# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 2 Tensor System Model for Multi-Domain Communication Systems
### A Generic System Model  
The modes of a tensor can represent various domains of transmission and reception in a communication system. The input (transmit signal) and output (received signal) of a multi-domain communication system with \(N\) transmit domains and \(M\) receive domains can be defined as order \(N\) and \(M\) tensors, respectively. Let the input be represented by \(\mathcal{X}\in\mathbb{C}^{I_{1}\times\cdots\times I_{N}}\) where the dimension (size) of the \(i\)th input domain is \(I_{i}\). Let the output be represented by \(\mathcal{Y}\in\mathbb{C}^{I_{1}\times\cdots\times I_{M}}\) where the dimension of the \(j\)th output domain is \(j_{j}\). The channel \(\mathcal{H}\), defined as a multi-linear operator spanning all the transmit and receive domains, can be represented using a tensor of size \(j_{1}\times j_{2}\times\cdots\times J_{M}\times I_{1}\times I_{2}\times\cdots \times I_{N}\). Each element in the receive tensor \(\mathcal{Y}\) is a linear combination of all the elements of the transmit tensor \(\mathcal{X}\), where the coefficients of the combination are the elements of the channel tensor \(\mathcal{H}\). Hence, the output \(\mathcal{Y}\) can be written in terms of the Einstein product between the channel and the input as \(\mathcal{H}*_{N}\mathcal{X}\). In the presence of noise, we can represent a generic system model as:  
\[\mathcal{Y}=\mathcal{H}*_{N}\mathcal{X}+\mathcal{N} \tag{3}\]  
where \(\mathcal{N}\in\mathbb{C}^{I_{1}\times\cdots\times I_{M}}\) represents the order \(M\) received noise tensor. The conventional MIMO matrix model can be seen as a specific case of (3) where the input and output are order \(1\) tensors (vectors) and the channel is an order \(2\) tensor (matrix) [10]. In such a case, the Einstein product between the channel and the input by definition reduces to standard matrix multiplication. System models for three different cases (in the absence of noise) are illustrated in Figure 1. Each white block in the figure represents an input/output element, while the yellow blocks represent the channel components. In the first case, the input and output are order \(1\) tensors (vectors) of size \(2\), and the channel is an order \(2\) tensor (matrix) of size \(2\times 2\), as used in conventional MIMO systems. This model can be evolved further to incorporate additional domains as illustrated in the second and third cases. In the second case, the input and output are order \(2\) tensors of size \(2\times 2\) each, while the channel is an order \(4\) tensor of size \(2\times 2\times 2\times 2\). Further, in the third case, the input and output are order \(3\) tensors of size \(2\times 2\times 2\) and the channel is an order \(6\) tensor of size \(2\times 2\times 2\times 2\times 2\times 2\). The input indices are represented using \(i_{1},i_{2},i_{3}\) and the output indices using \(j_{1},j_{2},j_{3}\). The multi-domain nature of the tensor channel and its coupling with the input through the Einstein product in the system model allow us to perceive (3) as an evolution of the MIMO matrix channel model, where the latter is just a special case of the former.