 9 11 13 15

1 3 5 7 9 11 13 15 1 3 5 7 9 11 13 15 1 3 5 7 9 11

1 2 3 4 5 6

1 2 3 4 5 6 7 8

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6

ξ_k (%)

Fig. 6. A propagation of the pulse (1st row) through the shift register, DCT-II coefficients (2nd row), absolute values of the DFT (3rd row) and corresponding real (Re), imaginary parts (Im) (4th row). The 1st column shows the pulse (shape A), when two time bins are on the pedestal level, the 2nd one (shape B), when only the one time bin is still on the pedestal level, while the 3rd one (shape C) shows the pulse fully fulfilled the range of investigating shift registers. For a signal shape related to the exponential attenuation (shape C), the contribution of higher DCT coefficients is small and suitable for a trigger. When a peak appears in the declining signal (last column—shape D), the DCT coefficients immediately excesses assumed relatively narrow acceptance range for triggers. The DFT coefficients (Re and Im in 4th row) have similar structure as the DCT, however, for the pure exponentially declining signal the higher real DFT harmonics have relatively high values and they are not suitable for triggering. Absolute values of DFT components (3rd row) are clearly insensitive on discussed conditions.

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In fact, the FFT routine can be supplied in an interleaving mode, even samples treated as real data, odd samples as imaginary data.

However, the FFT algorithm in Ref.[11]has been implemented and optimized for only real samples from the ADC.

The DCT algorithm presented in the paper has a significant advantage in comparison to the FFT one. The structure of DCT coefficients are 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. 6). For the exponentially attenuated signals from the PMTs higher DCT coefficients (scaled to the 1st harmonics) 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. 6) 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. 6 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 pure attenuated signal is distorted by some peak on the tail and a trigger condition has been violated.

The plot in the 4th row and 3rd column in Fig. 6 shows a contribution of the DFT vs. the absolute value of the 1st harmonic.

For a exponential attenuated input signal (with the attenuation factor ¼b) the contribution of both real and imaginary coefficients decreases monotonically with a significant value for all real coefficients. From the DFT definition we get:

¯Xk¼AX^N1

n¼0

e^bⁿe^jð2^p^k=NÞn¼A 1 e^N^b

1 e^b^þjð2^p^=NÞk (3)

x¼Reð ¯XkÞ

k ¯X1k ¼ ð1 þ e^b^NÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 e^bcosfÞ²þe²^bsin²f

q (4)

wheref¼2

p

k=N.

Calculating (4) for boundary factorsb¼ ð0:28; 0:42Þ (seeFig. 7) and for k ¼ N=2 (as the lowest in a monotonically decreasing chain), we obtain for N ¼ 16 :x¼24% and 28%, respectively.

These values are too large to be use for triggering. Even an extension of the DFT size does not help very much. For N ¼ 32 : we get still large values:x¼17% and 23%.

Almost vanishing higher DCT coefﬁcients provide much natural trigger conditions. 32-point FFT (roughly equivalent to 16-point DCT) does not offer better stability.

5. Preliminary assumptions for a trigger

The trigger described in Ref.[11] is based on a 16-point FFT algorithm and is clocked with 40 MHz. The ‘‘length’’ of the analyzing sliding window equals to 400 ns, which is sufﬁcient for an analysis of the muonic bump, especially for the ‘‘old’’ showers.

For the new 100 MHz sampling, 16-point samples would correspond to a 150 ns sliding window only. An analysis of the Auger database (collected for 40 MHz sampling) showed that a

‘‘length’’ of FADC traces corresponding to 200–250 ns is desirable.

The analysis of the attenuation part of the Auger FADC traces gives an attenuation factor roughly independent of time (Fig. 7).

FADC traces of almost very inclined events are characterized by a dramatic jump from the pedestal level up to maximal value due to their origin related to the geometrical conﬁguration of PMTs in the tank (Fig. 3).

The 100 MHz sampling, however, should be already sensitive on a signal shape especially in the range of the rising edge of the analog signal, which corresponds to the beginning of the muonic shower. This region contains crucial information and should be sampled with full resolution. The tale of the signal, much longer than its front, contains less important information and can be sampled with a bigger grid. For ﬁrst eight samples in a 10 ns grid and next eight samples in a 20 ns grid, the ‘‘length’’ of an analyzing sliding window is 230 ns and seems to be sufﬁcient for our analysis (Fig. 8).

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 the undershoots.

Denoting cosine factors as

Fðk; nÞ ¼ cos k

p

N n þ1 2

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¯XkðpedÞ ¼^N1X

n¼0

ðxnþpedÞFðk; nÞ

¼ ¯XkþpedX^N1

n¼0

Fðk; nÞ ¼ ¯Xkþped W. (6)

Due to symmetry and parity of the cosine

W ¼ 2^N=21X

n¼0

cos k

p

2

cos

p

N n þ1 2

k k

p

2

¼

0; kFodd

2^N=21P

n¼0

Fðk; nÞ; kFeven.

8>

><

>>

:

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By a recursion, repeating (7) we get ﬁnally N=2 ¼ 2 and k ¼ 0; N=2.

For k ¼ N=2 X¹

n¼0

cos

p

2 n þ1 2

¼0. (8)

In a consequence for kX1 the DCT coefﬁcients ¯Xk are independent of the pedestal.

The DCT coefﬁcients normalized to the 1st harmonics

xk¼¯Xk

¯X1

(9)

are also invariant on the scaling of a signal amplitude. It means the set of ¯Xk(i.e. 1st–3rd graph in the 2nd row ofFig. 2) remains Fig. 7. Histogram of attenuation factors for January 2005 and 2008, respectively,

arebJan 2005 (0:344 0:062) andbJan 2008 (0:310 0:076).

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the same independently of the jump from the pedestal to a maximal values (if the amplitude of pulses were, i.e. 500 ADC-counts or so instead of 100 ADC-counts for shapes A–C inFig. 6).

As seen fromFig. 6, if ADC samples propagating in the shift registers, which drive the DCT routine, match the pattern of exponential attenuation, the contribution of higher x_k significantly drops down. The level ofxkcontributions can be used to for a preliminary spectral trigger. If in the propagating signal a sample, significantly different from an expecting shape, appears in the input shift register,xkdramatically change and stop fulfilling a trigger condition (Fig. 9).

Propagation of pulses for three different rise times and two attenuation factors (min: ¼ 0:28 and max: ¼ 0:42) is shown inFig.

10. These factors have been estimated upon the real events registered on the ﬁeld. We expect that the pulses from the very inclined and horizontal showers will be attenuated with a factor between estimated minimal and maximal limits. Thus the corresponding spectral coefﬁcientsxkare in very narrow ranges suitable for triggering.

Fig. 11 show the ranges of the xk coefﬁcients allowing a generation of the sub-triggers (seeFig. 14). If for the investigated shape of the pulse and for the index k, the spectral coefﬁcientxkis inside the range showing in the corresponding graphs A⁰;B⁰;C⁰;D⁰_1;2;3, Eq. (11) is true and the sub-trigger (k) is generated. For exponentially declining tails the xk set are very similar.

However, the analysis of the pure attenuated signals is the only part of trigger process. We are going to select pulses with a ﬁxed

rise time and the exponential attenuation tail with a dedicated circuit (‘‘engine’’) recognizing required shape. If the ‘‘engine’’ is ﬁxed for a recognition of the shape with, i.e. two time bins (graphs A3;B3;C3;D3), the circuit fromFig. 12is waiting for the set ofxk

corresponding to the graph A⁰₃. We do not require all xk

coefficients have to obey Eq. (11). We allow two or three coefficients could be outside the acceptance range. This mitigation should improve the trigger condition. A requirement when allxkhave to be inside the acceptance range seems to be too strong (in reality the signal contains also the noise) and could bias significantly a trigger rate.

If a number of xk are above the ‘‘occupancy’’ threshold (the shape A⁰₃has been recognized), the multiplexer changes the set of the expected limits for thexkand in the next clock cycle the circuit is waiting for the set ofxkcorresponding to the graph B⁰₃. If this shape is also recognized, the circuit in the next two clock cycles is waiting for the pure attenuated signal related to the graphs C⁰₃and D⁰₃, respectively. A similar analysis is performed for the shapes related to graphs A1;B1;C1;D1and A2;B2;C2;D2.

The trigger can be active for several time bins. The muonic front hitting a detector gives signals in three PMTs, which can be shifted between them due to a pure geometry of a shower, different characteristics of PMTs, analog channels, etc. However, a shift should not exceed 25 ns, as shows the analysis of the Auger database. The interval of 25 ns corresponds to 2–3 time bins for the 100 MHz sampling. Attenuation tail has a ‘‘length’’ of more than 10 time bins. So, we expect overlapping trigger conditions in three channels and a ﬁnal 3-fold coincidence triggerfor at least a single time bin.

0 20 40 60 80 100

0.28 0.35 0.42

time (ns)

0 000 000 600 000 1000

%

index (k)

-35 -30 -25 -20 -15 -10 -5 0

5

%

0.28 0.35 0.42

6 7 8 9 10 11 12 13 14 15

1 2 3 4

6 7 8 9 10 11 12 13 14 15 1 2 3 4 5

time (ns) 0 20 40 60 80 100 120 140 160 180200 220

0 20 40 60 80 100 120 140 160 180 200 220

Fig. 8. Investigated shapes of Auger ADC traces with attenuation factors from Fig. 7 and the corresponding set ofxk.x1not shown as equals to 100%. For different attenuation factors thexkalmost does not change. The main variation appears forx2on the level of 20% andx2;3;4on the level less than 5%. As expected for different amplitudes, the set ofxkremains almost (integer representation of numbers inside the FPGA) unchanged.

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An approach based on the exponentially declining signal (shape A) only may give too high trigger rate due to single muons.

However, this trigger veriﬁes a purity of the exponential shape of signal.

Fig. 12shows the structure of FPGA procedures generating the ﬁnal trigger. In this scheme three different shapes are investigated. This approach can be easily extended on numerous patterns being compared for consecutively clock cycles with, respectively,

0

time (ns) ADC

counts

0 50 100 150 200 250 300

0

counts

0 50 100 150 200 250 300

k (40 MHz) 0

index (k)

ξ_{k (%)} ξ_{k (%)}

2 3 4 5 6 7 8 9 10 11 12 13 14 15

k (40 MHz) 0

index (k)

2 3 4 5 6 7 8 9 10 11 12 13

0

0 20 40 60 80 100

-40 -20 0 20

0 20 40 60 80 100

counts

0 50 100

0

0 20 40 60 80 100

0 10 20 30 40

0 20 40 60 80 100

counts

0 50 100

Fig. 9. Comparison ofxkfor 40 (2nd row) and 100 MHz (5th row) sampling, respectively. Sets ofxkare signiﬁcantly different, an analysis of the Auger data for 100 MHz extrapolation requires a careful resampling. The 100 MHz sampling with a continuous 10 ns grid (3rd row) gives largerx_kranges (5th row) in comparison to an extended grid with 1st eight samples taken with 10 ns and the next with lower density 20 ns (4th row). Narrower ranges ofxkfor the extended mode as well as wider sliding window investigating the ADC trace give better trigger conditions than in a simple continuous sampling mode. The extended mode will be the next baseline for the DCT implementation.

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multiplexed thresholds in order to detect traces with more sophisticated shapes.

6. Implementation of the code into a FPGA

An internal FPGA routine calculating coefﬁcients of the discrete cosine transform is derived from a well-known algorithm of Arai–Agui–Nakajima[12], however, signiﬁcantly optimized to the implementation into the FPGA [13]. Full 16-point DCT internal structure of the FPGA routine is shown inFig. 13. Mathematical details are presented in a separated paper[14].

According to the assumptions from Section 5 the trigger should be generated if DCT coefﬁcients normalized to the 1st harmonics were in a arbitrary narrow range:

Thr^L_kpxk¼ ¯Xk

¯X1

¼

Z

_kHf ðkÞ

Z

₁H15pThr^Hk (10)

where Thr^L_k and Thr^H_k are lower and upper thresholds for each spectral index k, respectively.

Altera^s library of parameterized modules (LPM) contains a lpm_divide routine supporting a division of ﬁxed-point variables.

However, this routine needs huge amount of logic elements and is slow (calculation requires ca. 14 clock cycles in order to keep sufﬁciently high registered performance). DSP blocks also do not

support this routine. A simple conversion to

H15y^L_k¼H15

Z

1

Z

_k^Thr

L k

pHf ðkÞ

pH15

Z

₁

Z

_k^Thr

H k

¼H15y^H_k (11)

allows implementation of fast multipliers from the DSP blocks and calculation of products in a single clock cycle.y^L_kandy^H_k are lower and upper scaled thresholds, respectively, which are set as external parameters.

According to (11) the calculation of a sub-trigger needs two multipliers, two comparators and an AND gate (Fig. 14). The multiplier stage of an embedded multiplier block supports 9 9 or 18 18 bit multipliers. Depending on the data width or operational mode of the multiplier, a single embedded multiplier can perform one or two multiplications in parallel. Due to wide data busses embedded multiplier blocks do not use the 9 9 mode in any multiplication. Each multiplier utilizes two embedded multiplier 9-bit elements. The full DCT procedure needs the calculation of all coefﬁcients 70 DSP blocks. However, the scaling of ¯Xkin the last pipeline chain is no longer needed. It is moved to the thresholds according to (11). Removing last pipeline chain reduces amount of DSP blocks to 40. Sub-triggers routines (Fig. 14) need two DSP blocks each. The chip EP3C40F324I7 0

0 20 40 60 80 100

time (ns)

countsADC 0

0 20 40 60 80 100

time (ns) countsADC

k(100 MHz)

0

-50 -40 -30 -20 -10

0 index (k)

ξ_k (%)

continuous extended

k (100 MHz) 0

-3 0 3 6 9 12 15 18 21

index (k)

continuous extended

2 3 4 5 6 7 8 9 10 11 12 13 14 15

2 3 4 5 6 7 8 9 10 11 12 13

0 50 100 150

ξ_k (%)

Fig. 9. (Continued)

(11)

0

0.28 0.42

0

0.28 0.42

0 0

20 40 60 80 100

0 20 40 60 80 100

0.28 0.42

0

0.28 0.42

0 st

0.28 0.42

0 st 0

20 40 60 80 100

0 20 40 60 80 100

0.28 0.42

0 50 100 150 200 time (ns) 0 50 100 150 200 time (ns)