On Low-Complexity Implementation of Nonlinear Vector Precoding for Multi-Antenna Systems
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Wireless communication systems using multiple antennas at a transmitter, and many receivers, such as multiple-input multipleoutput (MIMO) broadcast systems, admit an application of a joint signal processing at the transmitter side. The algorithms that accomplish the task of joint signal processing that leads to enhanced performance of the MIMO broadcast systems are known as precoding algorithms. We consider a wireless MIMO broadcast system with white Gaussian noise, where the transmitter has multiple antennas, and each receiver is equipped with a single antenna. The knowledge of perfect channel state information (CSI) at the transmitter is assumed. The joint signal processing is admitted at the transmitter side, while the receivers are non-cooperative devices. The focus of this work is on the analysis of the MIMO system with finite number of transmit and receive antennas. In cellular systems, when the number of data streams is larger than the number of transmit antennas, and furthermore, the number of data streams exceeds a certain threshold number, the cellular systems become overloaded. In this work, an algorithm for vector precoding suitable for overloaded MIMO channels is proposed. It is shown that this algorithm can be used to increase the number of transmitted data streams significantly beyond the number of transmit antennas. The probability that this vector precoding algorithm, due to the channel singularity, fails to provide the data without crosstalk to the receiver is calculated analytically. This result is extended to the single-user MIMO channel in the presence of correlated Rayleigh fading, and more generally, arbitrary fading correlations. Non-overloaded MIMO system is considered in the rest of this work. The Hopfield neural network (HNN) is a powerful computational algorithm that belongs to the field of artificial neural networks (ANNs). A practical vector precoding algorithm using theHNNis derived, and analyzed by numerical simulations. The simulation results indicate that the HNN for vector precoding achieves the performance competitive to sphere encoder (SE) within wide system loads, where the system load is the ratio of the number of receive and transmit antennas. However, the degradation of the performance is exhibited in the MIMO broadcast system with the number of transmit antennas close to the number of receive antennas. Further improvement of the HNN for vector precoding algorithm is addressed by applying the Lenstra, Lenstra, Lov´asz lattice basis reduction (LLL-LR) algorithm. It is shown that the LLL-LR aided HNNfor vector precoding controls the energy penalty within the load range, where the sole HNN for vector precoding suffers from performance degradation. Computational complexity of the proposed algorithms is considered. The investigation of the number of iterations needed to be performed by the HNN algorithm until the convergence has been reached, indicates that small number of iterations is sufficient. The practical computational complexity of the HNN for vector precoding and LLL-LR aided HNN for vector precoding algorithms offers low-complexity alternatives to the SE within a wide range of loads. Characterization of the performance of vector precoding algorithms in terms of the spectral efficiency is the ultimate performance measure. We quantify the spectral efficiency of the minimum mean square error (MMSE) vector precoding algorithm by numerical simulations. Specifically, the performance metric is a normalized spectral efficiency (per transmit antenna). The discussion is restricted to the spectral efficiency of MMSE vector precoding with quadrature phase-shift keying (QPSK) signaling. The optimum system load that maximizes the spectral efficiency is obtained. The performance enhancement that MMSE vector precoding obtains in comparison to vector precoding with linear zero forcing (ZF) preprocessing, in the low to medium the energy per bit divided by the noise spectral density Eb N0 region, is quantified. We find that MMSE vector precoding does not significantly outperform its linear counterpart.