Nested Tucker tensor decomposition with application to MIMO relay systems using tensor space–time coding (TSTC)

• We present a new tensor model called a Tucker train decomposition or a nested Tucker decomposition (NTD).
• Assuming that the core tensors are known, we propose two algorithms for estimating the matrix factors of a NTD(4).
• We show that a NTD(4) allows to model a one-way two-hop MIMO relay system with TSTC at the source and the relay.
• We derive two semi-blind receivers allowing to jointly estimate the information symbols and the two individual relay channels.
• We demonstrate the effectiveness of the proposed receivers by means of Monte Carlo simulations.

The aim of this paper is twofold. In a first part, we present a new tensor decomposition that we call Tucker train decomposition or nested Tucker decomposition (NTD). NTD can be viewed as a particular case of tensor-train decomposition recently proposed for representing and approximating high-dimensional tensors in a compact way. NTD of a fourth-order tensor is more specially analysed in terms of parameter estimation and uniqueness issue. In a second part, we show that the use of a tensor space–time coding (TSTC) structure at both the source node and the relay node of a one-way two-hop multi-input multi-output (MIMO) relay communication system leads to a nested Tucker decomposition of the fourth-order tensor formed by the signals received at the destination. Two semi-blind receivers are then proposed for jointly estimating the transmitted information symbols and the two individual relay channels. The first one is iterative, based on a three-step alternating least squares (ALS) algorithm, whereas the second one, denoted 2LSKP, is a closed-form solution based on the LS estimations of two Kronecker products. Two supervised receivers are also derived by using a (short) pilot-assisted closed-form solution for calculating channel estimates. These estimates are exploited either for initializing the ALS receiver or for designing a zero-forcing (ZF) receiver. Extensive Monte Carlo simulation results are provided to demonstrate the performance of the proposed relay system.