Beamformer design

Transmitter design for multi-user MIMO broadcast channel has been extensively studied in the last few years. It is well known that the optimal precoding strategy involves non-linear interference pre-subtraction, commonly referred to as dirty paper coding. This technique is rarely used in practice due to computational restrictions and also because its optimality is restricted to Gaussian alphabets, which are never used in practice. Owing to their computational simplicity, linear precoding strategies are very widely employed in the downlink transmission.

The scenario where the base-station is equipped with $N$ transmit antennas and transmits to $K$ remote users, each equipped with $M$ antennas, is modelled as

\begin{equation} \textbf y_i=\textbf H_i \textbf x_i + \textbf z_i \quad \forall i \end{equation} where $\textbf x$ is an $N\times 1$ complex vector representing the transmit signal, $\textbf H_i$'s are the $M\times N$ complex channel matrices, $\textbf y_i$'s are $M\times 1$ complex vectors representing the received signal and $\textbf z_i$'s are the i.i.d additive complex Gaussian noise vectors with variance $\sigma^2/2$ on each of its real and imaginary components. For the case where the remote users are equipped with a single antenna, a linear beamforming strategy is commonly employed at the transmitter. In this scenario, $K$ beamforming vectors may be used to create $K$ data streams, one to each user. The design of the $K$ beamforming vectors is posed as a convex optimization problem that tries to minimize the total transmit power while satisfying certain SINR contraints. Due to the well known uplink-downlink duality, the optimal beamformers can be computed efficiently.

The SINR constraints are imposed so as to meet a certain target rate and error probability for every user. The SINR constraints implicitly assume the interference to be Gaussian in nature.