Using tensor contractions to derive the structure of slice-wise multiplications of tensors with applications to space–time Khatri–Rao coding for MIMO-OFDM systems

Abstract

The slice-wise multiplication of two tensors is required in a variety of tensor decompositions (including PARAFAC2 and PARATUCK2) and is encountered in many applications, including the analysis of multidimensional biomedical data (EEG, MEG, etc.) or multi-carrier multiple-input multiple-output (MIMO) systems. In this paper, we propose a new tensor representation that is not based on a slice-wise (matrix) description, but can be represented by a double contraction of two tensors. Such a double contraction of two tensors can be efficiently calculated via generalized unfoldings. It leads to new tensor models of the investigated system that do not depend on the chosen unfolding (in contrast to matrix models) and reveal the tensor structure of the data model, such that all possible unfoldings can be seen at the same time. As an example, we apply this new concept to the design of new receivers for multi-carrier MIMO systems in wireless communications. In particular, we consider MIMO-orthogonal frequency division multiplexing (OFDM) systems with and without Khatri–Rao coding. The proposed receivers exploit the channel correlation between adjacent subcarriers, require the same amount of training symbols as traditional OFDM techniques, but have an improved performance in terms of the symbol error rate. Furthermore, we show that the spectral efficiency of the Khatri–Rao-coded MIMO-OFDM can be increased by introducing cross-coding such that the “coding matrix” also contains useful information symbols. Considering this transmission technique, we derive a tensor model and two types of receivers for cross-coded MIMO-OFDM systems using the double contraction of two tensors.