# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 1 Introduction  
As Internet of Things (IoT) gains prominence, the massive increase in the number of devices requiring wireless connectivity along with extremely high data rates present significant challenges for future communication systems and networks, such as 5G and beyond. To meet such competitive objectives, various technologies are being proposed including Large multiple-input multiple-output (MIMO) [1], Millimeter-Wave [2], non-orthogonal multiple-access schemes [3, 4], machine-to-machine communications, and network densification [5]. These trends create the need for communication systems with transceivers that incorporate an eclectic mix of domains. Addressing physical layer issues, such as modulation, coding, waveform selection, equalization, etc., while keeping in mind the gamut of transmission domains available, will be a major challenge. Accounting for the distinct features of such domains in the design process of future communication systems will be crucial. Various multi-antenna and multi-carrier techniques are continuously being researched to improve spectral efficiency and link reliability through space-time-frequency coding methods that exploit diversity in all spatial, temporal, and frequency domains. A detailedsurvey of various multi-carrier candidate waveforms considered for 5G and beyond systems can be found in [6]. Furthermore, spectral efficiency can be increased with multiple access schemes such as Power Domain Non-orthogonal Multiple Access (PD-NOMA) or Sparse Code Multiple Access (SCMA) [7, 8]. Thus, in addition to space, time, and frequency, it could be useful to incorporate users or code sequences as additional domains in the transceiver design process. The associated signal processing and coding required at the transceivers will invariably span more than one domain of communication. Signals that span multiple domains can be mathematically represented using multi-way arrays, more commonly known as tensors [9]. A generic unified mathematical framework that allows the modeling of multi-domain communication systems can be developed using tensors. A traditional method of mathematically representing multiple domains is via concatenation of various domain signals into a vector and treating the associated channel as a matrix, thereby creating a virtual, albeit large, MIMO system model. However, the distinction between the domains is obscured in such a matrix-based degeneration and hence does not allow for easy identification of the mutual coupling that may exist between the various domains. A tensor-based approach maintains the natural multi-domain structure of the system, and the inter-domain interactions are therefore retained and easily identified. Recently, a tensor framework for multi-linear complex MMSE estimation has been developed in [10], which bypasses the need of any restructuring of the tensor signals for estimation purposes. In the past decade, the use of tensors for modeling communication systems has gained much attention for analysis and the improvement of system performance, as it allows a consolidated representation of multiple signaling domains [9, 10, 11, 12, 13, 14]. In this paper, we show that tensors can be used for signals and systems representation, and tools from multi-linear algebra, such as the [PERSON] product, can be employed for developing a generic system model. The framework developed using the Einstein product leads to multi-domain channel modeling and aids in developing simultaneous signal processing techniques across all domains. The [PERSON] product is a form of tensor contracted product [15]. Using the properties of the [PERSON] product, several well-known linear algebra relations can be extended to a multi-linear setting without the requirement of any tensor to vector/matrix transformation. The multi-linear algebra notions developed using the Einstein product preserves the natural tensor structure of the associated quantities and therefore has various applications in engineering disciplines [10, 15, 16, 17, 18].