## 2 System Model and Conventional Precoding Schemes  
MU-MIMO broadcasting system which is composed of one base station with \(N_{T}\) transmit antennas and \(K\) users shown in Figure 1 is considered. Rayleigh flat fading channel is assumed and the \(k\)th user has \(N_{k}\) receive antennas. The total number of receive antennas is \(N_{R}=\sum\limits_{k=1}^{K}N_{k}\). The transmit signal vector for the \(k\)th user is \(\mathbf{x}_{k}\in\mathbb{C}^{L_{k}\times 1}\), where \(L_{k}(\leq N_{k})\) is the number of data streams of the \(k\)th user. The total number of data streams is \(L_{\text{total}}=\sum\limits_{k=1}^{K}L_{k}\) and cannot be greater than \(N_{R}\). The received signal for the \(k\)th device can be expressed as follows,  
\[\mathbf{y}_{k}=\underbrace{\mathbf{H}_{k}\mathbf{P}_{k}\mathbf{x}_{k}}_{\text {intended signal}}+\underbrace{\sum\limits_{j=1,j\
otin k}^{K}\mathbf{H}_{k} \mathbf{P}_{j}\mathbf{x}_{j}}_{\text{IUI}}+\mathbf{n}_{k}, \tag{1}\]  
where \(\mathbf{H}_{k}\)\(\in\mathbb{C}^{N_{k}\times N_{T}}\), \(\mathbf{P}_{k}\)\(\in\mathbb{C}^{N_{T}\times L_{k}}\), \(j\) and \(\mathbf{n}_{k}\)\(\in\mathbb{C}^{N_{k}\times 1}\) are the channel matrix of the \(k\)th user, precoding matrix for the \(k\)th user, index of other users except the \(k\)th user and the additive white Gaussian noise vector of the \(k\)th user which has zero mean and variance \(\sigma_{n}^{2}\), respectively.  
The set of entire received signal vectors \(\mathbf{y}\)\(\in\mathbb{C}^{N_{R}\times 1}\) can be expressed as follows,  
\[\mathbf{y} =\left[\begin{array}{cc}\mathbf{y}_{1}^{T}&\cdots&\mathbf{y}_{K}^{T }\end{array}\right]^{T}\] \[=\left[\begin{array}{c}\mathbf{H}_{1}\\ \vdots\\ \mathbf{H}_{K}\end{array}\right]\left[\mathbf{P}_{1}\ \cdots\ \mathbf{P}_{K}\right]\left[ \begin{array}{c}\mathbf{x}_{1}\\ \vdots\\ \mathbf{x}_{K}\end{array}\right]+\left[\begin{array}{c}\mathbf{n}_{1}\\ \vdots\\ \mathbf{n}_{K}\end{array}\right]\, \tag{2}\] \[=\mathbf{\tilde{H}}\left[\begin{array}{c}\mathbf{x}_{1}\\ \vdots\\ \mathbf{x}_{K}\end{array}\right]+\left[\begin{array}{c}\mathbf{n}_{1}\\ \vdots\\ \mathbf{n}_{K}\end{array}\right]\]  
where \(\mathbf{\tilde{H}}\)\(\in\mathbb{C}^{N_{R}\times L}\) is the effective channel. The effective channel \(\mathbf{\tilde{H}}\) is obtained as follows,  
\[\mathbf{\tilde{H}}=\left[\begin{array}{cccc}\mathbf{H}_{1}\mathbf{P}_{1}& \mathbf{H}_{1}\mathbf{P}_{2}&\cdots&\mathbf{H}_{1}\mathbf{P}_{K}\\ \mathbf{H}_{2}\mathbf{P}_{1}&\mathbf{H}_{2}\mathbf{P}_{2}&\cdots&\mathbf{H}_{ 2}\mathbf{P}_{K}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{H}_{K}\mathbf{P}_{1}&\mathbf{H}_{K}\mathbf{P}_{2}&\cdots&\mathbf{H}_{ K}\mathbf{P}_{K}\end{array}\right]. \tag{3}\]  
Diagonal components in the effective channel matrix are MIMO channels for each user and off-diagonal components express the IUI.  
If a unit variance for the noise is assumed, the capacity region can be written as follows,  
\[C=\mathop{\cup}\limits_{P_{1},\cdots,P_{K}\mathbb{s},\Gamma_{k}\sum_{k}P_{l}=P} \left\{\left(R_{1},\cdots,R_{K}\right)\in\Re^{+K},R_{i}\leq\log_{2}\frac{ \det\left[\mathbf{I}+\mathbf{H}_{i}\left(\sum_{j\geq i}Q_{i}\right)\mathbf{H}_ {i}^{H}\right]}{\det\left[\mathbf{I}+\mathbf{H}_{i}\left(\sum_{j\geq i}Q_{i} \right)\mathbf{H}_{i}^{H}\right]}\right\}, \tag{4}\]  
where \(Q_{k}=\mathbb{E}\left(\mathbf{x}_{k}\mathbf{x}_{k}^{H}\right)\) and \(P_{K}=\text{Tr}\left(Q_{k}\right)\) are the covariance of signal vector for the \(k\)th user and the power allocated to the \(k\)th user.