# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 4 Harnessing the Domain Trade-Off
### Multi-Domain Beamforming (Precoding)
However, in a multi-domainsystem such as MIMO GFDM, corresponding to each antenna we have data symbols over \(K\) sub-carriers and \(M\) sub-symbols. The tensor framework defines the input \(\mathcal{D}\) as a third order tensor, and thereby its covariance as a sixth order tensor \(\mathcal{C}_{\mathcal{D}}=\mathbb{E}[\mathcal{D}\circ\mathcal{D}^{*}]\). Hence, the per antenna power constraints can be defined as a set of constraints on the pseudo-diagonal elements of the covariance:  
\[\sum_{ \begin{subarray}{c}k,m\\ \text{sum over all sub-carriers}\\ \text{and sub-symbols}\\ \text{corresponding to antenna 1}\end{subarray} }\leq P_{1}\quad\text{and}\quad \underbrace{\sum_{ \begin{subarray}{c}k,m\\ \text{sum over all sub-carriers}\\ \text{and sub-symbols}\\ \text{corresponding to antenna 2}\end{subarray} }}_{\text{corresponding to antenna 1}}\leq P_{2}. \tag{27}\]  
Subsequently, finding the optimal covariance with per antenna power constraints to achieve capacity requires finding a positive semi-definite covariance tensor \(\mathcal{C}_{\mathcal{D}}\) subject to (27) such that \(\log\det(\mathcal{H}_{E}\ast_{3}\mathcal{C}_{\mathcal{D}}\ast_{3}\mathcal{H}_ {E}^{H}+\mathcal{I})\) is maximized. Since the constraints from (27) are linear, finding the optimal \(\mathcal{C}_{\mathcal{D}}\) still remains a convex optimization problem that can be solved using the KKT conditions. Such solution yields a multi-domain beamformer with per antenna power constraints. Several beamforming techniques are suggested in [50] for MIMO GFDM systems using matrix SVD. The system model in [50] combines all the domains of transmission to form vectorized signals, and represents the channel using a matrix. Using such a model, ref. [50] suggests a scheme where beamforming is performed after GFDM modulation. The input, which is represented as a vector \(\mathbf{d}\in\mathbb{C}^{N\times 1}\) with \(N=KM\) symbols, is first modulated using a modulator matrix \(\mathbf{G}\in\mathbb{C}^{N\times N}\) and then multiplied by a transmit beamforming matrix \(\mathbf{B}_{T}\in\mathbb{C}^{N_{T}N\times N}\) to generate the transmit vector as \(\mathbf{x}=\mathbf{B}_{T}\mathbf{G}\mathbf{d}\) across all the \(N_{T}\) transmit antennas. The model implicitly assumes that the same GFDM modulator matrix is used for all the antennas. The beamforming matrix \(\mathbf{B}_{T}\) is chosen based on the SVD of the matrix channel \(\mathbf{H}\in\mathbb{C}^{N_{R}N\times N_{T}N}\), where \(\mathbf{H}\) represents the channel between the output of the GFDM modulator and the received signal. Another scheme presented in [50] performs the beamforming before the GFDM modulator, thereby accounting for not just the physical channel but also the modulator matrix. In this scheme, the modulator matrix is defined as a block diagonal matrix \(\mathbf{\tilde{G}}\in\mathbb{C}^{N_{T}N\times N_{T}N}\) where main diagonal blocks are \(\mathbf{G}\). The input is generated using the beamformer matrix as \(\mathbf{x}=\mathbf{\tilde{G}}\mathbf{B}_{T}\mathbf{d}\), where the matrix \(\mathbf{B}_{T}\) is obtained using the SVD of matrix \(\mathbf{H}\mathbf{\tilde{G}}\). In addition, a sub-optimal scheme presented in [50] considers single antenna selection for transmission without any beamforming across the sub-carriers and sub-symbols. The antenna selection is done such that the received SNR is maximized. The transmit beamformer matrix corresponding to the antenna selected is an identity matrix of size \(N\times N\), denoted by \(\mathbf{I}_{N\times N}\), and for every other antenna it is an all zero matrix of size \(N\times N\), denoted by \(\mathbf{0}_{N\times N}\). Hence, the structure of the beamformer matrix is given as \(\mathbf{B}_{T}=[\mathbf{0}_{N\times N},\dots,\mathbf{0}_{N\times N},\mathbf{I}_{N\times N},\bm {0}_{N\times N},\dots,\mathbf{0}_{N\times N}]^{T}\), and the transmit signal is generated as \(\mathbf{x}=\mathbf{\tilde{G}}\mathbf{B}_{T}\mathbf{d}\).