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

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.