## 4 Computational Complexity Analysis  
The GBD and MGBD need to eliminate IUI by multiplication of the post-coder. Therefore, the complexities for GBD and MGBD at the receiver are exactly same. However, since the proposed scheme uses simpler way not to use post-coder for removing IUI compared to GBD and MGBD, the proposed scheme has low complexity at the receiver. Therefore, even if the proposed scheme has to remove IUI at the receiver, the complexity at the receiver of the proposed scheme is same as the RBD. Since the operations at the receiver are more burden than the operations at the base station, the lower complexity at the receiver is great advantage. Since the complexity of the proposed scheme at the receiver is same as RBD, the comparison between the complexities at the base station of the proposed scheme and RBD is considered. RBD and the proposed scheme require two SVD operations to decompose the channel matrix. However, three SVD operations are required at the MGBD. The one more SVD at the MGBD is utilized for making post-coder. Therefore, since the complexity of the MGBD at the transmitter is extremely high, the complexity of the MGBD at the transmitter is also calculated and compared with the proposed scheme. The computational complexity of the SVD grows with the dimension of the channel matrix which is decomposed. Since the dimension of the channel matrix which is decomposed by using SVD at the proposed scheme is decreased by excluding the row vectors, the proposed scheme has low complexity at the transmitter compared to RBD. The numbers of flops of conventional schemes and proposed scheme are calculated to measure computational complexity according to [19]. A multiply followed by addition of the real number needs 2 flops. A multiply followed by addition with complex number needs 8 flops. The complexities of each operation for each scheme are shown in Tables 1-4, where \(\bar{N}_{k}=N_{R}-N_{k}\) and \(\bar{L}_{k}=L_{\text{total}}-L_{k}\) are considered. The complexities of the conventional schemes and the proposed scheme are derived deterministically. The numbers of flops of conventional schemes and the proposed scheme are calculated to measure computational complexity according to [19]. For example, the complexity of the SVD operation such as (7) and (9) in Table 2 is \(32\left(nm^{2}+2m^{3}\right)\) at [19] when the complexity of SVD for \(m\times n\) complex matrix is calculated. Therefore, since (7) is the SVD of \(\bar{N}_{k}\times N_{T}\) complex matrix, the complexity of (7) for each user is \(32\left(N_{T}\bar{N}_{k}^{2}+2\bar{N}_{k}^{3}\right)\). (9) is the SVD of \(N_{k}\times N_{T}\) complex matrix. Therefore, the complexity of (9) for each user is \(32\left(N_{T}N_{k}^{2}+2N_{k}^{3}\right)\)In (8), the complexity of \(\left(\mathbf{\Sigma}_{k}^{H}\mathbf{\Sigma}_{k}+\alpha\mathbf{I}\right)^{-1/2}\) is \(K\left(18N_{T}+\bar{N}_{i}\right)\) in [19]. Since the dimensions of \(\mathbf{\bar{V}}_{k}\) and \(\left(\mathbf{\hat{\Sigma}}_{k}^{H}\mathbf{\hat{\Sigma}}_{k}+\alpha\mathbf{I} \right)^{-1/2}\) are \(N_{T}\times N_{T}\), the complexity of multiplication between \(\mathbf{\bar{V}}_{k}\) and \(\left(\mathbf{\hat{\Sigma}}_{k}^{H}\mathbf{\hat{\Sigma}}_{k}+\alpha\mathbf{I} \right)^{-1/2}\) is \(8 KN_{T}^{3}\). Therefore, the total complexity of (8) can be expressed as \(\sum\limits_{k=1}^{K}\left(18N_{T}+\bar{N}_{k}+8N_{T}^{3}\right)\). The complexities at the receiver are shown in Table 1 when \(\mathbf{R}_{k}\) and \(\mathbf{y}_{k}\) are the post-coder of the \(k\)th user and the received signal of the \(k\)th user, respectively. GBD and MGBD need extra operation in (19) for multiplying the post-coder at the receiver. In contrast, since the RBD precoder eliminates IUI perfectly at the transmitter, extra operation at the receiver to remove remaining IUI is no needed. The proposed scheme also does not need extra multiplications to remove remaining IUI by excluding the received signal corresponding to remaining IUI.