## 2 System Model and Conventional Precoding Schemes
### Generalized Block Diagonalization  
When IUI is removed perfectly, all degrees of freedom are consumed by transmit beamforming, GBD [22] does not consume all degrees of freedom to eliminate IUI but acquires the transmit diversity gain by removing IUI imperfectly. Since GBD scheme does not eliminate all IUI but remains intended IUI, extra degrees of freedom that have equal number with decreased data streams are left. The transmit diversity gain is obtained since the data streams are transmitted with the transmit antennas which are more than the data streams. The intended IUI is eliminated at the receiver by using the extra degrees of freedom. When the GBD precoder is used, the transmit antenna resources are exploited not only to eliminate IUI partially but also to acquire the transmit diversity gain. Thus, (3) can be expressed as follows,  
\[\mathbf{\tilde{H}}=\left[ \begin{array}{cccc}\mathbf{H}_{1}\mathbf{P}_{1}& \mathbf{\Delta}_{1,2}&\cdots&\mathbf{\Delta}_{1,k}\\ \mathbf{\Delta}_{2,1}&\mathbf{H}_{2}\mathbf{P}_{2}&\cdots&\mathbf{\Delta}_{2, k}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{\Delta}_{k,1}&\mathbf{\Delta}_{k,2}&\cdots&\mathbf{H}_{K}\mathbf{P}_{ K}\end{array} \right], \tag{12}\]  
where \(\mathbf{\Delta}_{k,j}=\mathbf{H}_{k}\mathbf{P}_{j}\) indicates the remaining IUI that is formed by using partial nulling concept and reducing the number of total data streams. The received signal of the \(k\)th user is expressed as follows,  
\[\mathbf{y}_{k}=\mathbf{H}_{k}\mathbf{P}_{k}^{\text{GBD}}\mathbf{x}_{k}+\sum_{ j=1,j\
otin k}^{K}\mathbf{\Delta}_{k,j}\mathbf{x}_{j}+\mathbf{n}_{k}. \tag{13}\]  
The remaining IUI can be perfectly removed by post-coding at the receiver by spending extra degrees of freedom that are obtained by reducing the number of data streams and using partial nulling. The SVD of the channel matrix for the \(k\)th user is written as follows,  
\[\mathbf{H}_{k}=\mathbf{U}_{k}\mathbf{\Sigma}_{k}\mathbf{V}_{k}^{H}, \tag{14}\]  
where \(\mathbf{U}_{k}\), the diagonal elements of \(\mathbf{\Sigma}_{k}\), \(\mathbf{V}_{k}\), the \(j\)th left singular vector in \(\mathbf{U}_{k}\) and the \(j\)th right singular vector in \(\mathbf{V}_{k}\) are the set of left singular vectors, the singular values, the set of right singular vectors, \(\mathbf{u}_{k,j}\) and \(\mathbf{v}_{k,j}\), respectively. A case of \(L_{k}=N_{k}-1\) is considered for the sake of simplicity. Since the number of data streams is one less than the number of receive antennas, the receiver can spend one degree of freedom to suppress the IUI. To build proper \(\Delta_{k,j}\) that can be suppressed by using one degree of freedom, every column vector in \(\Delta_{k,j}\) needs to be parallel to certain vector \(\mathbf{q}\). The reasonable choice for \(\mathbf{q}\) is the \(N_{k}\)th left singular vector \(\mathbf{u}_{k,N_{k}}\) corresponding to minimum singular value since the intended IUI and the loss of channel gain become minimum. Thus, precoding matrix needs to be formed by using dominant channel components except the \(N_{k}\)th right singular vector \(\mathbf{v}_{k,N_{k}}\) so that all column vectors of \(\Delta_{k,j}\) are parallel to \(\mathbf{u}_{k,N_{k}}\). The general case of \(L_{k}=N_{k}-l_{k}\) is considered, where \(l_{k}\) is the number of reduced data streams for the \(k\)th device. The number of vectors that are parallel to every column vector in \(\Delta_{k,j}\) increases to \(l_{k}\). Thus, parallel vectors are left singular vectors from \(\mathbf{u}_{k,N_{k}-l_{k}+1}\) to the \(\mathbf{u}_{k,N_{k}}\).