# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 4 Harnessing the Domain Trade-Off
### Capacity of a Tensor Channel  
The MIMO matrix channel can be decomposed into parallel non-interfering scalar channels using matrix eigenvalue decomposition (EVD). The optimum input power allocation is done by the classical water-filling solution applied to the decomposed parallel channels to achieve capacity [43]. The unitary matrix obtained from the channel EVD is used to prescribe a transmit precoding. However, these power allocation and precoding operations are restricted to the transformation of transmit vectors in the antenna domain only. The tensor framework extends these operations to multiple domains. For the system model defined in (3), assuming \(\mathcal{X}\) and \(\mathcal{N}\) are zero mean independent and the channel tensor \(\mathcal{H}\) is known, the covariance tensor of the output \(\mathcal{Y}\) denoted as \(\mathcal{C}_{\mathcal{Y}}\in\mathbb{C}^{I_{1}\times\cdots\times I_{M}\times I _{1}\times\cdots\times I_{M}}\) can be written as:  
\[\mathcal{C}_{\mathcal{Y}}=\mathbb{E}[\mathcal{Y}\circ\mathcal{Y}^{*}]= \mathcal{H}*_{N}\mathcal{C}_{\mathcal{X}}*_{N}\mathcal{H}^{H}+\mathcal{C}_{ \mathcal{N}} \tag{21}\]  
where \(\mathcal{C}_{\mathcal{X}}\) and \(\mathcal{C}_{\mathcal{N}}\) are the input and noise covariance tensors, respectively. Assuming the noise tensor contains circular symmetric zero mean complex Gaussian entries with covariance tensor as identity, i.e., \(\mathcal{C}_{\mathcal{N}}=\mathcal{I}\), it can be shown (using ([25] Lemma 1)) that the mutual information between input and output tensors satisfies:  
\[I(\mathcal{X};\mathcal{Y})\leq\log\det\left(\mathcal{H}*_{N}\mathcal{C}_{ \mathcal{X}}*_{N}\mathcal{H}^{H}+\mathcal{I}\right) \tag{22}\]  
where equality is achieved only if \(\mathcal{Y}\) is Gaussian. Here we consider \(\log\) to the base 2, and the \(\det(. )\) operation is the tensor determinant of the enclosed entity, defined as the product of the tensor eigenvalues [16]. For a fixed channel, it can be shown that in the presence of zero mean circular symmetric Gaussian noise, the output \(\mathcal{Y}\) from (3) will also be zero mean circular symmetric complex Gaussian if \(\mathcal{X}\) is so. Hence, for maximizing the mutual information, we consider \(\mathcal{X}\) to be circular complex Gaussian with covariance \(\mathcal{C}_{\mathcal{X}}\). The capacity of the tensor channel can be then found through the following convex optimization problem:  
\[ \begin{array}{ll}\max\limits_{\mathcal{C}_{\mathcal{X}}}&\log\det\Bigl{(} \mathcal{H}*_{N}\mathcal{C}_{\mathcal{X}}*_{N}\mathcal{H}^{H}+\mathcal{I} \Bigr{)}\\ s.t.&\operatorname{tr}(\mathcal{C}_{\mathcal{X}})\leq P,\qquad\mathcal{C}_{ \mathcal{X}}\succeq 0,\end{array}  \tag{23}\]  
where \(\operatorname{tr}(\mathcal{C}_{\mathcal{X}})\) denotes the sum of the pseudo-diagonal elements of \(\mathcal{C}_{\mathcal{X}}\). The inequality constraint \(\operatorname{tr}(\mathcal{C}_{\mathcal{X}})\leq P\) represents a sum power constraint on the input, and \(\mathcal{C}_{\mathcal{X}}\succeq 0\) means that the input covariance tensor is positive semi-definite. Such an optimization problem and can be solved using the Karush-Kuhn-Tucker (KKT) conditions resulting in the optimal input covariance and the capacity as:  
\[\mathcal{C}_{\mathcal{X}}^{(opt)}=\mathcal{V}*_{N}\left(\mu^{-1}\mathcal{I}- \mathcal{D}^{-1}\right)^{+}*_{N}\mathcal{V}^{H} \tag{24}\]  
\[C=\sum_{i_{1},\ldots,i_{N}}\left(\log\left(\frac{d_{i_{1},\ldots,i_{N}}}{\mu} \right)\right)^{+} \tag{25}\]  
More details on the derivation of (24) and (25) can be found in [25]. The unitary tensor \(\mathcal{V}\) and the pseudo-diagonal tensor \(\mathcal{D}\) are obtained from the tensor EVD of \((\mathcal{H}^{H}*_{M}\mathcal{H})\), given as \((\mathcal{H}^{H}*_{M}\mathcal{H})=\mathcal{V}*_{N}\mathcal{D}*_{N}\mathcal{V}^ {H}\). Such decomposition can be computed using tensor EVD methods presented in [15; 16; 17]. The eigenvalues of \((\mathcal{H}^{H}*_{M}\mathcal{H})\) which are the pseudo-diagonal elements of \(\mathcal{D}\) are denoted by \(d_{i_{1},\ldots,i_{N}}\).