GESTOS EN LA LITURGIA

The current FPGA technology enabled the integration of a digital spectroscopy channel, including trigger and energy filters together with a long waveform buffer, into a single low-cost FPGA device. The energy resolution of the moving window deconvolution im- plementation for the GRT4 VME module exceeded the expectations, which is also due to the low noise of the GRT4 analog stage.

The supplementary modules generated for the energy filter, i.e. trigger filter and waveform buffer, performed as expected. For small systems the internal digital trigger allows a quick and remote setup of the GRT4, while for larger system, e.g. highly seg- mented HPGe detector, an external common trigger signal has to be fed onto the GRT4. The extended waveform buffer allowed for the investigation of the ADC non-linearities with slightly suboptimal filter parameters. For pulse shape analysis applications a sin- gle Block RAM, storing up to 3.2s of data, is sufficient and is therefore foreseen for the

final implementation [76].

The state machine allowed for an adjustment of the peak capture delay and the pile up rejection period, the proper operation was verified in various measurements. How- ever, the waveform memory was not integrated into the MWD state machine. If wave- form and energy are to be read in coincidence, then the pile up signal should be used to flag the events only, since the waveform buffer is stopped with each trigger. In order to enable the usage of the GRT4 in nuclear physics experiments, live and real time clocks have to be added to enable dead time correction if the implemented internal digital trig- ger algorithm is used.

5.2. PREPROCESSING ALGORITHMS: MWD AND GRT4 127 The energy resolution at high energies is mainly limited by the detector. The trape-

zoidal shaping eliminates the risetime effects due to the adjustable flat top period and the BLR allows a proper suppression of the pedestal of the energy filter and improves the energy resolution considerably. The dependence of the energy resolution on the trape- zoidal filter parameters was measured and the best energy resolution was obtained if the BLR averages the previous 256 s of energy filter data. At small energies the contri-

bution of noise from the electronics becomes more important, which can be compensated by using longer peaking timesL. The resolution at low energies is also very sensitive to

ADC non-linearities and an initial DNL correction showed promising results. Further tests have to be performed after the non-linearity correction has been fully implemented and tested.

The non-linearities of the AD6645 ADC were corrected using a small look-up table and the energy resolution improved significantly over the whole energy range. The ADC non-linearity has to be investigated further using a pileup pulser to determine the cor- rection coefficients for the full ADC range. First tests have already been performed with a different ADC confirming the observations [76]. If the observed non-linearities would be device dependent, each channel of the system would have to be calibrated individually and the obtained coefficients would have to be downloaded to the corresponding GRT4. In this case it would be advisable to evaluate ADCs from different manufacturers1.

The developed VHDL code enables the realization of a digital spectroscopy system at low cost and furthermore the conversion of commercially available waveform digi- tizers into a spectroscopy system. Since the VHDL code is largely independent of the FPGA architecture, the code can be synthesized for other XILINX FPGAs allowing the setup of a digital DAQ system based on commercially available digitizer modules with user FPGAs2. A more resource saving variant of the MWD code has been developed and simulated including 8 spectroscopic channels into a single Spartan 2 FPGA. Based on the existing code, the MWD algorithm can be tailored quickly to new detectors and applications.

For the characterization of the AGATA detector at the University of Liverpool, the GRT4 cards will be used. First tests were already performed with a highly segmented detector [76]. Based on the experience gained with the MWD and the GRT4, new elec- tronics are currently in the planning or design phase, a dedicated electronics for the

-ray tracking array AGATA [16] and a dedicated electronics for the smartPET detector

[78], a planar HPGe detector for medical imaging applications. Furthermore, it is envis- aged to build a low cost digital electronics for the DSSSD [79] detector of the MINIBALL experiment.

The developed VHDL implementation of the moving window deconvolution proved the advantage of trapezoidal shaping for HPGe detectors and - since the energy is the most important information of a -ray - enables creation of custom digital spectrometers

at a low cost in the future. This is of great importance for the AGATA array, further increasing the number of channels by a factor of 40 compared to the MINIBALL array.

1_{ADCs including calibration circuits are available for low sampling rates (1 MHz) only.}

2_{Nallatech [77] offers the BenADIC, a 20 channel ADC CompactPCI module with 6 user FPGAs and}

Appendix A

Introduction to Digital Signal

Processing

”The secret to creativity is knowing how to hide your sources.” - Albert Ein- stein

In this chapter, the fundamentals of digital signal processing necessary for the un- derstanding of the following chapters will be reviewed and the properties of various dig- ital filters will be discussed. For further details the reader is referred to the literature [80, 51, 81, 82, 83, 84, 85].

In the following a one-dimensional problem is assumed. The incoming analog signal is represented as a function of a single variable, usually the timet,f(t).

The most important system for digital signal processing is the linear time-invariant system (LTI), which is characterized by its impulse response. The output of such a sys- tem for any input signal can be derived from the knowledge of the impulse response.

The LTI system is defined in section A.1 and is followed by a description of the analog- to-digital conversion process in section A.2. Two important types of digital filters are presented, the IIR and the FIR filter. Afterwards, the numerical differentiation and the resampling process are presented, because of their importance for pulse shape analysis.

A.1

Discrete Linear Time-Invariant Systems

A discrete system is represented by the transformation of the input sequence x[n℄ into

the output sequencey[n℄

x[n℄ ! y[n℄

Æ[n℄ ! h[n℄;

where the last equation defines the response of the system to a single impulseÆ[n℄, the

impulse responseh[n℄, of the system.

The superposition property defines the linear system. If y 1 [n℄ is the response of a system to an inputx 1 [n℄andy 2 [n℄is the response tox 2 [n℄ x 1 [n℄ ! y 1 [n℄; x 2 [n℄ ! y 2 [n℄; 129

then a system is called linear if the response of the system to a superposition of two signals is given by 1 x 1 [n℄+ 2 x 2 [n℄ ! 1 y 1 [n℄+ 2 y 2 [n℄:

The time-invariant system is defined by the property that the output is delayed by the same amount of time as the input signal:

x[n+d℄!y[n+d℄;

which means, that the system is invariant under translation operations. If the input sequence can be expressed as

x[n℄=

X

k

x[k℄Æ[n k℄

the response of the LTI system is determined by the convolution [80] of the input se- quence with the impulse response

y[n℄ =

X

k

x[k℄h[n k℄

= x[k℄h[n k℄:

If two systems are connected in series then the overall impulse response is the convo- lution of the individual impulse responsesh

s [n℄=h 1 [n℄h 2 [n℄=h 2 [n℄h 1 [n℄. The inverse systemh i is defined by Æ[n℄=h[n℄h i [n℄=h i

[n℄h[n℄meaning that the overall response

to an input sequencex[n℄is the input sequence itself.