We report on the first direct measurement of the basic features of microwave radio emission from extensive air showers. Using a trigger provided by the KASCADE-Grande air shower array, the signals of the microwave antennas of the CROME (Cosmic-Ray Observation via Microwave Emission) experiment have been read out and searched for signatures of radio emission by high-energy air showers. Microwave signals have been detected for more than 30 showers with energies above $3times10^{16}$,eV. The observations presented in this Letter are consistent with a mainly forward-beamed, coherent and polarised emission process in the GHz frequency range. An isotropic, unpolarised radiation is disfavoured as the dominant emission model. The measurements show that microwave radiation offers a new means of studying air showers at very high energy.
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