Performance of dual-hop amplify-and-forward beamforming and its equivalent systems in rayleigh fading channels

Combining a dual-hop relaying with multi-input multi-output (MIMO) transmission is a natural extension to overcome the channel impairments. Transmit beamforming (TBF) and maximal ratio combining (MRC) are widely accepted ones, which maximize the signal-to-noise ratio (SNR) at the receiver when channel state information is available. With these methods, there are four possible combinations in constructing dual-hop transmission: TBF-TBF, MRC-MRC, MRC-TBF, and TBF-MRC, respectively. We provide optimal amplify-and-forward (AF) weights at a relay, which maximize the end-to-end SNR for the four systems, respectively, and show the equivalence of the four systems in terms of the SNR. Using relaxed AF weights from the optimal ones, we provide a probability density function (PDF) and a moment generating function (MGF) for the end-to- end SNR per bit with an assumption of an equal number of diversity branch for each hop, which is used to obtain the BER performance for M-ary QAM and PSK constellations, respectively. Numerical results show that the BERs with the relaxed AF weights provide tight lower bounds for those with optimal AF weights. We also compare the BER performance of above AF relaying with that of dual-hop decode-and-forward (DF) relaying.