FPGA based triggers in the Pierre Auger Observatory

Z. Szadkowskia for the Pierre Auger Collaboration

aDivision of Experimental Physics, University of #Lodz, Pomorska 151, 90-236 #Lodz, Poland.

ABSTRACT

The Pierre Auger Observatory is an international project for research on ultra-high energy cosmic rays (UHECR)above 1019 eV and currently the world largest “hybrid” experiment. The hybrid technique uses both the detectionof particles at the ground in water Cherenkov detectors (WCD) and measurements of fluorescence light generatedin the atmosphere by the extensive air showers. This technique offers unique possibilities for cross-calibration andreduction of measurement uncertainties. The surface array will contain 1600 surface detector stations distributedover 3000 km2 in the Southern Hemisphere (under construction in Argentina) and 4000 WCD distributed over10000 km2 at a Northern Hemisphere site in Colorado (deployment planned for 2008). Four fluorescence stationseach containing 6 wide-angle Schmidt telescopes and each equipped with 440 PMTs with 1.5◦ angle resolutionand 30 · 30◦ of field of view are located on the border of the surface array in the Southern Site.

The paper presents implementations of the FPGA used for triggering. The use is illustrated with currentlyworking schemes as well as other possible ideas especially promising for the surface detector in which currentlyused triggers seems to be not effective enough for very inclined and horizontal showers. Proposed new triggersenlarge a simple 1-bit threshold detection to the online analysis of a FADC trace shape. This corresponds toa charge from PMTs or signal power estimation. The next category is spectral triggers analyzing sequentialsamples in the frequency domain. Digital sigma-delta filters immunize the trigger circuitry on the pedestalvariation as results of huge temperature swings. Presented DCT trigger provides native pedestal independence.

Keywords: Pierre Auger Observatory, triggers, fluorescence detector, surface detector, FPGA, FFT, DCT

1. INTRODUCTION

The Fluorescence Detector (FD) is one part of the Pierre Auger Observatory. The atmospheric fluorescencemeasurements, although only available on clear moonless nights, play an essential role by providing a subset ofshowers measured both laterally and longitudinally. The primary role of the fluorescence detector is to measurethe longitudinal profile of each shower. The FD is built from a set of mirrors with a matrix of PMTs (22x20).Each camera observes a 30◦ sector of the sky. The observation angle of each PMT is relatively narrow, only 1.5◦.Such fine angular resolution leads to a narrow track image on the PMT surface, often below the transversal sizeof PMT. The task of the 2nd level trigger is to recognize if the PMT signals from the matrix correspond to areal shower track or to a noise event. The FD built from a set of cameras with a matrix of PMTs collects lightproduced from nitrogen fluorescence activated by ultra-relativistic showers in the atmosphere.

Each surface detector (SD) is a solar powered water Cherenkov detector instrumented with low power elec-tronics. It communicates with the Central Data Acquisition Station (CDAS) via a custom radio network. Lowgain and high gain (32 times larger) signals from three downward facing 9” PMTs are sent to the front-endelectronics, through a 5 pole anti-aliasing filter with a cutoff at 20 MHz, and digitized at 40MHz by 10-bitFADC. Samples are given to the trigger/memory system (TMS). The TMS investigates the FADC samples forinteresting trigger patterns, temporarily stores the data in the buffer memory and interrupts the detector stationmicro-controller if a trigger condition occurs. The station controller sends trigger packets and, when requested,event data to the CDAS via the wireless network. A hierarchical event trigger is used to select events of interestand reject uninteresting events in order not to exceed the rate constraints imposed by the station micro-controller,the communications link bandwidth and the CDAS.1

Further author information: (Send correspondence to Z. Szadkowski)E-mail: zszadkow@kfd2.phys.uni.lodz.pl, Telephone: +48 42 635 56 59

Figure 1. The map of locations of the fluorescence detectors (four stations: Los Leones, Coihueco, Los Morados and LomaAmarilla) and 1600 surface detectors (1400 operational in April 2007) in the Southern Site. Dedicated detectors in AugerSouth (Infill Array) with smaller spacing, targeted to lower EAS energy range located close to Los Morados FD station.

The large area covered, the limited power budget necessitated by solar power operation, the restricted databandwidth available through the stations’ wireless network connections, and the temperature extremes inherentin an uncontrolled environment, impose special constraints on the surface detector trigger system. Severalapproaches to the implementation of the first level trigger were pursued during the engineering phase of theproject, both in the lab and in the field.

2. TRIGGERS IN THE FLUORESCENCE DETECTOR

Each FD telescope consists of the matrix of 440 PMT and 440 1st level triggers controlling each PMT as well asa single 2nd level trigger investigating the structure of active PMTs on the entire matrix. The light coming fromthe fluorescence nitrogen excited by the EAS forms ”tracks” of ”fired” pixels on the PMT matrices. The signalproduced by each PMT has a trapezoidal shape (Figure 2 - left), with durations varying from a few hundreds ofnanoseconds to several microseconds. Its intensity depends both on the shower size (e.g. number of electrons)at that particular point being observed and on the distance of observation, as r−2 dependence as well as anatmospheric light scattering contribute to attenuation of the signal. Simulations show that the dynamic rangefor showers in the range 1019 to 1021 eV seen from a distance of 7 to 30 km is of about 15 bits. The trackis more/less the straight line (sometimes with gap, if the traveling spot of light does not reach sufficiently thecenter of the PMT).2

The 1st level trigger is located on the front-end board and decides on the pixel status. It is built by fourFPGAs (Altera R© FLEX6KTM) each controlling six channels. The ADC values are integrated by boxcar averagingthe last 10 samples. The signal to noise ratio is improved by a factor of

√10. The threshold is tuned in the

Figure 2. Signal formation in the PMT. The light spot traveling through a center of the PMT gives strong signal, close tothe edge signal decreases and may be below the threshold. Although the light spot ”touches” the PMT, the pixel is notfired. There is a gap in a formed track (left). The 1st level trigger with boxcar integration and the rate control (right).

calibration process to achieve ≈ 200 Hz per pixel. This value has been chosen as a compromise between accuracyand a density of data stream. The pixel trigger signal is generated for a programmable coincidence time (holdofftime ≈ 10 µs) whenever the Boxcar running sum10 (Figure 2 - right) exceeds an adjustable threshold (ThSum).The hit rate of each channel is measured over a 5 s interval using the synchronization signal of the GPS clock.Larger deviations of this rate from the demand value of 200 Hz will modify the threshold to maintain a constantrate and to compensate varying light intensities.3

A group of PMT signals generated by a real track forms more/less collinear patterns. The 1st level trigger”fires” the pixel if the signal is above the threshold. The 2nd level trigger is generated from a set of approximatellycollinear patterns. They may be generated also by sky background, point sources like aircraft as well as by randomelectronic noise. The probability of accidental coincidences is significantly reduced for four-fold coincidences of”fired” pixels. If the light falling on a PMT is under the threshold, a gap or missing end point will occur on thetrack. 4-fold patterns created by the removal of one PMT from 5-fold patterns have been chosen for the 4-foldlogic. In all cases a pattern is characterized by the horizontal and vertical indices of the pixels whose signals areabove the threshold.4

The task of the 2nd level trigger is a generation of a trigger on the pure geometrical criteria (programmedseveral topologically different patterns). Patterns of 4 or 5 adjacent pixels overlapping during the hold off timeare accepted as valid trigger provided the overlap persists no longer than a programmed time. 4-fold patternevents rate is estimated on the level 0.2 Hz per matrix. The algorithm of the 2nd level trigger allows recognitionof a probable track coming from the EAS and significantly suppresses noise from random firing pixels spreadover a large matrix as well as allows investigation track registered on two neighboring cameras.

Contrary to the off-line investigation of huge database of possible events to select tracks really coming fromthe showers, on-line recognition of EAS tracks significantly improves a quality and performance of the system.It allows on-line synchronization of data acquisition from the surface detector in order to freeze data in selectedwater Cherenkov tanks to get data as hybrid events.

The five topologically different 5-fold patterns are considered:

Five PMTs in a single row (5),

One PMT deflected from the main direction (4 1),

Two PMTs deflected from the main direction from both ends (1 3 1),

3 PMTs lying on one row and the next two on the neighboring one (3 2),

PMTs lying on three neighboring rows 2+2+1, respectively (2 2 1).

All patterns are obtained from the basic 5-fold patterns by horizontal or vertical translation as well as bymirror reflection or by rotation, taking into consideration the 60◦ geometry. Possible gaps degenerate the set ofpatterns. The same 4-fold pattern with gap may be obtained from different parents. Redundant patterns are

5 4_1 1_3_1 3_2 2_2_1

Basic pattern

60° rotated

120° rotated

X-axis mirrored

Y-axis mirrored

Figure 3. Types of topologically different 5-fold patterns (left-up), Symmetry transformation of the basic pattern (left-middle), samples of patterns with the same set of indices, however coming from different parents (left-down), tracks of ashower on two PMT matrices as a stereo event (registered simultaneously by two eyes). The left graphs on the right sideshow a shower track on the PMT matrix created by fired PMTs; right graphs show FADC traces of fired PMTs.

Figure 4. Three-dimensional view of the first quadriple event.5 The array is viewed from the North-East corner; ie. thebottom right FD is Loma Amarilla (left). Track view on two neighboring cameras for Coihueco (right).

removed. Pattern of 4 or 5 adjacent pixels overlapping during the hold off time are accepted as valid triggerprovided the overlap persists no longer than a programmed time.

A sliding window with 5-pixel width scans the matrix of 440 PMTs. It allowed reducing a simultaneousanalysis to sub-set of patterns and in the consequence selecting smaller and cheaper PLD chip.

The 2nd level trigger has been implemented into the single FLEX10KTM Altera R© chip EPF10K100QEC240.

3. CURRENTLY USED SD TRIGGERS

The TMS in the surface detector was originally planned as an ASIC, containing all functions in a single chip,together with the memories for fast and for slow channels. Fast channels are for shower profile investigationsand slow channels for investigation of the muon component and for calibrations. Subsequent developmentof Programmable Logic Devices (PLDs) allowed use of commercial chips and an implementation of advancedalgorithms in them. The flexibility of the PLD configuration reduced the risks if some algorithms were not fully

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Figure 5. A single bin (left) and Time over Threshold (right) trigger implementation.

Thresholds ≈ 1.75VEM

Thresholds ≈ 90 ADC-counts above pedestals

Thresholds ≈ 0.2VEM

Thresholds: ≈ 10 ADC- counts above pedestals

Figure 6. Real data from the tanks triggered by threshold trigger (left) and ToT trigger (right).

optimized and allowed easier modifications in the future if some new measurements were desired. The Altera R©

APEXTM EP20K200RI240-2 chip supported by a dual-port RAM IDT70V3569S5DRI has been successfully usedfor the TMS in the Auger Engineering Array.6 However, the cost of the PLD chip significantly exceeded theexpected budget. The Altera R© ACEXTM chips EP1K100QI208-2 reduced significantly the total cost, but at theexpense of splitting signal processing into two parallel working chips that were synchronized with additional gluelogic.7 ∼800 SD have been equipped with the ACEX design. Two years later it became possible to implementnew Altera R© CycloneTM chips EP1C12Q240I7, merging functions of both ACEXTM chips.8 Resource utilizationin ACEXTM chips reached 70%. Use of a chip with larger capacity improves the reliability of the TMS due tobetter PLD network optimization, as well as allowing implementation of new algorithms.

Selection of the physical events is performed in a hierarchical way:

T1: FPGA based local trigger (station level) = Threshold trigger (rate ≈ 100 Hz) or Time over Threshold(ToT) trigger (rate ≈ 1 Hz),

T2: Software local trigger (station level) = Threshold trigger (reduced rate to ≈ 20 Hz) or ToT trigger withthe same rate as in T1,

T3: Central trigger (array level),

T4: Physics trigger (off-line),

T5: Quality trigger (off-line).

Two different triggers are currently implemented at the 1st level. The first is a single-bin trigger generatedat 3-fold coincidence of a simple 1.75 IV EM threshold. The estimated current for a Vertical Equivalent Muon(IV EM ) is the reference unit for the calibration of FADC traces signals.9 This trigger is relatively noisy, witha rate of about 100 Hz. It is used mainly to detect fast signals, which correspond to the muonic componentgenerated by horizontal showers. The single bin trigger is generated when the input signal is above the fixedthresholds calculated in the micro-controller during the calibration process. It is the simplest trigger useful forhigh-level signals. The second trigger is the Time over Threshold (ToT) trigger that requires at least 13 timebins above a threshold of 0.2 IV EM in a 120 bin sliding window with for a coincidence on any two PMTs. Apre-trigger (”fired” time bin) is generated if the signal is above the threshold of 0.2 IV EM simultaneously inany two channels (two-fold coincidence). The amount of pre-triggers is analyzed in the interval of 120 time-bins(sliding window). If the number of pre-triggers exceeds the occupancy threshold (= 13) the final ToT trigger isgenerated. This trigger has a relatively low rate of about 1.6 Hz, which is the expected rate for double muonsfor an Auger tank. It is relatively efficient for selecting small but spread-in-time signals, typical for high energydistant EAS or for low energy showers, while ignoring the single muon background. However, currently usedtriggers are not dedicated to very inclined showers because the threshold 1.75 IV EM is relatively high and thedetection efficiency decreases with the zenith angle. Those showers should be triggered using other signatures,e.g. curvature of the shower front. Hadron induced showers with dominant muon component give an early peakwith a typical rise time of about 15 ns and decay time of the order of 80 ns.10 The expected shape of FADCtraces suggests using a spectral trigger investigating on-line a contribution of spectral components, instead ofpure threshold analysis.

T3, T4 and T5 software triggers are described in details in.9

4. SIGMA-DELTA COMPENSATION OF PEDESTALS

Large ambient temperature swings are observed between day and night, and there are additional systematicseasonal changes. These induce large temperature variations in the electronic box, which may reach more than50C. It causes also a significant variation of the pedestal from both dynodes and anodes. The seasonal andshort-time pedestal variation of selected tank is shown in Figure 7. The variation may reach more than 5 ADCcounts over the full temperature range.11 Pedestal drifts would significantly affect the trigger, especially for

Channel 1 in the tank "Luz" #131

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Figure 7. An example of long-time variation of pedestal for high-gain channel in the 131st tank within 2005 - 2006 (left) andFADC drifts a function of temperature measured in the climate chamber (right). The uncompensated drift range amountsto ≈5 ADC-channels. The compensated one by the preliminary sigma-delta routine does not exceed 1 ADC-channel

the ToT trigger where the thresholds (0.2 VEM ≈ 10 ADC-counts) are not greatly above the pedestals. Inthe currently used algorithm, the calibration channel modifies thresholds in order to keep required the triggerconditions stable. However, keeping a stable pedestal seems to be better approach than threshold tuning via theslow calibration loop.

Additional routines could be introduced to compensate for the observed temperature drift of the pedestals.Practically the Sigma- Delta algorithm11 has proven very useful for this purpose. The long-term drift changes thepedestal significantly. Short-term fluctuation may also appear. The Sigma-Delta algorithm requires an additionalcircuit controlling an effective pedestal by increasing or decreasing a single bin depending on a comparison ofan ADC value and the current pedestal. The length of the counter corresponds to the compensation responsetime. In order to compensate mainly the long-term drift, a 38-bit counter has been chosen. For 10-bit FADCinput, the fractional part is 28 bits. The additional integrator smoothes the quantization teeth, coming from anapproximation of input ADC integer values. For continuous counting with 40 MHz sampling and 5-bit integratorthe 28-bit counter will be overloaded after about T ≈ 15 s. Thus, if there is long-term drifting, the pedestal willbe compensated at a rate of ∼ 4 ADC-count/min, sufficiently slow in order not to introduce short-time (relatedto the shower profile interval = 19.2 µs) distortions.

5. NEW TRIGGERS IN THE TIME DOMAIN

Currently used ToT trigger analyses FADC traces by counting the number of FADC samples exceeding a fixedthreshold within a sliding time window of consecutive samples. It is a simple 1-bit analysis.

An extension of the ToT idea is the integral of all FADC values in a sliding window. The integral of thetime bins corresponds to the charge from PMTs and to the Cherenkov flux collected in the tank. A secondestimator is the sum of the squared time bins in a sliding window. Here the idea is to distinguish a contributionof a signal from significantly high-level time bins from low-level ones close to noise. A third estimator evaluatedis the on-line calculated Area/Peak ratio. The next category is spectral triggers evaluating FADC shapes on acontribution of Fourier or DCT coefficients.

5.1 Area over Threshold (AoT) trigger

The first two generations of the 1st level SD trigger based on the APEXTM and in particular ACEX R© chips didnot contain enough resources to implement more sophisticated triggers other than those currently used.

From a 38-bit counter of the digital filter compensating the pedestal variations, effective 16-bit output valuesare taken next for data processing, related to 10-bit integer FADC input samples with additional 6-bit fractionalpart. The performance of the full 16-bit bus significantly increases the accuracy of a final trigger.

128-word shift register ADC[15..0]

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Figure 8. Structure of the AoT trigger routine (left). Pipeline registers are not shown. The length of the shift register(128 time bins = 3.2 µs) corresponds approximately to the currently used size of the sliding window in the ToT trigger(3 µs). An implementation of the shift register with a length = 2k, (where k is an integer number) is most effective. Thestructure of the 16-bit square device (right). 16-bit outputs from Dual-Port RAM do not have overlapping bits and couldbe connected to the same A-port of the adder. Output from a multiplier is connected to B-port with 9-bit shift

The running sum of the 16 bit compensated ADC values in a sliding window of 128 consecutive samples iscalculated by means of two adders, one of which is driven by a 128 word shift register (see Figure 8). Next, thesum is compared with a preset threshold to produce the AoT trigger.8

AoT :∑len

i=0(Ai − pedi) > ThrA

The length of the delay chain in the AoT is fixed, contrary to the ToT trigger, where the length of the slidingwindow could be tuned dynamically.

5.2 Power over Threshold (PoT) trigger

Signal (digitized by Flash ADC) corresponds to the charge of photoelectrons converted by photomultipliers fromthe Cherenkov photons. However, if we distract from the origin of the signal, the FADC traces can be consideredas some rare signal spread in time and on the noise background, which the power estimator could select. ToTtrigger can be generated by rather accidental conditions. The relatively narrow but high bump may be toonarrow to exceed an occupancy threshold and could not be registered by ToT trigger, as well as could be alsotoo low for relatively high single bin threshold and finally could be rejected. On the other hand, relatively weaksignals spread in time and coming from low-energy showers generate ToT, which rather should be rejected asinsignificant.

A possible trigger may base on a power signal estimator:8

PoT :∑len

i=0(Ai − pedi)2 > ThrP

The implementation of the algorithm is similar to the AoT case. The CycloneTM FPGA unfortunately doesnot contain embedded DSP blocks and a multiplication routine has to be implemented as a soft routine.

5.3 Area/Peak Ratio trigger

The area/peak ratio of the waveform is another useful estimator but is calculated at present only offline. Prelim-inary results show that the values of the area/peak ratio are contained in a relative narrow dynamic range. Thatparameter could be used also for trigger purposes. The area is already calculated in the AoT routine. A parallelroutine additionally calculates the peak value in a sliding window. Instead of online calculation of an area andpeak ratio using a relatively slow and resources spendthrift routine dividing of two integers a peak value can bemultiplied by a expected factor and compared with an area.

AoT, PoT and Area/Peak ratio triggers have been implemented and tested in a preliminary fashion underreal field conditions. Additional flags indicated the type of triggers.

Figure 9. Development of showers generated early and deeply in the atmosphere.

Figure 10. Samples of traces for different tanks hit by the horizontal shower. All traces in 36 hit tanks have similar shapes,very narrow sharp peak.

6. SPECTRAL TRIGGERS

The extremely rare flux of UHECR requires sophisticated detection techniques. Standard methods used fortypical events may not be sensitive enough to capture unusual events, which may be important to understandthe current data or to confirm/reject some new hypothesis. Currently used triggers are not optimized to thehorizontal and very inclined showers that are interesting as potentially generated by neutrinos. Those showerscould be triggered using signatures related to the curvature of the shower front, the rise time of traces, or theirmuon component giving early peaks for ”old” showers.12

The structure of signals detected in water Cherenkov tanks and generated by horizontal showers dependstrongly on the point of the EAS initialization. Old showers generated by hadrons early in the atmosphere giveflat muonic front; showers generated by deeply interacting neutrinos are characterized by a curved front (radiusof curvature of a few km), a large electromagnetic component and with particles spread over a few microsecondsinterval. In both cases muonic front produces a bump, which can be a starting signature of horizontal showers.The bump for the old showers is shorter and sharper than for the young ones and results in a larger contributionin higher Fourier coefficients. For young showers, with relatively smooth shape of a signal profile, the lowerFourier components should dominate. The online analysis of the Fourier components may trigger specific events.

In the last few years, radio detection of cosmic ray air showers is developing successfully and becoming manifestin a number of new experiments. In particular, the LOPES project has implemented modern interferometermethods to measure the radio emission from extensive air showers.13 LOPES has confirmed that the emissionis coherent and of geo-magnetic origin, as expected by the geo-synchrotron mechanism, and has demonstratedthat a large scale application of the radio technique has great potential to complement current measurements ofultra-high energy cosmic rays. New triggers in the frequency domain are being developed for applications of theradio technique in the framework of the Pierre Auger Observatory.

An enhancement of the Auger acceptance at full efficiency down to 1017 eV is planned by a deployment ofan SD graded infill at smaller distances over an area much smaller than Auger. Area of 23.5 km2 is sufficient toregister a reasonable flux.14 Apart from the Auger tanks at the 1.5 km spacing, extra 42 SD’s will be installed.The aim of the new AMIGA15 project (Auger Muons and Infill for the Ground Array) is to show that theaddition of a small area (due to the rapidly decaying flux with energy) of SD stations with nearby buried muoncounters would allow a dramatic increase of the physical scope of this Observatory, reaching lower energies at

which the transition from galactic to extragalactic sources is expected. Lowering the Auger energy thresholdby more than one order of magnitude will allow a precise measurement of the cosmic ray spectrum in the veryinteresting regions of the second knee and the ankle.

Each one of the proposed muon detectors will have an area of 30 m2, and will be buried at the side of each SDstation at a depth of approximately 2.5 m. The distance to the station will be large enough to avoid shadowingfrom the water tank, guaranteeing a uniform shielding, but small enough to represent the same physical pointinside the shower front and to be able to share GPS time signals and telecommunications with its associatedsurface detector. Muon detectors will be equipped with plastic scintillators already developed and used by theMINOS experiment.

From the beginning the Pierre Auger Observatory was planned to observe both hemispheres, for completesky coverage at the highest energies. One expects the Auger South will be completed in 2007. Designing of theNorthern site has been already started. The SD in the Auger North will sample showers at ground level using4000 Cherenkov detectors placed in a rectangular one-mile grid. The modified configuration with respect to theAuger South is motivated by the aim of improving the acceptance at the highest energies. The Cherenkov lightis detected in the Auger South by three large PMTs viewing the water tank from the top. Similar PMT willbe used in the Auger North. However, in order to reduce cost and simplify the design, only one large PMT,placed in the middle of the tank, will be installed. In order to extend the dynamic range the main PMT will besupported by the smaller one. The significant modification of the light sensors and a reduction of independentchannels corresponding to PMTs bounds the use of coincidence techniques.16

R&D of the surface detector system for Auger North has extended the array area from 3000 km2 in AugerSouth to 10000 km2 in Auger North, changed a surface detectors grid from triangular to square, and changed thePMT design from three PMTs to 2 PMTs only, one with a large photo-cathode surface working in a high-gainchannel, the other a smaller tube targeted to ultra-high energy showers, where high-gain channel have beensaturated.16 Significant design modifications, huge progress in electronics, and new challenges and expectationsof the physics has inspired considerations of new types of triggers, which now could be easily implemented inmodern FPGAs but were unattainable in earlier versions used in the Auger South.

New projects and challenges require newer approach to the triggering. The 4th generation of the Front-End17

is being developed for the proposed AMIGA project as well as being as a test platform for radio-detection purposesand for the Auger North. It will be clocked at 160 MHz. The proposed sampling rate for the Auger North is 100MHz. The resolution is still 10-bit. The new Front-End is developed as universal one, providing 10-bit/200 MSps(AD9216) for AMIGA and Auger North or 14-bit/130 MSps (AD9248) for radio-detection. The new powerfulAltera R© StratixIITM FPGA with numerous DSP blocks and new architecture allows an implementation andtesting of new algorithms needed for much more sophisticated detection requirements than in the past.

6.1 16-point FFT trigger

The existing software procedures, available as commercial IP routines, are too slow to be used as a trigger workingin the real time. Online triggering requires the hardware implementation calculating multipoint DFT with asufficient speed. In the FFT algorithm (Figure 1) two first pipeline stages utilize adders and differential circuitsonly, and can be implemented as the single bin routines; however, multipliers need at least two clock cycles toassure a sufficient registered performance. Some signals (i.e. B8 = A8), which do not require any mathematicalcalculation, have to be delayed in appropriate registers in order to assure the correct synchronization.

The algorithm has been implemented for a 10-bit FADC bus into the Cyclone EP1C12Q240I7 Altera R© chip,with 40MHz clock (as in the 1stfirst level SD trigger). It requires 12 multipliers, 28 adders, 29 differential circuitsand 12 delaying routines (simple registers) for synchronization (totally ∼2000 logic elements).18 Fixed-pointmultiplication provides accuracy on the level 0.05% (multiplication of two 12- bit factors gives as result 24-bit value, however, the result from multiplier is shortened to 15 bits only (in order to save resources), whichprovides an additional 3 bits of accuracy (in comparison with the pure cut-off of the input 12-bit signed databy the scaling factor). In order to minimize the resource occupancy the AHDL code utilizes the Library ofParameterized Modules (LPM) with unsigned parameters, where possible and the fixed- point multiplicationstructure, where the size of the fractional part is arbitrarily fixed. The intermediate results of multiplications are

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ααααB11

C13Α13Α13Α13Α

====

ααααB13

C15Α15Α15Α15Α

====

ααααB15

C9Β9Β9Β9Β

====

ββββB9

C11Β11Β11Β11Β

====

ββββB11

C13Β13Β13Β13Β

====

ββββB13

C15Β15Β15Β15Β

====

ββββB15

C8=B8

C12=B12

D13=C13B+C15Α15Α15Α15Α

D11=C11A+C9Β9Β9Β9Β

D15=C13A-C15B

D9=C9A-C11B

D8 = C8+C14141414

D14= C8-C14

D0 = C0 + C1111

D1 = C0 - C1

Re X5= D14+ D13131313

Re X7 = D8-D15

Re X3 = D14-D13

Im X7= D10-D11

Im X3= -(D12+D9999))))

Im X5=D12-D9

Re X1= D8+D15151515

Im X1= -(D10+D11111111))))

Re X4 = D2 D2 = C2

Im X4 = - D3 D3 = C3

D5 ==== γγγγC5

D7 ==== γγγγC7

D4 = C4

D6 = C6

Re X2 = D4+D7777

Re X6 = D4-D7777

Im X2= - (D6+D5555))))

Im X6 = D6-D5555

B12=A12

D10 = C10+C12121212

D12=C10-C12

Re X0 = D0

Re X8 = D1

Figure 11. Global pipeline internal structure of the 16-point FFT

approximated on the next pipeline stage after next sum/difference. Re and Im coefficients are finally calculatedwith 10-bit resolution.

The very high registered performance of the FFT routine (≥ 200 MHz) allows a utilization of the samelogical structure for three 10-bit FADC channels multiplexing with three times the frequency of the global clock(generated internally by PLL)). The FFT code has been successively merged with the Auger code. The Augerinput data from three shift registers (corresponding to three high-gain channels) are multiplexed with 120 MHzand are fed into the FFT routine as a whole. The output FFT coefficients, calculated inside the FFT routine,are next de-multiplexed and stored in separate registers. They are then available for each channel independentlyand can be used for a trigger with the standard clock 40 MHz. The initial resources occupancy increased from54% to 78%. The final registered performance of the global code is on the level 140 MHz, which gives a sufficientsafety margin

A 16-point Discrete Fourier Transform based on the Radix-2 FFT algorithm analyzing the shape of FADCtraces might be used as a spectral trigger, estimating “power spectral densities”: ξ = {Re Xk}2 + {Im Xk}2

which can be calculated by the “square devices” (see Figure 8). Unfortunately the Cyclone chips used in the 3rd

generation of the triggers do not contain DSP blocks. Multiplications have to be implemented in logic elements.This is resource-consuming solution especially for wide buses. Additional implementation of a dual-port RAMas a bank of 8-bit “squares” significantly reduces resource utilization in comparison with a direct multiplicationof 16x16 input data. The FFT implementation provides a powerful tool to trigger events on the characteristicsof their traces in the frequency domain .

6.2 16-point DCT trigger

For real signal xn

XN2 +k =

N−1∑n=0

xne−j 2πN (N

2 +k)n =N−1∑n=0

xn(−1)n[ej 2π

N kn]∗

= X∗N2 −k

(1)

N2

thspectral line of Xk , k = 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 N2 + 1 spectral lines for k =

0,1,...,N/2 corresponding to frequencies fk = k · f0 = k 1N∆t , changing from zero to fsmpl

2 with fsmpl

N grid. DCTfor real signal xn gives independent spectral coefficients for k = 0,1,...,N-1, changing fk also from zero to fsmpl

2 ,but with fsmpl

2N grid. DCT vs. DFT gives twice better resolution.

The sequence of 8-point DCT is effective in image data compression. Since 1988 the 8-point Arai-Agui-Nakajima (AAN) algorithm derived from the 16-point DFT and calculating only the real parts of the DFT iswidely used.19 A minimization of multiplications amounts is one of a fundamental goal in long-term numericalcalculations. Reduction of product terms significantly speed up sophisticated calculations, because a single mul-tiplication requires several clock cycles of processor. Multiplications in powerful FPGA chips can be performedin very fast dedicated DSP blocks in a single clock cycle. Signals processed in parallel threads in a hardwareimplementation of a pipeline design have to be synchronized to each other. Pipeline approach requires additionalshift registers for synchronization also for signal currently not being processed. However, such synchronizationneeds additional resources.

The original AAN algorithm has been optimized for a FPGA environment. A simple implementation of apart of DCT8 algorithm for X4, ..., X7 shows that a reduction of a single pipeline stage at the expense of oneadditional multiplication performed in a dedicated DSP block significantly decreases a logic elements utilizationas well as significantly improves a registered performance, because a multiplication in a DSP block is much fasterthan performed in standard logic elements.20

Multiplications of integer signals coming from the FADC by real scaling factors (multipliers in Figure 12) givefloating-point results. In order to keep possible high speed of calculation and not to utilize resources spendthriftthe fixed-point algorithm of processing has been chosen. Integer values were approximated on each pipelinestages. This introduced approximation errors, however mostly in a LSB (apart of X15, which approximationerror is much higher due to big η15 coefficient).

A 16-point sampling, at the 160 MHz, would correspond to 100 ns. A “length” of spikes coming from “old”showers is expected less than 200 ns. So, for 160 MHz sampling rate, an undersampling is considered also for anoptimization.

The analysis of very inclined showers from the Auger database gives an estimation of FADC traces as

xk =A e−αtk + ped, with the attenuation factor α ≈ -0.4 ± 0.1 , where tk is taken with 25 ns grid.

Simulation performed for the factor α estimated above, random amplitude: A ≤ 255 and a random pedestal= (50 ± 2) ADC-counts give

0 ≤ Xk/X1 ≤ 0.12 (k = 9, .., 14) (2)

for more than 99% of events.

For k ≤ 8 ratios of Xk/X1 are spread on relatively wide range and do not introduce relevant information. Thelimits estimated above can be taken as boundary conditions in order to select online very inclined showers witha big efficiency. If the only selected Xk (k=9,..,14) are used for a trigger, a utilization of resources significantlydrops down.

Altera R© LPM contains actually a lpm divide routine supporting a division of fixed-point variables, howevera comparison of X1 with a Xk multiplied by a suitable factor corresponding to the assumed thresholds is muchmore effective than a direct ratio calculation for a future comparison.

x15

x14

x13

x12

x11

x10

x9

x 8

x7

x6

x5

x4

x3

x2

x 1

x 0

A15=x0-x15

A14=x1-x14

A13=x2-x13

A12=x3-x12

A11=x4-x11

A10=x5-x10

A9=x6-x9

A8=x7-x8

A7=x7+x8

A6=x6+x9

A5=x5+x10

A4=x4+x11

A3=x3+x12

A2=x2+x13

A1=x1+x14

A0=x0+x15

B15 = A15

B14=A4+A15

B13=A13+A14

B12=A12+A13

B11=A11+A12

B10=A10+A11

B9=A9+A10

B8=A8+A9

B7=A0-A7

B6=A1-A6

B5=A2-A5

B4=A3-A4

B3=A3+A4

B2=A2+A5

B1=A1+A6

B0=A0+A7

C15 = B15

C14 = B14

C213 = S2B13

C12=B12+B14

C11 = S4B11

C10=B10+B12

C29 = S2B9

C8=B8+B10

C7 = B7

C6=B6+B7

C5=B5+B6

C4=B4+B5

C3=B0-B3

C2=B1-B2

C1=B1+B2

C0=B0+B3

C613 = S6B13

C69 = S6B9

D15====C15+C11

D14 = C14

D212 = S2C12

D11=C15-C11

D9=C613–C2

9

D28 = S2C8

D7 = = = = C7

D26 = S2C6

D5 = S4C5

D24 = S2C4

D3 = C3

D2=C2+C3

D1=C0-C1

D0=C0+C1111

D612 = S6C12

D68 = S6C8

D64 = S6C4

D66 = S6C6

E15=D15+D13131313

E14=D14+D10

E13=D15-D13131313

E12=D212+D6

8

E11=D11+D9

E10=D14-D10

E9=D11-D9

E8=D612–D2

8

E7=D5+D7777

E6=D26+D6

4444

E5=D7-D5555

E4=D66-D

24444

E3 = D3

E2 = S4D2

E1 = D1

E0 = D0

F15 = E15

F14=E14+E12

F13 = E13

F12=E14-E12

F11 = E11

F10=E10+E8

F9 = E9

F8=E10-E8

F7= E7-E6666

F6= E7+E6666

F5= E5+E4444

F4= E5-E4444

F3=E3+E2

F2=E3-E2

F1 = E1

F0 = E0

G15 = F15

G14= γγγγ1F14

G13 = F13

G12= γγγγ7F12

G11 = F11

G10= γγγγ3F10

G9 = F9

G8= γγγγ5F8

G7 = F7

G6 = F6

G5 = F5

G4 = F4

G3 = F3

G2 = F2

G1 = F1

G0 = F0

H7=ZS7X14 = G7

H6=ZS1X2 = G6

H5=ZS3X6 = G5

H4=ZS5X10 = G4

H3=ZS2X4 = G3

H2=ZS6X12= G2

X8=H1/4

X0=H0/4

H15=ZS1X1=G15+G14141414

H14=ZS15X15=G15-G14141414

H13=ZS7X7=G13+G12121212

H12=ZS9X9=G13-G12121212

H11=ZS3X3=G11+G10101010

H10=ZS13X13=G11-G10101010

H9=ZS5X5=G9+G8888

H8=ZS11X11=G9-G8888

X1=ηηηη1H15

X15=ηηηη15H14

X7=ηηηη7H13

X9=ηηηη9H12

X3=ηηηη3H11

X13=ηηηη13H10

X5=ηηηη5H9

X11=ηηηη11H8

X14=ηηηη14141414H7

X2=ηηηη2H6

X6=ηηηη6H5

X10=ηηηη10H4

X4=ηηηη4H3

X12=ηηηη12H2

X8

X0

D13====C213+C6

9

D10 = S4C10

Figure 12. Global pipeline internal structure of the 16-point DCT.

Denoting spectral thresholds as Thrk (corresponding to a value from (2), however not necessary the same forall selected indices) the inequality (2) can be simplified as follows:

H15 ≥ ξkHp (3)

where (k=9,..,14) correspond to (p=12,4,8,2,10,7) respectively and

ξk =ηk

η1Thrk(4)

A simple set of parallel comparators selecting ranges of Xk with a serial AND gate will produce a finalspectral trigger. It will be generated at least 24 clock cycles later in comparison with the Threshold trigger forsufficiently high amplitude corresponding to the sample x0 (16 clock cycles of propagation in the shift registersand at least 8 clock cycles of performance in a DCT chain). If the amplitudes of samples are sufficiently highfor a Threshold trigger generation, the next triggers are inhibited for 768 time bins necessary to fulfill memorybuffers (see Fig.7 in 7). This bounds a spectral trigger generation to a relatively low amplitudes.

In the currently used algorithm, the calibration channel modifies thresholds in order to keep required triggerconditions. However, the response from the calibration channel is relatively slow. DCT coefficients for k ≥ 1are independent of pedestal. The implementation of the trigger with native pedestal independence seems to bebetter approach than thresholds tuning via the slow calibration loop.

7. SUMMARY

Sophisticated algorithms selecting very rare events can be currently implemented into new powerful FPGAs.These events are crucial to accept/reject various hypothesis, to fix discrepancies between different experimentsand to understand physics beyond the standard status. Huge progress in electronics, the collective experienceand intellectual potential of the Auger team aims to solve some of the mysteries of the universe.

ACKNOWLEDGMENTS

This work was funded by the Polish Committee of Science under KBN Grant No. 2 P03D 001 24

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