Optimal Multiuser Transmit Beamforming: A Difficult Problem with a Simple Solution Structure [Lecture Notes]

International audienceTransmit beamforming is a versatile technique for signal transmission from an array of N antennasto one or multiple users [1]. In wireless communications, the goal is to increase the signal power atthe intended user and reduce interference to non-intended users. A high signal power is achieved bytransmitting the same data signal from all antennas, but with different amplitudes and phases, suchthat the signal components add coherently at the user. Low interference is accomplished by making thesignal components add destructively at non-intended users. This corresponds mathematically to designingbeamforming vectors (that describe the amplitudes and phases) to have large inner products with thevectors describing the intended channels and small inner products with non-intended user channels.If there is line-of-sight (LoS) between the transmitter and receiver, beamforming can be seen as forminga signal beam toward the receiver; see Figure 1. Beamforming can also be applied in non-LoS scenarios,if the multipath channel is known, by making the multipath components add coherently or destructively.Since transmit beamforming focuses the signal energy at certain places, less energy arrives to otherplaces. This allows for so-called space-division multiple access (SDMA), where K spatially separatedusers are served simultaneously. One beamforming vector is assigned to each user and can be matchedto its channel. Unfortunately, the finite number of transmit antennas only provides a limited amount ofspatial directivity, which means that there are energy leakages between the users which act as interference.While it is fairly easy to design a beamforming vector that maximizes the signal power at the intendeduser, it is difficult to strike a perfect balance between maximizing the signal power and minimizingthe interference leakage. In fact, the optimization of multiuser transmit beamforming is generally anondeterministic polynomial-time (NP) hard problem [2]. Nevertheless, this lecture shows that the optimaltransmit beamforming has a simple structure with very intuitive properties and interpretations. Thisstructure provides a theoretical foundation for practical low-complexity beamforming schemes