# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 3 Simultaneous Signal Processing Across Domains
### Tensor-Based Receiver Designs  
Tensor equalization has been considered in [9] as a technique of reducing interference in multi-domain communication systems. In particular, [9] presents multi-linear minimum mean square error (MMSE) and zero forcing equalizers, while [31] also presents tensor-based decision feedback equalizers. Such techniques can be used to design receivers which simultaneously combat the effects of interference from different domains (Multi-Domain Interference) such as inter-carrier interference (ICI), inter-symbol interference (ISI), inter-antenna interference (IAI), and inter-user interference (IUI). An application of such a tensor-based equalizer in a multi-user MIMO GFDM receiver is considered in [9], where it is shown that such tensor-based equalizers outperform the per-domain equalizers which do not exploit the information on interfering terms from other domains.  
\begin{table}
\begin{tabular}{c c c} \hline
**Component** & **Figure 3** & **Figure 4** \\ \hline \(\mathcal{F}_{1,1,1,1}\) & \(1\) & \(1+D\) \\ \(\mathcal{F}_{1,1,1,2}\) & \(0.5\) & \(0\) \\ \(\mathcal{F}_{1,1,2,1}\) & \(0.25\) & \(0.5(1+D)\) \\ \(\mathcal{F}_{1,1,2,2}\) & \(0.125\) & \(0\) \\ \(\mathcal{F}_{1,2,1,1}\) & \(0.5\) & \(0.5(1-D)\) \\ \(\mathcal{F}_{1,1,2,2}\) & \(1\) & \((1-D)\) \\ \(\mathcal{F}_{1,2,2,1}\) & \(0.125\) & \(0.125\) \\ \(\mathcal{F}_{1,2,2,2}\) & \(0.25\) & \(0.25\) \\ \(\mathcal{F}_{2,1,1,2}\) & \(0.125\) & \(0.125\) \\ \(\mathcal{F}_{2,1,2,1}\) & \(1\) & \((1-D^{2})\) \\ \(\mathcal{F}_{2,1,2,2}\) & \(0.5\) & \(0.5(1-D^{2})\) \\ \(\mathcal{F}_{2,2,1,1}\) & \(0.125\) & \(0\) \\ \(\mathcal{F}_{2,2,1,2}\) & \(0.25\) & \(0.5(1-D^{4})\) \\ \(\mathcal{F}_{2,2,2,1}\) & \(0.5\) & \(0\) \\ \(\mathcal{F}_{2,2,2,2}\) & \(1\) & \((1-D^{4})\) \\ \hline \end{tabular}
\end{table}
Table 2: Structure of the TPRS system.  
Figure 4: TPRS with inter- and intra-tensor interference.  
The tensor framework can also be used to develop estimation techniques for receiver design in communication systems. In (3), if the channel tensor is known, a multi-linear receiver which operates on the received signal tensor \(\mathcal{Y}\) producing \(\hat{\mathcal{X}}=\mathcal{G}*_{M}\mathcal{Y}\), can be used to provide an estimate \(\hat{\mathcal{X}}\) of the transmitted tensor \(\mathcal{X}\). We can find the tensor \(\mathcal{G}\in\mathbb{C}_{I}^{I_{1}\times\cdots\times I_{N}\times I_{1}\times \cdots\times I_{M}}\) such that the mean square error between \(\mathcal{X}\) and \(\hat{\mathcal{X}}\), defined as  
\[\text{MSE}=\mathbb{E}[||\mathcal{X}-\hat{\mathcal{X}}||^{2}]=\mathbb{E}[|| \mathcal{X}-\mathcal{G}*_{M}\mathcal{Y}||^{2}] \tag{17}\]  
is minimized. The notation \(||.||\) denotes Frobenius norm. This leads to a tensor multi-linear MMSE receiver which is specified by:  
\[\mathcal{G}=\mathcal{C}_{\mathcal{X}}*_{N}\mathcal{H}^{H}*_{M}(\mathcal{H}*_{ N}\mathcal{C}_{\mathcal{X}}*_{N}\mathcal{H}^{H}+\mathcal{C}_{\mathcal{N}})^{-1} \tag{18}\]  
where \(\mathcal{C}_{\mathcal{X}}=\mathbb{E}[\mathcal{X}\circ\mathcal{X}^{*}]\) and \(\mathcal{C}_{\mathcal{N}}=\mathbb{E}[\mathcal{N}\circ\mathcal{N}^{*}]\) are order \(2N\) and \(2M\) covariance tensors of input and noise, respectively (assuming that \(\mathcal{X}\) and \(\mathcal{N}\) are zero mean). A detailed derivation of (18) can be found in [19]. An application of such a tensor-based receiver in a MIMO OFDM system is considered in [10] where a comparison with per sub-carrier estimation which ignores the inter-carrier interference terms for estimation is presented. It is shown that as interference from other sub-carriers becomes dominant, the performance of per sub-carrier receiver deteriorates significantly, while the tensor receiver's is significantly better since it makes use of the interference terms for data estimation.