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A random beamforming (RBF) technique is proposed to achieve omnidirectional coverage for broadcast channels in multiple-antenna systems. In the proposed scheme, the time and frequency resources for communication are divided into many time-frequency blocks (TFBs). A random pattern is employed on each TFB, with the average power in all directions over many TFBs being equal. The design criteria and detailed design of the random pattern sequence are given. We propose a basic RBF scheme and an Alamouti-enhanced scheme, as well as the corresponding structures of the transmitter and the receiver. Simulations are carried out for an eight-element unitary linear array with half-waveform separation in a fast-fading channel, showing that the proposed scheme greatly outperforms the cyclic delay diversity scheme and is robust to calibration errors and failures of radio-frequency chains.