# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 4 Harnessing the Domain Trade-Off
### Numerical Examples
The input has a single domain of dimension 8, and output has a variable number of domains. It can be observed that better MSE performance is achieved as the number of domains at the receiver is increased. When the output has a single domain of dimension 2, the MSE is fairly high as the number of elements in the output are far less than the number of transmit elements. As the number of domains at the receiver increases, we get better MSE performance. Further, the performance of a system with three domains of dimension 2 at the receiver is the same as the performance of a system with a single domain of dimension 8 at the receiver. Hence, performance improvements can be achieved by  
Figure 7: Capacity vs. SNR for different domain configurations at the receiver. addition of domains rather than having to increase the dimensions of individual domains. Such domain trade-off is exploited through the tensor multi-linear MMSE receiver which jointly estimates the signal across all the domains. So far in these examples, we have observed that the individual domain's dimensions can be flexibly interchanged if the overall size of input and output remains constant. However, it is important to note that such a behavior is observed over an average of 100 iterations when all the tensor channel entries are i.i.d. Gaussian, which has been the case in our numerical examples so far. For any two given channel tensors with the same overall size but different dimensions of individual domains, the capacity may not always be exactly same. The exact nature of such a domain trade-off would depend on the specific tensor channel eigenvalues. The example of a channel with i.i.d. Gaussian entries without tagging specific physical meaning to the domains incorporated was put forward only to illustrate the basic idea of trade-off. Further, to understand such trade-off in a more realistic set-up, we consider the example of a MIMO GFDM system from (4), where the channel is represented as an order 6 tensor of size \(5\times K\times M\times S\times K\times M\). The system model follows the representation from (4) in Section 2.2. The model is further illustrated in Figure 9, where order 3 tensors are denoted using cubes and higher order tensors are denoted using double edge squares with the order mentioned on top right corner. Figure 8: MSE vs. SNR for different domain configurations at the receiver. Figure 9: Tensor system model for MIMO GFDM. Let the number of transmit and receive antennas be \(N_{T}\) and \(N_{R}\), respectively, with \(N_{T}=N_{R}=S\). Such an order 6 tensor channel associated with MIMO GFDM is considered in [25], where capacity behavior with respect to the different pulse shaping filter options is explored. Here, we use the simulation set-up from [25] and analyze the capacity for different number of data streams, sub-carriers, and sub-symbols, denoted by \(S,K\), and \(M\), respectively, to explore the trade-off between these domains. The channel is generated as a cascade of transmit filter, physical channel and receive filter. In this example, we use a raised cosine transmit pulse shaping filter with roll off factor 1 at the transmitter. The receive filter is matched to the transmit filter and the entries of the physical channel are generated using i.i.d. complex Gaussian with zero mean and unit variance. Furthermore, the entries of the equivalent channel \(\mathcal{H}\) are normalized to ensure that the average received power is same as the transmit power \(P\). The noise tensor \(\mathcal{N}\) contains zero mean and unit variance circular symmetric Gaussian entries such that noise covariance is \(N_{0}\mathcal{I}\) with \(N_{0}=1\). With channel gain normalized to one, the signal to noise ratio is defined as \(SNR=(N_{T}E_{s})/N_{0}\)[44], where \(E_{s}\) is the transmit energy per element defined as \(P/(N_{T}KM)\). We assume that \(\mathcal{H}\) is known at the receiver and the transmitter.