## 3 Proposed Partial Nulling RBD  
Figure 2 is the flowchart of the proposed scheme. The proposed scheme is based on the RBD precoder and applies partial nulling concept at the same time. Thus, when all users cannot get diversity gain, the proposed scheme has the same precoder as RBD. The proposed scheme adopts partial nulling concept by selecting the row of the channel matrix which is not nullified depending on the power of the channel. The proposed scheme does not remove IUI corresponding to the rows of the channel matrix which have relatively low power to remain the transmit antenna resources. The number of the rows which are selected not to nullified is same as the number of decreased data streams. Since the number of the data streams is reduced and is less than the number of transmit antennas, the transmit diversity gain is achieved. The performance of the proposed scheme is better than RBD due to the transmit diversity gain. The proposed scheme also can achieve better performance compared to GBD and MGBD when the number of data streams is slightly reduced. The remaining IUI is removed by consuming the extra degrees of freedom at the receiver. However, the way to remove IUI in the proposed scheme is simpler than the way that uses post-coder at the user. Since the extra multiplication at the receiver is not needed by using simpler way in the proposed scheme, the proposed scheme has same complexity as RBD even if the proposed scheme has to remove IUI at the receiver. In other words, the complexity of the proposed scheme at the receiver is lower than GBD and MGBD which have to multiply post-coder to remove IUI at the receiver. A case of \(L_{\text{total}}=N_{R}-1\) is considered for the sake of simplicity. When the channel from the \(i\)th transmit antenna to the \(j\)th receive antenna is \(h_{j,i}\), the power of the channel to the \(j\)th receive antenna is expressed as follows,  
\[P_{j}=\sum\limits_{i=1}^{N_{T}}{{{\left|{h_{j,i}}\right|}^{2}}}. \tag{28}\]  
The reasonable choice for row that is not nullified is the row with the smallest power to minimize the loss of channel gain. When the index of receive antenna with the smallest power of the channel is defined as \(m\), the \(m\)th row of the channel matrix with the smallest power \(P_{m}\) is chosen as the row that is not nullified. The \(m\)th receive antenna corresponding to the \(q\)th device is considered. When \(m_{q}=m-\sum\limits_{k=1}^{q-1}{N_{k}}\), the \(m_{q}\)th row of the channel matrix for the \(q\)th device is not nullified. In other words, the number of data streams for the \(q\)th device is decreased to \(N_{q}-1\). The channel matrix of the \(q\)th device except the \(m_{q}\)th row can be defined as follows,  
\[{{\bf{H^{\prime}}}_{q}}={{\left[{{\bf{h}}_{q,1}^{T}\cdots{\bf{h}}_{q,m_{q}-1}^ {T}{\bf{h}}_{q,m_{q}+1}^{T}\cdots{\bf{h}}_{q,N_{q}}^{T}}\right]}^{T}}, \tag{29}\]  
where \({\bf{h}}_{i,j}\) is the \(j\)th row vector of channel matrix for the \(i\)th device. When \(k=q\), the channel matrix \(\bar{\mathbf{H}}_{k}\) which is composed of the channel matrix of all devices except the \(k\)th device is denoted as follows,  
\[\bar{\mathbf{H}}_{k}=\Big{[}\mathbf{H}_{1}^{T}\cdots\mathbf{H}_{k-1}^{T} \mathbf{H}_{k+1}^{T}\cdots\mathbf{H}_{K}^{T}\Big{]}^{T}. \tag{30}\]  
When \(k\
eq q\), the row vector with the smallest power is excluded from the channel matrix \(\bar{\mathbf{H}}_{k}\). Thus, the channel matrix \(\bar{\mathbf{H}}_{k}\) is denoted as follows,  
Figure 2: Flowchart of the proposed scheme. \[\bar{\mathbf{H}}_{k}=\left[\mathbf{H}_{1}^{T}\cdots\mathbf{H}^{\prime T}_{q}\cdots \mathbf{H}_{k-1}^{T}\mathbf{H}_{k+1}^{T}\cdots\mathbf{H}_{k}^{T}\right]^{T}. \tag{31}\]  
Since the number of data streams is one less than the number of receive antennas, the number of chosen row vector to be excluded is one. A case of \(L_{\text{total}}=N_{R}-l_{total}\) is considered where \(l_{total}\left(<N_{R}\right)\) is the total number of reduced data streams.