From the feedback of the channel state information (CSI), precoding techniques improve the performance of multiple-input multiple-output (MIMO) systems by optimizing various criteria. In this correspondence, an efficient precoder that maximizes the minimum distance (dmin) of two received vectors is studied. This criterion leads to a nondiagonal precoding scheme and allows achieving a full diversity order. However, the optimized solution for MIMO systems using a high-order QAM modulation is rather complex and changes for different constellations. Therefore, we propose herein a general form of minimum Euclidean distance based precoders for all rectangular QAM modulations. It is shown that the new solution optimizes the distance dmin for small and large dispersive channels.

Minimum Euclidean Distance Based Precoders for MIMO Systems Using Rectangular QAM Modulations

Through a feedback link, the channel state information acquired at the receiver is known at the transmitter and MIMO precoding techniques can be applied according to various criteria. In this paper, an efficient linear precoder based on the maximization of the minimum Euclidean distance between two received data vectors for three data-stream MIMO spatial multiplexing systems is proposed. By using a singular value decomposition, the precoding matrix can be parameterized as the product of a power allocation matrix and an input shaper matrix. According to this representation, the optimal precoders are proposed for the case of three independent data-streams. Simulation results over Rayleigh fading channels show considerable bit-error-rate improvement with the proposed solution in comparison with other precoding strategies.

Minimum Euclidean Distance-Based Precoding for Three-Dimensional Multiple Input Multiple Output Spatial Multiplexing Systems

The objective of maximizing the minimum Euclidean distance between two received data vectors via real-valued linear precoding is considered for multiple-input multiple-output (MIMO) systems with arbitrary dimensions. Assuming that perfect channel state information (CSI) is available at both the transmitter side and the receiver side, a novel low-complexity precoding algorithm is proposed to recursively construct the full-rank, the rank-deficient and the rank-one precoders with respect to particular channel realization. From the lattice theoretical perspective, the full-rank precoder is generated by using well-known dense packing lattices, which can also be recursively involved into the construction of higher-dimensional rank-deficient precoders. Moreover, the optimal solution is clearly figured out for the specific case of rank-one precoder. The closed-form expression of the achievable minimum distance is obtained and the analytical results reveal the performance bounds for different rank-constraint precoders. Simulation results validate the efficiency of the proposed precoder as compared with the state of the art.

Recursive Construction of Minimum Euclidean Distance-Based Precoder for Arbitrary-Dimensional MIMO Systems

This letter proposes a new MIMO precoding method for dual-stream data transmission based on the criterion of minimum Euclidean distance. The problem is optimally solved by expanding an ellipse within a 2-dimensional lattice to the maximum extent. A closed-form expression of minimum Euclidean distance is derived for square quadrature amplitude modulation (QAM) constellations. The ellipse expanding method (EEM) allows for very efficient implementation by checking a small lookup table with respect to the condition number of channel matrix. Illustrative examples validate the promising performance of the proposed method in comparison with existing schemes.

Ellipse Expanding Method for Minimum Distance-Based MIMO Precoder

This correspondence deals with the construction of minimum Euclidean distance precoders for multiple-input multiple-output (MIMO) systems with up to four transmit antennas. By making use of a state-of-the-art technique for optimization over the unitary group, we can numerically optimize the MIMO precoders. The correspondence then proceeds by identifying the obtained precoders as well-known lattices (square Z2, Schlafli D4, D6, Gosset E8). With three transmit antennas, the results are slightly different compared with other numbers of transmit antennas since the obtained precoder is not an instance of the densest 6-dimensional lattice. The overall conclusions of the correspondence are that the found precoders for MIMO transmission are highly structured and that, even with small constellations, lattice theory can be used for the design of MIMO precoders.

Construction of Minimum Euclidean Distance MIMO Precoders and Their Lattice Classifications

This paper proposes an efficient sub-optimal MIMO linear precoder based on the maximization of minimum distance for three virtual suchannels. A new virtual MIMO channel representation with two channel angles allows the parameterization of the linear precoder and the optimization of the distance between signal points at the received constellation. To illustrate the optimization process, a precoder is derived for BPSK and QPSK modulation following the max-SNR approach, which consists in pouring power only on the most favored virtual sub-channel. Simulation results over a Rayleigh channel confirm the interest of this new precoder in terms of bit-error-rate.

O. Berder

Papers

3-D Minimum Euclidean Distance Based Sub-Optimal Precoder for MIMO Spatial Multiplexing Systems