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In this work, we propose a differential beamforming (DBF) scheme for bi-directional communication between two single-antenna terminals via a multi-antenna relay station (RS). The proposed scheme utilizes differential phase-shift keying modulation to enable beamforming at the RS without knowledge of the instantaneous channel state infor mation (CSI) at any entity in the network. In our differential scheme, receive and transmit beamforming at the RS is performed based on the implicit CSI contained in the received signals in the preceding time slots.Thus, the DBF scheme is applicable even if the communication channels are time-variant. For time-invariant channels, we show that our DBF scheme is associated with a performance penalty of 3 dB as compared to the ideal amplify-and-forward relaying scheme, which requires perfect CSI. Our simulation results confirm the analytical results for time invariant channels. For time-variant channels, the simulation demonstrate a high performance gain of the DBF scheme compared to schemes of the literature.Index Terms-Amplify-and-forward relaying, bi-directional relaying, differential phase-shift keying, differential beamforming, relay networks

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