# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 4 Harnessing the Domain Trade-Off
### Numerical Examples  
We find the capacity of a tensor channel with i.i.d zero mean and unit variance circular complex Gaussian entries, under total power constraint. We assume that the channel realizations are known at the transmitter and receiver. The results presented are averaged over 100 different channel realizations. Furthermore, we assume additive white Gaussian noise (AWGN) with zero mean. The signal to noise ratio (SNR) is defined as the ratio of the transmit power to the noise variance. Capacity is calculated at 8 dB SNR using (25). Both input and output are order 2 tensors of size \(X\times Y\) each, where \(X\) and \(Y\) denote the dimensions of the individual domains. Thereby the channel is a fourth order tensor of size \(X\times Y\times X\times Y\). Figure 6 presents the capacity of such a fourth order tensor channel against \(X\) and \(Y\). As \(X\) and \(Y\) increase, the capacity of the tensor channel increases. It can be further observed that different configurations of \(X\) and \(Y\) can achieve same capacity, allowing a trade-off between available resources in the two domains. For instance, an input and output tensor of \(12\times 1,1\times 12,6\times 2,2\times 6,4\times 3\) or \(3\times 4\), all achieve the same capacity. Each line curve on the surface plot in Figure 6 represents different configurations of \(X\times Y\) which achieve similar capacity. The design of any multi-domain communication system needs to consider the restrictions on the size of individual domains due to bandwidth or antenna limitations or other domains specific constraints. Figure 6: Capacity vs. dimensions \((X\times Y)\) of order two input and output tensors. The domain trade-off is further apparent in Figure 7 which illustrates the behavior of the capacity under sum-power constraint against SNR. The transmitter employs a vector input, i.e., a single domain of dimension 8 and the output is a tensor of various number of domains. The channel capacity curves show an increase in the capacity achieved when the number of receive domains is increased. It can also be observed that the capacity achieved by an \(8\times 8\) channel is the same as that of the capacity achieved by an \(8\times 2\times 2\times 2\) channel. This implies that even with limited resources in one domain, increasing the number of domains can give higher capacity by exploiting the trade-off between multiple domains. Similar trade-off can be observed between domains in the mean square error (MSE) performance at the receiver when joint signal processing is performed across all the domains using the tensor multi-linear MMSE approach. Figure 8 shows the MSE results for different configurations of input and output. The transmit tensor \(\mathcal{X}\) has i.i.d. components drawn from a 4-QAM constellation with unit energy such that the transmit covariance is a scaled identity tensor \(\sigma_{s}^{2}\mathcal{I}\), with \(\sigma_{s}^{2}=1\). The channel entries are i.i.d zero mean unit variance circular complex Gaussian. Further, the channel is normalized to make the average received signal power the same as the total transmit power \(P=\text{tr}(\sigma_{s}^{2}\mathcal{I})=L_{T}\sigma_{s}^{2}\), where \(L_{T}\) denotes the number of elements in the transmit tensor. We assume the noise tensor has i.i.d. zero mean and variance \(\sigma_{n}^{2}\) circular complex Gaussian entries. Since the channel is normalized to provide unit power gain, the signal to noise ratio is defined as \(SNR=P/(L_{R}\cdot\sigma_{n}^{2})=(L_{T}\cdot\sigma_{s}^{2})/(L_{R}\cdot\sigma_ {n}^{2})\), where \(L_{R}\) denotes the number of elements in the output tensor. A tensor-based multi-linear MMSE receiver from (18) is employed to estimate the transmitted signal. With input \(\mathcal{X}\) and its estimate \(\hat{\mathcal{X}}\), the normalized mean square error is defined as MSE=\(||\mathcal{X}-\hat{\mathcal{X}}||^{2}/L_{T}\). The plotted results are averaged over 100 different channel realizations, with 500 different noise and input realizations for each channel.