  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 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 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 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 "random 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 randomly coded MIMO-OFDM systems
using the double contraction of two tensors.
