A general duality theory for uplink and downlink beamforming

We assume a multi-user downlink scenario, where the SINR at each mobile is controlled by adjusting spatial pre-filters and transmission powers at the base station, prior to transmission. In this context, an interesting duality between uplink and downlink beamforming was observed throughout the last decade. This duality allows the joint downlink optimization problem to be solved efficiently by considering the equivalent uplink problem instead, which is easier to handle. In this paper we characterize the interference scenarios and noise levels for which the SINR requirements can be supported for a given power constraint. Moreover, we provide necessary and sufficient conditions for duality. Finally, it is shown how duality can be applied in order to derive optimal transmission strategies for combined downlink beamforming and Costa pre-coding.