PhD defence of Divyanshu Pandey - Information theoretic aspects of tensor based multi-domain communication systems
Most modern communication systems employ several domains for transmission and reception such as space, time, frequency, users, code sequences, and transmission media. Thus, the signals and systems involved in information transfer have an inherent multi-domain structure which can be well represented using tensors. A tensor is a multi-way array which can be seen as a higher order generalization of vectors or matrices. A unified mathematical framework capable of intuitively modelling multi-domain communication systems can be developed with the help of tensors. The use of tensors to characterize, analyze, and build multi-domain communication systems is proposed in this thesis. A generic system model is defined in this work for multi-domain communication systems with N input domains and M output domains. The multi-linear channel between such higher order input and output signals is defined as an order M+N tensor, which couples the input and output through the Einstein product. The suggested framework is generic, where the physical interpretations of the domains can vary depending on the specific system being modelled.
An information theoretic analysis of multi-domain communication systems is considered by deriving the Shannon capacity and input power allocation for a fixed higher order tensor channel under a family of power constraints. Owing to the multi-domain nature of the input signals, the power constraints in multi-domain communication systems can span one or more domains. This thesis demonstrates the tensor framework's ability to mathematically represent a variety of such power constraints. Shannon capacity of tensor channels under such family of power constraints is derived. Water-filling is extended from a matrix setting to higher domains in such a tensor-based formulation, encapsulating the impact of various domains and allowing collaborative multi-domain precoding and power allocation. It is also shown that as the number of domains increases, the multiplexing gain for a tensor channel can increase exponentially, indicating the ability of the tensor-based communication systems to offer the enormous information transmission rates required for beyond 5G systems. In addition, this thesis illustrates how the tensor framework can be used to characterize the capacity and rate regions of multi-user MIMO channels. The tensor-based technique leads to a coordinated users transmission scheme. The tensor framework treats the multi-domain interference terms as information bearing entities, and thus ensures higher achievable sum rates as compared to the independent users transmissions.
Further, the Einstein Product of tensors is used to develop a framework for minimum mean square error (MMSE) estimation for multi-domain signals and data. Both proper and improper complex tensors are addressed by the framework. The traditional linear and widely linear MMSE estimators are extended to the tensor setting, resulting in multi-linear and widely multi-linear MMSE estimation. Further, a relation between the MMSE error covariance tensor and the gradient of the mutual information is extended from a vector setting to tensors, known as the tensor I-MMSE relation. Furthermore, the tensor I-MMSE relation is used to find the capacity of tensor channels when the input is drawn from arbitrary distributions. In the presence of circularly symmetric Gaussian noise and under no constraint on the input constellation, an input drawn from a circularly symmetric Gaussian distribution achieves the channel capacity. However, under practical scenarios, the input is often drawn from discrete signalling constellations which are far from Gaussian distributed. By making use of the tensor I-MMSE relation, an iterative precoder is developed in this thesis which achieves capacity of the tensor channels when the input is limited by the choice of signalling constellations.