# An Enhanced Precoder for Multi User Multiple-Input Multiple-Output Downlink Systems
## 2 Multi User MIMO Downlink System Model  
This paper considers a multi user MIMO downlink system model with \(N_{T}\) transmit antennas at base station and \(N_{R}\) receive antennas, as shown in Figure 1. In this system, the \(i\)-th UE has \(N_{R_{i}}\) receive antennas and \(N_{R}=\sum\limits_{j=1}^{K}N_{R_{j}}\) is the total number of receive antennas, where \(N_{R}\leq N_{T}\). The \(r\) data streams are transmitted from the base station to \(K\) separate UEs via complex Rayleigh flat fading channel. The BS precodes a data signal based on CSI to weaken IUI and transmits the precodedsignal. Each UE receives a signal with mitigated IUI and postcodes the received signal to eliminate residual IUI.  
In this system model, the received signal vector \(\mathbf{Y}=\left[\begin{array}{cccc}\mathbf{y}_{1}^{T}&\mathbf{y}_{2}^{T}& \cdots&\mathbf{y}_{K}^{T}\end{array}\right]^{T}\in\mathbb{C}^{r\times 1}\) is represented as follows,  
\[\mathbf{Y}=\mathbf{G}\left(\mathbf{H}\mathbf{\tilde{X}}+\mathbf{W}\right), \tag{1}\]  
where \(\mathbf{\tilde{X}}\in\mathbb{C}^{N_{T}\times 1}\) is transmitted signal vector expressed by \(\mathbf{\tilde{X}}=\mathbf{F}\mathbf{X}\) and \(\mathbf{W}=\left[\begin{array}{cccc}\mathbf{w}_{1}^{T}&\mathbf{w}_{2}^{T}& \cdots&\mathbf{w}_{K}^{T}\end{array}\right]^{T}\in\mathbb{C}^{N_{R}\times 1}\) is additive white Gaussian noise (AWGN) vector with independent and identically distributed (i.i.d) components of variance \(\sigma_{W}^{2}\) and zero mean. \(\mathbf{X}=\left[\begin{array}{cccc}\mathbf{x}_{1}^{T}&\mathbf{x}_{2}^{T}& \cdots&\mathbf{x}_{K}^{T}\end{array}\right]^{T}\in\mathbb{C}^{r\times 1}\) and \(\mathbf{F}=\left[\begin{array}{cccc}\mathbf{F}_{1}^{T}&\mathbf{F}_{2}^{T}& \cdots&\mathbf{F}_{K}^{T}\end{array}\right]^{T}\in\mathbb{C}^{N_{T}\times r}\) are data signal vector which is uniformly distributed with \(E\left\{\mathbf{x}_{i}\mathbf{x}_{i}^{H}\right\}=\mathbf{I}_{r_{i}}\) and precoding matrix. \(r_{i}\) is the number of data streams for the \(i\)-th UE and \(r\) is defined as follows,  
\[r=\sum_{j=1}^{K}r_{j}. \tag{2}\]  
Also, it is assumed that \(E\left[\left\|\mathbf{\tilde{X}}\right\|^{2}\right]=P_{T}\) and \(\mathbf{W}\) is independent of transmitted signal \(\mathbf{X}\). The joint channel matrix is represented as \(\mathbf{H}=\left[\begin{array}{cccc}\mathbf{H}_{1}^{T}&\mathbf{H}_{2}^{T}& \cdots&\mathbf{H}_{K}^{T}\end{array}\right]^{T}\in\mathbb{C}^{N_{R}\times N_{ T}}\). The decoding matrix \(\mathbf{G}\in\mathbb{C}^{r\times N_{R}}\) which is a diagonal matrix of postcoding matrices for all UEs is defined as follows,  
\[\mathbf{G}=\left[\begin{array}{cccc}\mathbf{G}_{1}&\mathbf{0}&\cdots& \mathbf{0}\\ \mathbf{0}&\mathbf{G}_{2}&\cdots&\mathbf{0}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}&\mathbf{0}&\cdots&\mathbf{G}_{K}\end{array}\right]. \tag{3}\]  
Then, the received signal of the \(i\)-th UE \(\mathbf{y}_{i}\) can be written as follows,  
\[\mathbf{y}_{i}=\mathbf{G}_{i}\left\{\underbrace{\mathbf{H}_{i}\mathbf{F}_{i} \mathbf{x}_{i}}_{\text{desiredsignal}}+\underbrace{\mathbf{H}_{i}\sum_{k=1, k\
eq i}^{K}\mathbf{F}_{k}\mathbf{x}_{k}}_{\text{IUSignal}}+\mathbf{w}_{i}\right\}, \tag{4}\]  
where \(\mathbf{G}_{i}\in\mathbb{C}^{r_{i}\times N_{R_{i}}}\), \(\mathbf{H}_{i}\in\mathbb{C}^{N_{R_{i}}\times N_{T}}\), \(\mathbf{F}_{i}\in\mathbb{C}^{N_{T}\times r_{i}}\), \(\mathbf{x}_{i}\in\mathbb{C}^{r_{i}\times 1}\) and \(\mathbf{w}_{i}\in\mathbb{C}^{N_{R_{i}}\times 1}\) are decoding matrix, Rayleigh flat fading channel matrix, precoding matrix, data signal vector and AWGN vector for the \(i\)-th UE respectively. The desired signal is interfered with by the IUI signal. Also, the equivalent single user (SU) model of the \(i\)-th UE is represented in Figure 2.