# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 2 Tensor System Model for Multi-Domain Communication Systems
### Examples of Practical Systems  
A few examples of domains that may exist in a system include space, time, frequency, users, spreading sequence, channel multipath, and device index. Incorporating any additional domain of transmission in the unified system model can be carried out in an intuitive and comprehensible manner using the tensor framework. For instance, it is shown in [19] that a MIMO Orthogonal Frequency Division Multiplexing (OFDM) system can be represented using order \(2\) input and output tensors where both antenna and sub-carrier domains are considered. Thereby, the effective channel can be represented using an order \(4\) tensor with its domains corresponding to receive antenna, receive sub-carrier, transmit antenna, and transmit sub-carrier. The output is then given using (3) where the Einstein product is taken over the two transmit modes (\(*_{2}\)) between the channel and the input. Such a representation allows us to jointly consider both inter-carrier and inter-antenna interference in a single framework. Similarly, a MIMO Filter Bank multi-carrier (FBMC) system can also be represented using order \(2\) input and output tensors and an order \(4\) channel tensor where the antenna and sub-carrier domains are considered [9]. The system model can further be used for Generalized Frequency Division Multiplexing (GFDM) as well. GFDM is a flexible filtered multi-carrier modulation scheme where out of band radiation of the transmit signal is controlled using pulse shaping filters on each sub-carrier. GFDM uses a block-based transmission scheme where \(N=KM\) data symbols are transmitted over a time-frequency block having \(K\) sub-carriers and \(M\) timeslots called sub-symbols [23]. The traditional OFDM can be seen as a special case of GFDM when \(M=1\) and the modulator matrix is a DFT matrix. GFDM can also be seen as a generalization of single carrier FDM under suitable pulse shaping filter options [23]. Since GFDM is a generalization of other FDM schemes, it is called 'Generalized' FDM. A detailed comparison of GFDM with OFDM and single carrier FDM can be found in [24]. For single antenna transmission, since components in each GFDM transmit block have separate dependence on the sub-carrier and sub-symbol indices, it is more convenient to represent the \(N=KM\) data symbols using an order 2 tensor of size \(K\times M\). Further, in a MIMO GFDM system, the input and output can be represented as order 3 tensors indexed by antenna/stream, sub-carrier, and sub-symbol indices. Consider a MIMO GFDM system where \(S\) independent streams of data are transmitting \(KM\) data symbols each. The transmit data symbols can be arranged in an order 3 tensor \(\mathcal{D}\in\mathbb{C}^{S\times K\times M}\) where \(\mathcal{D}_{s,k,m}\) corresponds to the complex symbol on the \(k\)th sub-carrier, \(m\)th sub-symbol, and \(s\)th transmit stream. Similarly the received signal and noise tensors can be represented using third order tensors \(\tilde{\mathcal{D}}\in\mathbb{C}^{S\times K\times M}\) and \(\mathcal{N}\in\mathbb{C}^{S\times K\times M}\). Subsequently, the channel that couples the input  
Figure 1: The tensor system model and its evolution with the increase in order. with the output \(\tilde{\mathcal{D}}\) can be seen as an order 6 tensor \(\mathcal{H}_{E}\in\mathbb{C}^{S\times K\times M\times S\times K\times M}\). Let \(N_{T}\) and \(N_{R}\) denote the number of transmit and receive antennas, respectively. The tensor channel considered here is the equivalent channel obtained from the cascading of the transmit filter tensor \(\mathcal{H}_{T}\in\mathbb{C}^{N_{T}\times N\times S\times K\times M}\), physical channel \(\mathcal{H}_{C}\in\mathbb{C}^{N_{R}\times N\times N_{T}\times N}\), and the receive filter tensor \(\mathcal{H}_{R}\in\mathbb{C}^{S\times K\times M\times N_{R}\times N}\), i.e., \(\mathcal{H}_{E}=(\mathcal{H}_{*}*_{2}\mathcal{H}_{C}*_{2}\mathcal{H}_{T})\in \mathbb{C}^{S\times K\times M\times S\times K\times M}\).