  Sparse digital antenna arrays constitute a promising detection technique for
future large-scale cosmic-ray observatories. It has recently been shown that
this kind of instrumentation can provide a resolution of the energy and of the
shower maximum on the level of other cosmic-ray detection methods. Due to the
dominant geomagnetic nature of the air-shower radio emission in the traditional
frequency band of 30 to 80 MHz, the amplitude and polarization of the radio
signal strongly depend on the azimuth and zenith angle of the arrival
direction. Thus, the estimation of the efficiency and subsequently of the
aperture of an antenna array is more complex than for particle or
Cherenkov-light detectors. We have built a new efficiency model based on
utilizing a lateral distribution function as a shower model, and a
probabilistic treatment of the detection process. The model is compared to the
data measured by the Tunka Radio Extension (Tunka-Rex), a digital antenna array
with an area of about 1 km$^2$ located in Siberia at the Tunka Advanced
Instrument for Cosmic rays and Gamma Ray Astronomy (TAIGA). Tunka-Rex detects
radio emission of air showers using trigger from air-Cherenkov and particle
detectors. The present study is an essential step towards the measurement of
the cosmic-ray flux with Tunka-Rex, and is important for radio measurements of
air showers in general.
