Tunka-Rex is a radio detector for cosmic-ray air showers in Siberia, triggered by Tunka-133, a co-located air-Cherenkov detector. The main goal of Tunka-Rex is the cross-calibration of the two detectors by measuring the air-Cherenkov light and the radio signal emitted by the same air showers. This way we can explore the precision of the radio-detection technique, especially for the reconstruction of the primary energy and the depth of the shower maximum. The latter is sensitive to the mass of the primary cosmic-ray particles. In this paper we describe the detector setup and explain how electronics and antennas have been calibrated. The analysis of data of the first season proves the detection of cosmic-ray air showers and therefore, the functionality of the detector. We confirm the expected dependence of the detection threshold on the geomagnetic angle and the correlation between the energy of the primary cosmic-ray particle and the radio amplitude. Furthermore, we compare reconstructed amplitudes of radio pulses with predictions from CoREAS simulations, finding agreement within the uncertainties.
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