## 3 Proposed Partial Nulling RBD
In this case, since the number of data streams is \(l_{total}\) less than the number of receive antennas, the number of chosen row vectors not to be nullified is \(l_{total}\). The \(l_{total}\) row vectors which are not nullified are the row vectors with the relatively small power. Since the criterion for choosing the \(l_{total}\) row vectors of the channel is the power of the channel, the number of selected row vectors of the channel for each device can be unfair. However, the proposed scheme can achieve highly performance improvement by using unfair selection of the row vectors for the channel with the low power relatively and minimizing the loss of the channel gain. The \(l_{total}\) row vectors which are not nullified are omitted from the channel matrix \(\bar{\mathbf{H}}_{k}\). The precoder of the proposed scheme is formed like RBD precoder by utilizing \(\bar{\mathbf{H}}_{k}\) which does not contain the row vectors with relatively small power. The precoder of the proposed scheme can be expressed as follows,  
\[\mathbf{P}_{k}=\mathbf{P}_{k}^{\mathbf{a}}\mathbf{P}_{k}^{\mathbf{\beta}}. \tag{32}\]  
The channel matrix \(\bar{\mathbf{H}}_{k}\) can be expressed by using SVD as follows,  
\[\bar{\mathbf{H}}_{k}=\bar{\mathbf{U}}_{k}\bar{\mathbf{\Sigma}}_{k}\bar{ \mathbf{V}}_{k}^{H}, \tag{33}\]  
where \(\bar{\mathbf{U}}_{k}\), the diagonal elements of \(\bar{\mathbf{\Sigma}}_{k}\) and \(\bar{\mathbf{V}}_{k}\) are the set of left singular vectors, the singular values and the set of right singular vectors, respectively. Since \(\bar{\mathbf{H}}_{k}\) has smaller dimension compared to \(\bar{\mathbf{H}}_{k}\) due to excluding the row vectors, SVD operation at (33) has lower computational complexity. The first precoder \(\mathbf{P}_{k}^{\mathbf{a}}\) can be written as follows,  
\[\mathbf{P}_{k}^{\mathbf{a}}=\bar{\mathbf{V}}_{k}\bigg{(}\bar{\mathbf{\Sigma}}_ {k}^{H}\bar{\mathbf{\Sigma}}_{k}+\alpha\mathbf{I}\bigg{)}^{-1/2}. \tag{34}\]  
(34) has low computational complexity due to \(\bar{\mathbf{\Sigma}}_{k}\) which has smaller dimension like \(\bar{\mathbf{H}}_{k}\). The effective channel is expressed as follows,  
\[\mathbf{H}_{k}\mathbf{P}_{k}^{\mathbf{a}}=\bar{\mathbf{U}}_{k}\mathbf{\Sigma} _{k}\Big{[}\mathbf{V}_{k}^{(1)}\bar{\mathbf{V}}_{k}^{(0)}\Big{]}^{H}, \tag{35}\]  
where \(\bar{\mathbf{U}}_{k}\), the diagonal elements of \(\mathbf{\Sigma}_{k}\), \(\bar{\mathbf{V}}_{k}^{(1)}\) and \(\bar{\mathbf{V}}_{k}^{(0)}\) are 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, respectively. The second precoder \(\mathbf{P}_{k}^{\mathbf{\beta}}\) is denoted as follows,  
\[\mathbf{P}_{k}^{\mathbf{\beta}}=\bar{\mathbf{V}}_{k}^{(1)}. \tag{36}\]  
The precoder of the proposed scheme for the \(k\)th device is written as follows,  
\[{{\bf{H^{\prime}}}_{q}}={{\left[{{\bf{h}}_{q,1}^{T}\cdots{\bf{h}}_{q,m_{q}-1}^ {T}{\bf{h}}_{q,m_{q}+1}^{T}\cdots{\bf{h}}_{q,N_{q}}^{T}}\right]}^{T}}, \tag{29}\]0  
Since the row vectors which are not included in \(\bar{\mathbf{H}}_{k}\) are not belonged to the null space of the precoder, IUI which cannot be nullified by precoder is left. In other words, the IUI remains in the channels corresponding to the selected row vectors. The proposed scheme can obtain the diversity gain by remaining IUI. The received signal of the \(k\)th device can be expressed as follows,  
\[{{\bf{H^{\prime}}}_{q}}={{\left[{{\bf{h}}_{q,1}^{T}\cdots{\bf{h}}_{q,m_{q}-1}^ {T}{\bf{h}}_{q,m_{q}+1}^{T}\cdots{\bf{h}}_{q,N_{q}}^{T}}\right]}^{T}}, \tag{29}\]1  
where \(y_{i,j}\) is the received signal at the \(j\)th antenna for the \(i\)th device. A case of \(L_{\text{total}}=N_{R}-1\) is considered for the sake of simplicity. When the \(m_{q}\)th row vector of the channel matrix for the \(q\)th device is excluded from \(\overset{\rightharpoonup}{\mathbf{H}}_{k}\), \(y_{q,m_{q}}\) is the received signal through the channel with remaining IUI.