**1. Introduction**

14 Will-be-set-by-IN-TECH

378 Applications of Digital Signal Processing

as well as to understand how each factor affects the response variable in a model based on an orthonormal system. Moreover, it is already shown that the analysis of variance can also be performed in a model based on an orthonormal system. Hence, it is clear that two main procedures in the experimental design, that is, the estimation of the effects and the analysis of variance can be executed in a description of experimental design on the basis of an

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**6. References**

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 10<sup>18</sup> eV 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 10<sup>19</sup> eV is extraordinarily low: on the order of one 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 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 4 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 a "hybrid" measurements allow 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.

Experiments 3

<sup>381</sup> 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

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. 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 0.2 *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

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

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 FADC traces characteristic for the traces of very inclined showers. 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 (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

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.

*xncos <sup>π</sup>*

*<sup>N</sup>* (*<sup>n</sup>* <sup>+</sup>

1 2 )*k* 

(1)

ignoring the single muon background (Abraham J. et al., 2010).

a PMT as spread in time and relatively slow increasing.

of "gaps" in the array of activated surface detectors.

**3. Discrete Fourier Transform vs. Discrete Cosine Transform**

*X*¯ *<sup>k</sup>* = *α<sup>k</sup>*

*N*−1 ∑ *n*=0

**2. Triggers**

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. 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 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.

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.
