A Tensor Based Framework for Multi-Domain Communication Systems

The demand for mobile data is likely to grow at a pace more than envisaged in the coming years. Further, as applications such as the Internet of Things (IoT) come to fruition, there will be increased diversity in the types of devices demanding Internet connectivity and their requirements. Significant increase in data rate requirements is also expected due to services such as Ultra High Definition (UHD) video streaming and cloud computing. To meet all these demands, physical layer waveform candidates for future generations of communications need to be robust and inherently capable of extending into multiple domains (space, time, frequency, users, transmission media, code etc.) to ensure efficient utilization of resources. Multiple domains can be innately integrated into the design process of modulation schemes by using tensors, which are multi-way arrays. This paper introduces a unified tensor framework, providing a foundation for multi-domain communication systems that can be used to represent, design and analyse schemes that span several domains. Transmitted signals are represented by <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>th order time function tensors which are coupled, using a system tensor of order <inline-formula> <tex-math notation="LaTeX">$N+M$ </tex-math></inline-formula>, with the received signals which are represented by another tensor of order <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> through the contracted convolution. We begin with the continuous time representation of the tensor system model and present both the strict multi-domain generalization of the Nyquist criterion for zero interference (inter-tensor and intra-tensor interference) as well as a relaxation. We present an equivalent discrete time system model, and as an example of using the tensor framework we derive tensor based linear equalization methods to combat multi-domain interference. An application to multi-user MIMO-GFDM illustrates the utility of this novel framework for derivation of joint domain signal processing techniques.