## 2 System Model and Conventional Precoding Schemes
### Regularized Block Diagonalization  
RBD [10; 20; 21] suppresses IUI perfectly by using null space of the channels of any other users like BD and the null space is obtained from SVD operation. Specifically, the SVD of an \(m\times n\) real or complex matrix \(\mathbf{M}\) is a factorization of the form \(\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{H}\), where \(\mathbf{U}\) is an \(m\times m\) real or complex unitary matrix, \(\mathbf{\Sigma}\) is an \(m\times n\) rectangular diagonal matrix with non-negative real numbers on the diagonal, and \(\mathbf{V}\) is an \(n\times n\) real or complex unitary matrix. The exact calculation method and examples of SVD are shown on pages 367 to 370 of [24]. RBD scheme considers the noise term in contrast with BD. Therefore, the performance of the RBD scheme is better than the BD precoding when the noise is the dominant factor.  
RBD precoder is expressed as follows,  
\[\mathbf{P}_{k}^{\text{RBD}}=\mathbf{P}_{k}^{\mu}\mathbf{p}_{k}^{b}. \tag{5}\]  
To form the precoding matrix of the \(k\)th user, the channel matrix \(\mathbf{\bar{H}}_{k}\)\(\in\)\(\mathbb{C}^{\bar{N}_{k}\times N_{T}}\) which is composed of the channel matrices of all users except the \(k\)th user is denoted as follows,  
\[\mathbf{\bar{H}}_{k}=\left[\mathbf{H}_{1}^{T}\ \cdots\ \mathbf{H}_{k-1}^{T}\ \mathbf{H}_{k+1}^{T}\ \cdots\ \mathbf{H}_{K}^{T}\right]^{T}, \tag{6}\]  
where \(\bar{N}_{k}\) is the number of receive antennas of all devices except the \(k\)th device. The channel matrix \(\mathbf{\bar{H}}_{k}\) is expressed by using SVD as follows,  
\[\mathbf{\bar{H}}_{k}=\mathbf{\bar{U}}_{k}\mathbf{\bar{\Sigma}}_{k}\mathbf{ \bar{V}}_{k}^{H}, \tag{7}\]  
where \(\mathbf{\bar{U}}_{k}\in\mathbb{C}^{\bar{N}_{k}\times\bar{N}_{k}}\), the diagonal elements of \(\mathbf{\bar{\Sigma}}_{k}\)\(\in\)\(\mathbb{C}^{\bar{N}_{k}\times N_{T}}\), \(\mathbf{\bar{V}}_{k}\)\(\in\)\(\mathbb{C}^{N_{T}\times N_{T}}\) and \(\left(\bullet\right)^{H}\) are the set of left singular vectors, the singular values, the set of right singular vectors and hermitian operator, respectively. The first precoder \(\mathbf{P}_{k}^{a}\) can be expressed as follows,  
\[\mathbf{P}_{k}^{a}=\mathbf{\bar{V}}_{k}\Big{(}\mathbf{\bar{\Sigma}}_{k}^{H} \mathbf{\bar{\Sigma}}_{k}+\alpha\mathbf{I}\Big{)}^{-1/2}, \tag{8}\]  
Figure 1: System model of the multi-user multiple input multiple output (MU-MIMO) downlink channel.  
where \(\alpha\) and \(\mathbf{I}\) are the ratio of the total noise power to the total transmit power, i.e., \(\alpha=N_{R}\sigma_{k}^{2}/P_{total}\) and the identity matrix, respectively. The SVD of the effective channel for the \(k\)th user is obtained as follows,  
\[\mathbf{H}_{k}\mathbf{P}_{k}^{a}=\mathbf{\tilde{U}}_{k}\mathbf{\tilde{\Sigma}} _{k}\Big{[}\mathbf{\tilde{V}}_{k}^{(1)}\mathbf{\tilde{V}}_{k}^{(0)}\Big{]}^{H}, \tag{9}\]  
where \(\mathbf{\tilde{U}}_{k}\), the diagonal elements of \(\mathbf{\tilde{\Sigma}}_{k}\), \(\mathbf{\tilde{V}}_{k}^{(1)}\)\(\in\mathbb{C}^{N_{T}\times N_{k}}\) and \(\mathbf{\tilde{V}}_{k}^{(0)}\)\(\in\mathbb{C}^{N_{T}\times(N_{T}-N_{k})}\) indicate the set of left singular vectors, the singular values, the set of right singular vectors corresponding to non-zero singular values and the set of right singular vectors corresponding to zero singular values. The second precoder \(\mathbf{P}_{k}^{b}\) is obtained as follows,  
\[\mathbf{P}_{k}^{b}=\mathbf{\tilde{V}}_{k}^{(1)}. \tag{10}\]  
Finally, the precoding matrix of the RBD for the \(k\)th user can be written as follows,  
\[\mathbf{P}_{k}^{\text{RBD}}=\mathbf{\tilde{V}}_{k}\Big{(}\mathbf{\tilde{ \Sigma}}_{k}^{H}\mathbf{\tilde{\Sigma}}_{k}+\alpha\mathbf{I}\Big{)}^{-1/2} \mathbf{\tilde{V}}_{k}^{(1)}. \tag{11}\]  
RBD scheme eliminates IUI perfectly. The performance of the RBD is better than the BD precoding by taking noise term into account. However, the transmit diversity gain cannot be obtained since all transmit antenna resources are used to make precoder that eliminates channel components of other users.