GF(q)-based precoding: information theoretical analysis and performance evaluation

Abstract

In this paper, a new precoding scheme that is based on the operations in Galois field of size q = 2m(GF(q)) is proposed. Generally, precoding is a processing technique at transmitters to match the input signal to the channel in order to achieve optimal channel capacity through fully utilizing space, time, and frequency diversity. Precoding schemes can be divided into two main categories: linear precoding and nonlinear precoding. It has been shown from an information theoretical aspect that both the linear and nonlinear precoding schemes can achieve the optimal channel capacity. Our proposed GF(q)‐based precoding scheme is a nonlinear precoding technique, and the idea originates from finite inputs of the modulated symbols. When representing the modulated symbols and the elements in precoding matrix with the finite elements in Galois field of size q, and applying the operations defined in GF(q), we can obtain the precoded symbols that contains information of the original symbols. Starting from binary symmetry channel to additive Gaussian white noise channels, we have demonstrated that the proposed GF(q)‐based precoding schemes can enhance the system mutual information when the original finite inputs are not uniformly distributed. In addition, inspired by the mutual information analysis in binary symmetry channel, we investigated the selection of the precoding matrix in GF(q)‐based precoding schemes. As mutual information indicates the information about the source carried by the symbols of the channel output, greater mutual information enables the receivers to recover more information about the original source. To further utilize the greater mutual information brought by the proposed GF(q)‐based precoding schemes, we proposed a novel‐receiving structure by exchanging soft information between the GF(q) decoding block and channel decoding block. Simulation results show that the proposed iterative receiver improves the system bit error rate performance by 1 and 2 dB at the bit error rate level of 10 − 6 with binary phase shift keying and quadrature phase shift keying modulations, respectively. Inspired by the encouraging results of greater mutual information and better bit error rate performance, we are convinced that the proposed GF(q)‐based precoding schemes can be extended to fading channels and multiple input–multiple output systems to further approach channel capacity. Copyright © 2016 John Wiley & Sons, Ltd.