Average transmit power of adaptive ZF very large multi-user and multi-antenna systems

In this paper, we investigate adaptive zero-forcing uplink transmission for very large multi-user multi-antenna systems in Rayleigh fading environments. We assume that the number of antennas at the base station (denoted as $$M$$M) is not less than the number of users (denoted as $$K$$K) with each having single antenna, and power control can be done at the transmitter(s) as channel condition changes. Under constraints of individual rates and maximum transmit powers, we adopt the optimal transmit strategy of minimizing the total average transmit power (ATP). We derive and give individual ATP expressions for each link with short- and long-term rate constraints, respectively. Numerical results show that the individual ATP for each link with short term rate constraint is quite close to its long term counterpart when $$M-K$$M-K is large, and its corresponding outage probability can be designed to be nearly zero at the same time. Finally, we present two simple adaptive transmission schemes with constant transmit power satisfying short- and long-term rate constraints, respectively. Both of them are easy to implement, and asymptotically optimal when $$M-K$$M-K grows without bound.