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In this paper, a novel transmission technique for the multiple-input multiple-output (MIMO) broadcast channel is proposed that allows simultaneous transmission to multiple users with limited feedback from each user. During a training phase, the base station modulates a training sequence on multiple sets of randomly chosen orthogonal beamforming vectors. Each user sends the index of the best beamforming vector and the corresponding signal-to-interfence-plus-noise ratio for that set of orthogonal vectors back to the base station. The base station opportunistically determines the users and corresponding orthogonal vectors that maximize the sum capacity. Based on the capacity expressions, the optimal amount of training to maximize the sum capacity is derived as a function of the system parameters. The main advantage of the proposed system is that it provides throughput gains for the MIMO broadcast channel with a small feedback overhead, and provides these gains even with a small number of active users. Numerical simulations show that a 20% gain in sum capacity is achieved (for a small number of users) over conventional opportunistic space division multiple access, and a 100% gain (for a large number of users) over conventional opportunistic beamforming when the number of transmit antennas is four. Index Terms--Beamforming, multiple-input multiple-output (MIMO), space-division multiple access (SDMA).

The ergodic capacity of a fading vector channel with multiple transmit antennas and a single receive antenna is explored. Perfect channel information is assumed to be available at the receiver while the transmitter has only partial knowledge of the direction of the user's channel vector based on quantized feedback. We present necessary and sufficient conditions for the optimality of beamforming in such systems. The conditions are applicable to all quantized feedback scenarios regardless of the channel distribution, number of transmit antennas, number of quantization vectors or transmit power. The optimality conditions are closely related to the iteration conditions of the Lloyd algorithm, revealing an interesting link between the optimality of beamforming and the optimality of the vector quantizers. Using the conditions, we prove the capacity optimality of beamforming for several quantized feedback scenarios such as the antenna-selection scheme. We also point out examples of quantized feedback scenarios where beamforming is not optimal. We find that for the independent identically distributed Rayleigh fading channel with more than a single bit of quantized feedback, there is no capacity benefit from increasing the number of antennas beyond the number of quantization vectors. Extensions of the necessary and sufficient optimality condition to the multiple-input multiple-output case are also provided. Index Terms--Beamforming, ergodic capacity, multiple-input, multiple-output (MIMO) channels, partial/limited feedback, quantized feedback.

In this letter, we study two techniques, known as single and multiple beamforming, to exploit the perfect channel state information (CSI) available both at the transmitter and the receiver of a multiantenna wireless system. Assuming N and M are the number of antennas at the transmitter and the receiver, respectively, we show that single beamforming (transmission of a single symbol from all transmit antennas at the same time, employing the best subchannel) can achieve the maximum spatial diversity order in the channel (NM). We extend our analytical results to multiple beamforming (transmission of S symbols simultaneously, S > 1) and calculate that the diversity order achievable for this system is (N - S + 1)(M - S + 1). Index Terms--Beamforming, diversity, multi-input multi-output (MIMO) systems, pairwise error probability (PEP).