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The downlink communication of a multiuser multiple-input multiple-output (MIMO) system is comprised of a set of data streams that, unlike in the well-known point-to-point MIMO scenario, need to be detected independently and without any cooperation between the user terminals. Nevertheless, the separation of the user streams can be achieved by performing a signal shaping stage at the transmitter. The data preprocessing step performed prior to the transmission of the signal is also known as precoding and it can follow a linear (Joham, Utschick, & Nossek, 2005; Peel, Hochwald, & Swindlehurst, 2005) or non-linear (Hochwald, Peel, & Swindlehurst, 2005; Schmidt, Joham, & Utschick, 2008) approach.
Linear precoders are usually designed to reverse the distortion that the channel inflicts on the transmitted data signals. This way, the information symbols are pre-multiplied by a precoding matrix which can be designed based on different criteria. If a zero-forcing (ZF) approach is followed, the removal of all the interference between the users’ streams is required, which leads to the precoding matrix being reduced to a simple channel inversion (Peel, Hochwald, & Swindlehurst, 2005). The main drawback of this straightforward approach is the considerable increment in the power of the precoded signal, especially for ill-conditioned channels, which results in a substantial noise enhancement and a poor signal-to-noise ratio (SNR) at the receivers. This problem can be overcome by allowing for some interference among the users’ streams. The relaxation in the design constraints derives in the insertion of a regularization term in the channel inversion. The resulting linear precoder, also known as the regularized inversion precoder, outperforms the ZF approach especially in the low SNR regime (Peel, Hochwald, & Swindlehurst, 2005). It is also possible to design the precoding matrix following a Wiener filter approach in such a way that the mean square error (MSE) is minimized. Having a lower MSE, this approach outperforms the previous ones in terms of bit error rate (BER) performance.
A considerable performance improvement over linear precoding techniques can be achieved by including non-linear signal processing algorithms at both ends of the communication link. Nevertheless, the enhanced performance is a consequence of an extra added complexity, which may be prohibitive for real-time practical systems. This is the case of the well-known and capacity achieving dirty paper coding (DPC) technique presented in Costa (1983). Despite its relevance in the theoretic assessment of the capacity of multiuser broadcast channels, the large amount of required channel state information at the transmitter side and the great sensitivity to possible imperfections in this information render it unsuitable for practical implementation. This has led to the development of other non-linear precoding algorithms that aim at achieving a similar performance with a more reasonable complexity. One of the most noteworthy methods of non-linear pre-processing is Tomlinson-Harashima Precoding (THP), which was originally developed in Harashima (1972) and Tomlinson (1971) to mitigate the effect of intersymbol interference and was later adapted for transmission over MIMO channels in Fischer, Windpassinger, Lampe, and Huber (2002). The most remarkable feature of this precoding approach is the insertion of a modulo operation at both ends of the communication link in order to reduce the power of the precoded symbols. The modulo operator at the transmitter side can be equivalently replaced by the addition of a perturbing signal which can be optimized directly following various design criteria, such as minimizing the transmit power of the precoded symbols or optimizing the overall MSE. This leads us to the concept of vector precoding (VP) (Hochwald, Peel, & Swindlehurst, 2005), which improves the performance of THP at the expense of a greater complexity.