The Tensor Multi-Linear Channel and Its Shannon Capacity

Abstract

Tensors are multi-way arrays which can be used to model systems spanning many domains. This work proposes to use tensors for characterizing, analyzing, and designing multi-domain communication systems. Most modern day communication systems make use of coding and modulation across different domains such as space, frequency, time. Hence a unified mathematical framework characterizing such a multiple-domain system in an intuitive manner is well needed. In this paper, we present such a unified framework that characterizes a communication system with N input domains, M output domains and an M + N domains multi-linear tensor channel. The proposed framework is generic where the physical interpretation of the domains is system specific. We illustrate a few examples from multi-antenna multicarrier and multi-user systems that fit the proposed framework. Assuming a fixed tensor channel, we provide an information theoretic analysis by deriving its Shannon capacity and input power allocation under a variety of power constraints. In this paper we show how the tensor framework's suitability to mathematically describe a family of power constraints can be used to design and analyze various multiple domain communication systems. The tensor based approach extends water-filling from a matrix setting to tensors, encapsulating the effects of multiple domains thereby allowing joint multi-domain precoding. We show that the capacity pre-log for a tensor channel increases exponentially in the number of domains, indicating the potential of tensor based multi-domain communication systems to provide the large information transmission rates envisaged for 5G and beyond systems. We also show the application of the tensor framework in characterizing the capacity and rate regions of multi-user MIMO channels. Both multiple access and interference channels are considered where the tensor based approach leads to a coordinated users transmission scheme. Such a scheme ensures higher achievable sum rates as compared to independent user transmissions.INDEX TERMS Tensors, MIMO channels, multi-user MIMO, Shannon capacity, tensor SVD and EVD.

Capacity Performance of Tensor Multi-Domain Communication Systems With Discrete Signalling Constellations

“…This paper uses several tensor algebra results such as the Einstein product, outer product, tensor inverse, Hermitian. A detailed review of all such relevant tensor algebra results can be found in [27], [43].…”

“…The received signal is modelled as a fourth order tensor for MIMO OFDM in [24]. Similar application of tensors for signal representation in several multi-domain systems such as Filter bank-based Multi-carrier (FBMC) [25], cellular networks [26], MIMO multiple access and interference channels [27], Nonorthogonal Multiple Access (NOMA) systems [28], have also been investigated.…”

“…The tensor model retains the domain interpretability and their mutual effects, thereby provides a structured mathematical framework to deal with such multidomain systems. In [27] a discrete version of the tensor modelled multi-domain communication system from [1] is presented using the Einstein product. Further, [27] considers the Shannon capacity of higher order tensor channels under a family of power constraints with unconstrained input signals.…”

“…In [27] a discrete version of the tensor modelled multi-domain communication system from [1] is presented using the Einstein product. Further, [27] considers the Shannon capacity of higher order tensor channels under a family of power constraints with unconstrained input signals. However, in practical systems, input is often constrained to be drawn from discrete constellations.…”

“…Also, per antenna constraints are common in distributed MIMO systems where antennas do not share the same power source [40]. In a multi-domain system, power constraints can span beyond the antenna domain, such as per user power constraints in a multi-user MIMO system [27]. In order to mathematically represent such domain specific power constraints, conventional matrix approaches fall short since any lower order representation of the inherently higher order channel loses the domain distinction.…”