# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 2 Tensor System Model for Multi-Domain Communication Systems
### Examples of Practical Systems
The overall system model is represented using (3) as  
\[\tilde{\mathcal{D}}=\mathcal{H}_{E}*_{3}\mathcal{D}+\mathcal{N} \tag{4}\]  
A detailed derivation of this system model for MIMO GFDM along with characterization of \(\mathcal{H}_{T}\), \(\mathcal{H}_{C}\) and \(\mathcal{H}_{R}\) can be found in [25]. The model in (4) shows the innate ability of the tensor framework from (3) to incorporate additional domains in the system model. In addition to spatial, temporal, and frequency domains, different users can also be considered as a domain in the system model. Consider the uplink of a multi-user MIMO system where a base station (BS) equipped with \(N_{R}\) antennas is receiving information from \(K\) users, each having \(N_{T}\) antennas. Using the standard matrix notation, the received signal vector \(\boldsymbol{y}\in\mathbb{C}^{N_{R}\times 1}\) at the base station can be written as  
\[\boldsymbol{y}=\sum_{k=1}^{K}\boldsymbol{H}^{(k)}\boldsymbol{x}^{(k)}+ \boldsymbol{n} \tag{5}\]  
where \(\boldsymbol{x}^{(k)}\in\mathbb{C}^{N_{T}\times 1}\) is the signal transmitted by the \(k\)th user, \(\boldsymbol{n}\in\mathbb{C}^{N_{R}\times 1}\) is the received noise vector, and \(\boldsymbol{H}^{(k)}\in\mathbb{C}^{N_{R}\times N_{T}}\) is the channel matrix between the \(k\)th user and the BS. Such a system can be equivalently represented in the tensor framework using a third order channel tensor. We define the multi-user MIMO tensor channel as an order 3 tensor \(\mathcal{H}\in\mathbb{C}^{N_{R}\times N_{T}\times K}\) where each \(\boldsymbol{H}^{(k)}\) forms a slice of the third order tensor as \(\mathcal{H}_{:,:,k}=\boldsymbol{H}^{(k)}\). The input signal can be defined as an order 2 tensor \(\boldsymbol{X}\in\mathbb{C}^{N_{T}\times K}\), where each \(\boldsymbol{x}^{(k)}\) forms a column of the matrix \(\boldsymbol{X}\). Hence, the system model in (5) can be equivalently represented as:  
\[\boldsymbol{y}=\mathcal{H}*_{2}\boldsymbol{X}+\boldsymbol{n} \tag{6}\]  
Furthermore, in cellular networks, the notion of domains in the system model can be expanded to include the cell index as well. In such a case, the channel tensor will incorporate terms corresponding to the inter-cell interference also. Consider a \(K\) cell MIMO interfering broadcast channel (IBC) as in [26], where each cell consists of a base station with \(M\) antennas and \(U\) users with \(L\) antennas each. The channel matrix between the \(k\)th base station to the \(u\)th user in the \(i\)th cell is denoted by a matrix \(\boldsymbol{H}^{(k,i,u)}\in\mathbb{C}^{L\times M}\). Let \(\boldsymbol{s}^{(k)}\in\mathbb{C}^{M\times 1}\) be the broadcast transmitted signal vector by the \(k\)th base station intended to be received by all the users within its cell. Then, the received signal vector at the \(u\)th user in the \(i\)th cell is given as [26]:  
\[\boldsymbol{y}^{(i,u)}=\sum_{k=1}^{K}\boldsymbol{H}^{(k,i,u)}\boldsymbol{s}^{ (k)}+\boldsymbol{z}^{(i,u)} \tag{7}\]  
where \(\boldsymbol{z}^{(i,u)}\in\mathbb{C}^{L\times 1}\) denotes additive noise vector for the \(u\)th user in the \(i\)th cell. Note that the summation in (7) includes the desired term (corresponding to \(k=i\)), and also the interference terms received by a user from a base station outside its cell (inter-cell interference corresponding to \(k\
eq i\)). The system model in (7) can be represented using the tensor framework from (3). Consider the transmit signal corresponding to \(K\) base stations, each with \(M\) antennas as an order 2 tensor \(\boldsymbol{S}\in\mathbb{C}^{M\times K}\), where each vector \(\boldsymbol{s}^{(k)}\) forms a column of the matrix \(\boldsymbol{S}\). Similarly, the output and noise received by \(U\) users in each of the \(K\) cells with \(L\) antennas per user can be defined as order 3 tensors \(\mathcal{Y}\in\mathbb{C}^{L\times K\times U}\) and \(\mathcal{Z}\in\mathbb{C}^{L\times K\times U}\), respectively such that \(\mathcal{Y}_{:,i,u}=\boldsymbol{y}^{(i,u)}\) and \(\mathcal{Z}_{:,i,u}=\boldsymbol{z}^{(i,u)}\).