# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 3 Simultaneous Signal Processing Across Domains  
Signals and systems representation using tensors leads to the notion of tensor-based multi-domain filtering. The structure of the tensor plays a key role in such operations. A _square_ tensor is an even order tensor of order \(2N\) where the dimensions of the first \(N\) domains are the same as that of the last \(N\) domains. A square tensor of size \(I_{1}\times I_{2}\times\cdots\times I_{N}\times I_{1}\times I_{2}\times\cdots \times I_{N}\), whose elements are indexed by variables \(i_{1},i_{2},\ldots,i_{N},i_{1},i_{2},\ldots,i_{N}\), is called _pseudo-diagonal_ if non-zero elements occur only when \(i_{1}=j_{1},i_{2}=j_{2},\ldots,i_{N}=j_{N}\). Note that this structure is different from a diagonal tensor, where non-zero elements only occur when all the indices are equal. Hence, diagonal elements are a subset of the pseudo-diagonal elements. An illustration of the pseudo-diagonal elements of a fourth order \(3\times 3\times 3\times 3\) tensor is presented in Figure 2. The green blocks, where \(i_{1}=j_{1}\) and \(i_{2}=j_{2}\), are the pseudo-diagonal elements. The diagonal elements, where \(i_{1}=j_{1}=i_{2}=j_{2}\), are represented by green blocks marked with a white X. More details on pseudo-diagonal tensors can be found in [9]. An _identity_ tensor denoted as \(\mathcal{I}\) is a pseudo-diagonal tensor with all non-zero entries being 1. The pseudo-diagonal structure helps us classify TPRS and understand its spectrum and cross-spectrum shaping capabilities, as discussed in the next sub-section. As a precursor to TPRS, we first explain the notion of spectrum and cross-spectrum in tensor framework. Consider a tensor sequence \(\mathcal{X}[k]\in\mathbb{C}_{k}^{l_{1}\times\ldots\cdot l_{N}}\) which is a function tensor with discrete argument \(k\). The \(D\) transform of \(\mathcal{X}[k]\) is a tensor function defined as:  
\[\hat{\mathcal{X}}(D)=\sum_{k}\mathcal{X}[k]D^{k} \tag{9}\]  
with components \(\hat{\mathcal{X}}_{i_{1},\ldots,i_{N}}(D)=\sum_{k}\mathcal{X}_{i_{1},\ldots,i_ {N}}[k]D^{k}\), where \(D\) is the delay operator. A tensor \(\mathcal{X}\in\mathbb{C}^{l_{1}\times\ldots\times l_{N}}\) is said to be _random_ if its components \(\mathcal{X}_{i_{1},\ldots,i_{N}}\) are random variables. Similarly, a tensor sequence \(\mathcal{X}[k]\in\mathbb{C}_{k}^{l_{1}\times\ldots\times l_{N}}\) is called a random tensor sequence if its components are random processes indexed by variable \(k\). The expected value (mean) of such a tensor sequence \(\mathcal{X}[k]\) is defined as \(\mathcal{M}[k]=\mathbb{E}[\mathcal{X}[k]]\) where \(\mathcal{M}_{i_{1},\ldots,i_{N}}[k]=\mathbb{E}[\mathcal{X}_{i_{1},\ldots,i_{ N}}[k]]\). In addition, the auto-correlation of the tensor sequence \(\mathcal{X}[k]\) denoted as \(\mathcal{R}_{\mathcal{X}}[k,i]\in\mathbb{C}_{(k)}^{l_{1}\times\ldots\times l _{N}\times l_{N}\times\ldots\times l_{N}}\) is defined as an order \(2N\) function tensor where each component is a function of two discrete variables \(k\) and \(i\). It can be described using the outer product as:  
\[\mathcal{R}_{\mathcal{X}}[k,i]=\mathbb{E}[\mathcal{X}[k]\circ\mathcal{X}^{*}[ k-i]] \tag{10}\]  
which can be written component wise as:  
\[\mathcal{R}_{\mathcal{X}i_{1},\ldots,i_{N},i_{1}^{\prime},\ldots,i_{N}^{ \prime}}[k,i]=\mathbb{E}[\mathcal{X}_{i_{1},\ldots,i_{N}}[k]\cdot\mathcal{X}_ {i_{1}^{\prime},\ldots,i_{N}^{\prime}}^{*}[k-i]] \tag{11}\]  
where \(()^{*}\) denotes complex conjugate. Subsequently, a tensor sequence is called wide sense stationary (WSS) if its mean \(\mathcal{M}[k]\) is independent of \(k\) and its auto-correlation \(\mathcal{R}_{\mathcal{X}}[k,i]\) depends only on \(i\). The spectrum tensor of a WSS \(\mathcal{X}[k]\in\mathbb{C}_{k}^{l_{1}\times\ldots\cdot l_{N}}\) is an order \(2N\) tensor function \(\mathcal{S}_{\mathcal{X}}(\omega)\in\mathbb{C}_{\omega}^{l_{1}\times\ldots \cdot l_{N}\times l_{1}\times\ldots\cdot l_{N}}\) where each component is a function of the variable \(\omega\).