# On the Virtual Cell Transmission in Ultra Dense Networks
## 3 Transmission Schemes Based on Virtual Cell Merging
### Virtual Cell Merging with IA Transmission
#### 3.2.2 IA Transmission  
After virtual cell merging, we consider IA within every cluster to eliminate strong interference. We first consider IA with one cluster consisting of three virtual cells that have the strongest interference strength with each other. Figure 3 illustrates the procedure used for designing the transmit beamforming matrices for the clusters. As shown in Figure 3, the interference can be aligned on a lower dimensional subspace at each receiver, then the desired signals can be transmitted on the interference-free dimensions by IA. We assume that perfect channel state information is available to both the transmitter and receiver. There are three virtual cells (i.e., V1, V2 and V3), and the corresponding users are user1, user2 and user3 in the cluster. Assume that each user is equipped with \(D\) antennas, \(D=N_{0}\). Consider the interference from V1, V2 to user3; the following conditions are obtained:  
\[\text{span}(\mathbf{H}_{3,1}\mathbf{W}_{1})=\text{span}(\mathbf{H}_{3,2} \mathbf{W}_{2}) \tag{20}\]  
where \(\text{span}(\cdot)\) denotes the subspace spanned by the column vectors of a matrix, \(\mathbf{H}_{i,j}\in\mathbb{C}^{N_{0}\times D}\) is the channel matrix from the \(j^{th}\) RBA to the \(i^{th}\) user, which are all equipped with multi-antennas, \(\mathbf{W}_{1}\) and  
Figure 3: Interference alignment algorithm with one cluster. Dashed line arrows denote the interference from Virtual Cell 3 (V3) to V1 and V2. Dotted dashed line arrows denote the interference from V2 to V1 and V3. Solid line arrows denote the interference from V1 to V2 and V3. At the two-dimensional coordinate system, the interference from other virtual cells can be aligned into a lower dimensional subspace at each receiver. \(\mathbf{W}_{3}\) denote the transmit beamforming matrix for V1 and V3, respectively. What is more, the first \(\mathbf{h}_{k,\mathcal{B}_{m}}\) in the system model is a sector, because users are equipped with a single antenna. Multi-antenna users can be easily obtained from the model extended from the single-antenna user, where the single-antenna users can be seen as multiple streams of multi-antenna users. Therefore, in the IA transmission, we use \(\mathbf{H}_{i,j}\) as a matrix, where users and RBAs are all equipped with multi-antennas. Then, consider the interference to user1 and user2 from other two virtual cells as:  
\[\text{span}(\mathbf{H}_{2,1}\mathbf{W}_{1})=\text{span}(\mathbf{H}_{2,3} \mathbf{W}_{3}) \tag{21}\]  
\[\text{span}(\mathbf{H}_{1,3}\mathbf{W}_{3})=\text{span}(\mathbf{H}_{1,2} \mathbf{W}_{2}) \tag{22}\]  
From (21) and (22), we can get:  
\[\text{span}(\mathbf{W}_{1})=\text{span}(\mathbf{H}_{2,1}^{-1}\mathbf{H}_{2,3} \mathbf{W}_{3}) \tag{23}\]  
\[\text{span}(\mathbf{W}_{2})=\text{span}(\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3} \mathbf{W}_{3}) \tag{24}\]  
Using (20), (23) and (24), we can obtain the transmit beamforming matrices of \(\mathbf{W}_{3}\) as follows:  
\[\mathbf{w}_{3}^{i}=\text{eigvec}(\mathbf{H}_{2,3}^{-1}\mathbf{H}_{2,1} \mathbf{H}_{3,1}^{-1}\mathbf{H}_{3,2}\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3}) \tag{25}\]  
where \(i=\{1,2,\ldots,N_{0}/2\}\) denotes different data streams. Then, \(\mathbf{W}_{3}\) can be expressed as arbitrary \(N_{0}/2\) eigenvectors of matrix \(\mathbf{H}_{2,3}^{-1}\mathbf{H}_{2,1}\mathbf{H}_{3,1}^{-1}\mathbf{H}_{3,2} \mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3}\).