# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 2 Tensor System Model for Multi-Domain Communication Systems  
Arrays whose elements are indexed by multiple indices are called tensors. The various indices of a tensor are also known as _modes_. The _order_ of a tensor is the number of its modes. Thus, matrices where elements are indexed by 2 indices (row and column modes) can be seen as an order 2 tensor, and vectors where elements are indexed by a single index can be seen as order 1 tensors. A quick summary of basic tensor notations which we will use in this paper are included in Table 1.  
The _[PERSON] product_ of tensors is a form of contracted product where two tensors of orders \(P+N\) and \(N+M\) contract over their \(N\) common modes to generate a tensor of order \(P+M\). For any \(N\), the [PERSON] product is defined using the symbol \(*_{N}\) as [15]:  
\[(\mathcal{A}*_{N}\mathcal{B})_{i_{1},\ldots,i_{p},j_{1},\ldots,j_{M}}=\sum_{k _{1},\ldots,k_{N}}\mathcal{A}_{i_{1},i_{2},\ldots,i_{p},k_{1},\ldots,k_{N}} \mathcal{B}_{k_{1},\ldots,k_{N},i_{1},i_{2},\ldots,j_{M}} \tag{1}\]  
where \(\mathcal{A}\in\mathbb{C}^{I_{1}\times\cdots\times I_{p}\times K_{1}\times \cdots\times K_{N}}\) and \(\mathcal{B}\in\mathbb{C}^{K_{1}\times\cdots\times K_{N}\times I_{1}\times \cdots\times I_{M}}\). For \(\mathcal{X}\in\mathbb{C}^{I_{1}\times I_{2}\times\cdots\times I_{N}}\) and \(\mathcal{Y}\in\mathbb{C}^{I_{1}\times I_{2}\times\cdots\times I_{M}}\), the outer product \(\mathcal{X}\circ\mathcal{Y}\) is an order \(N+M\) tensor defined as:  
\[(\mathcal{X}\circ\mathcal{Y})_{i_{1},\ldots,i_{N},i_{1},\ldots,j_{M}}=\mathcal{ X}_{i_{1},\ldots,i_{N}}\mathcal{Y}_{i_{1},\ldots,i_{M}} \tag{2}\]  
Using properties of the [PERSON] product, many linear algebra concepts such as inversion, identity, hermitian, trace, determinant, rank, and eigenvalue decomposition (EVD) can be extended to a multi-linear setting. A detailed treatment of such tensor algebra results can be found in [9; 15; 16; 17; 19; 20; 21]. The [PERSON] product can be effectively used for representing multi-linear systems of equations, and has been recently employed to develop the notion of multi-linear system theory [18; 22]. In the following section, we use the [PERSON] product to develop a system model for a multi-domain communication system.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Explanation** \\ \hline \(\mathcal{X}\in\mathbb{C}^{I_{1}\times\cdots\times I_{N}}\) & Order \(N\) tensor of size \(I_{1}\times\cdots\times I_{N}\) with complex entries. \\ \(\mathcal{X}\in\mathbb{C}^{I\times J}\) and \(\mathbf{x}\in\mathbb{C}^{N\times 1}\) & Matrix of size \(I\times J\) and vector of size \(N\times 1\), respectively. \\ \(\mathcal{X}_{i_{1},\ldots,i_{N}}\) & Individual entries of tensor \(\mathcal{X}\) denoted using indices in subscript. \\ \(\mathcal{X}_{:,i_{2},\ldots,i_{N}}\) & Colon in subscript indicates all the elements of a mode (first mode in this case) corresponding to fixed other modes (\(i_{2},\ldots,i_{N}\) in this case). \\ \(\mathcal{X}[k]\in\mathbb{C}^{I_{1}\times\cdots\times I_{N}}_{t}\) & Sequence of order \(N\) tensors where each element is a function of the discrete variable \(k\). \\ \(\mathcal{X}(t)\in\mathbb{C}^{I_{1}\times\cdots\times I_{N}}_{t}\) & Function tensor where each element of an order \(N\) tensor is a function of continuous variable \(t\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: Tensor notations used in this paper.