The decompositions of higher-order tensors can be viewed as generalizations of matrix decompositions to orders higher than two. Figure 1 depicts the parallel factor (PARAFAC) decomposition, also known as the canonical polyadic decomposition (CPD), perhaps the most popular tensor decomposition.

In several signal processing applications for wireless communications, the transmitted and received signals and the communication channel have a multidimensional nature and may exhibit a multilinear algebraic structure. Tensor decompositions have been the subject of numerous works in the past two decades. The key characteristics of signal processing based on tensor decompositions, not covered by matrix-based signal processing, are the following. It does not require the use of training sequences nor the knowledge of channel impulse responses and antenna array responses. Moreover, it does not rely on statistical independence between the transmitted signals. Instead, tensor-based receiver algorithms are usually deterministic and exploit the multilinear algebraic structure of the signals. Tensor-based algorithms act on data blocks (instead of using a sample-by-sample processing approach). They are generally based on a joint detection of the transmitted signals (either from different users/sources or multiple transmit antennas).

The wireless communication channel usually spans several physical “dimensions” such as space, time, frequency, polarization, etc. Numerous works have successfully used tensor decompositions as a mathematical formalism to describe the algebraic structure of the wireless channel.

Tensors have recently been exploited to solve problems in reconfigurable surfaces-assisted communications, including channel estimation, active/passive beamforming design, and feedback control signaling.

Tensor decompositions have proven useful for designing different transmission structures for MIMO-OFDM systems with built-in blind detection. Transmission schemes combining diversity, multiplexing, and spatial reuse of codes have been formulated by explicitly exploiting the multilinear algebraic structure of tensor decompositions at both the transmitter and the receiver. On the one hand, modeling flexible space-time-frequency (STF) transceivers requires more general tensor decomposition structures beyond the well-known PARAFAC model. On the other hand, efficient algorithms are required for channel estimation and blind symbol decoding with guaranteed uniqueness.

In cooperative MIMO relaying systems, channel estimation involves estimating channel matrices at the receiver and/or relay. Starting from this setup and its generalizations (multiple relays, three-hop, and multi-hop cases), tensor decompositions have been largely applied to solve the channel estimation and the joint channel-symbol estimation problems for tensor-coded MIMO relaying systems.

When high-dimensional systems are modeled and optimized, linear methods often perform unsatisfactorily due to their slow convergence and the high number of parameters to estimate, which brings high computational and storage complexities.