How to Fully Exploit the Degrees of Freedom in the Downlink of MISO Systems With Opportunistic Beamforming

Abstract: The opportunistic beamforming in the downlink of multiple-input single-output (MISO) systems forms $N$ transmit beams, usually, no more than the number of transmit antennas $N_t$. However, the degrees of freedom in this downlink is as large as $N_t^2$. That is, at most $N_t^2$ rather than only $N_t$ users can be simultaneously transmitted and thus the scheduling latency can be significantly reduced. In this paper, we focus on the opportunistic beamforming schemes with $N_t<N\le N_t^2$ transmit beams in the downlink of MISO systems over Rayleigh fading channels. We first show how to design the beamforming matrices with maximum number of transmit beams as well as least correlation between any pair of them as possible, through Fourier, Grassmannian, and mutually unbiased bases (MUB) based constructions in practice. Then, we analyze their system throughput by exploiting the asymptotic theory of extreme order statistics. Finally, our simulation results show the Grassmannian-based beamforming achieves the maximum throughput in all cases with $N_t=2$, 3, 4. However, if we want to exploit overall $N_t^2$ degrees of freedom, we shall resort to the Fourier and MUB-based constructions in the cases with $N_t=3$, 4, respectively.