# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 3 Simultaneous Signal Processing Across Domains
### Tensor Partial Response Signaling  
Correlative coding, otherwise known as Partial Response Signaling, is a transmission method where correlation is introduced between successive symbols through the addition of a controlled amount of inter-symbol interference [29]. The objective is to shape the spectrum of the signal using correlative codes to achieve desirable properties. Frequency domain correlative coding has been also applied in MIMO OFDM systems to mitigate the effects of time-selective fading [30]. In such a system, the correlation introduced is restricted to one domain. Using the tensor framework, it is possible to extend this concept to a multi-domain system and control both the spectrum as well as the cross-spectrum of the transmitted signal tensor. Such a method, dubbed Tensor Partial Response Signaling (TPRS), makes use of tensor correlative codes to introduce a controlled amount of inter-tensor interference (between successive tensor symbols) and intra-tensor interference (between the domains of one tensor symbol). Consider a WSS discrete sequence of zero mean uncorrelated data tensors \(\mathcal{D}[m]\in\mathbb{C}_{m}^{I_{1}\times\cdots\times I_{N}}\) with the corresponding \(\mathcal{S}_{\mathcal{D}}(D)\) as an identity tensor of order \(2N\). To introduce correlation and alter the input spectrum tensor, we pass \(\mathcal{D}[m]\) through a linear discrete time invariant system \(\mathcal{F}[m]\in\mathbb{C}_{m}^{I_{1}\times\ldots I_{N}\times I_{1}\times \ldots I_{N}}\). Hence, the output of the system, denoted by \(\mathcal{X}[k]\in\mathbb{C}_{k}^{I_{1}\times\cdots\times I_{N}}\) is given using discrete contracted convolution between \(\mathcal{F}[m]\) and \(\mathcal{D}[m]\), defined as [9]:  
\[\mathcal{X}[k]=\sum_{m}\mathcal{F}[k-m]*_{N}\mathcal{D}[m] \tag{14}\]  
which in the \(D\) transform domain can be written as:  
\[\mathcal{\hat{X}}(D)=\mathcal{\hat{F}}(D)*_{N}\mathcal{\hat{D}}(D) \tag{15}\]  
The objective of \(\mathcal{\hat{F}}(D)\) is to introduce correlation in the transmitted data tensor; hence, we refer to \(\mathcal{\hat{F}}(D)\) as a TPRS system. Since the data tensor sequence is assumed WSS, we can relate the tensor \(\mathcal{\hat{S}}_{\mathcal{X}}(D)\) to \(\mathcal{\hat{F}}(D)\) by:  
\[\mathcal{\hat{S}}_{\mathcal{X}}(D)=\mathcal{\hat{F}}(D)*_{N}\mathcal{\hat{S}} _{\mathcal{D}}(D)*_{N}\mathcal{\hat{F}}^{H}(D^{-1}) \tag{16}\]  
A detailed derivation of (16) can be found in [31]. The relation in (16) when evaluated at \(D=e^{i\omega T}\) shows that the spectrum tensor of the transmit data tensor can be manipulated to alter both the spectrum and the cross spectrum among all the components by suitably changing the structure of the TPRS system \(\mathcal{\hat{F}}(D)\). The data streams in most digital communication systems consist of independent and identically distributed (i.i.d) symbols drawn from a specific constellation, implying that such sequences have a flat spectrum. For a tensor system with data tensors whose components are i.i.d, all components of the spectrum are the same. The cross-spectrum is zero as the individual components are not correlated, making \(\mathcal{\hat{S}}_{\mathcal{D}}(D)\) a multiple of identity tensor. In this case, (16) reduces to \(\mathcal{\hat{S}}_{\mathcal{X}}(D)=\mathcal{\hat{F}}(D)*_{N}\mathcal{\hat{F}} ^{H}(D^{-1})\). Hence, by using a TPRS system, it is possible to shape the spectrum and cross-spectrum of the different components of the data tensor by employing distinct codes. This means that the components of the transmitted signal tensor can have different spectral shapes even though the original data tensor has components that are i.i.d. Depending on how the interference is introduced, TPRS systems can be broadly classified into different classes. Controlled interference from within the same tensor symbol (intra-tensor interference) changes the level of the spectrum and cross-spectrum while maintaining a flat spectral power density.