# 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 the normalization, we can get:  
\[\mathbf{w}_{3}^{i}=\frac{\text{eigvec}(\mathbf{H}_{2,3}^{-1}\mathbf{H}_{2,1} \mathbf{H}_{3,1}^{-1}\mathbf{H}_{3,2}\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3})}{ \left\|\mathbf{H}_{2,3}^{-1}\mathbf{H}_{2,1}\mathbf{H}_{3,1}^{-1}\mathbf{H}_{ 3,2}\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,2}\right\|} \tag{26}\]  
\[\mathbf{w}_{1}^{i}=\frac{\text{eigvec}(\mathbf{H}_{2,1}^{-1}\mathbf{H}_{2,3} \mathbf{w}_{3}^{i})}{\left\|\mathbf{H}_{2,1}^{-1}\mathbf{H}_{2,3}\mathbf{w}_{ 3}^{i}\right\|} \tag{27}\]  
\[\mathbf{w}_{2}^{i}=\frac{\text{eigvec}(\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3} \mathbf{w}_{3}^{i})}{\left\|\mathbf{H}_{1,2}^{-1}\mathbf{H}_{1,3}\mathbf{w}_{ 3}^{i}\right\|} \tag{28}\]  
After determining the transmit beamforming matrices of \(\mathbf{W}_{1}\), \(\mathbf{W}_{2}\) and \(\mathbf{W}_{3}\), we can design the receive beamforming matrices \(\mathbf{U}_{k}\) for all users in order to cancel out the interference signals from the other virtual cells. With the decision variables \(\mathbf{U}_{k}\), an optimization problem can be formulated as minimizing the mean square error as:  
Figure 3:0  
We can get the normalized \making the network ultra dense) RBAs uniformly and independently distributed across \(\mathcal{A}\). Here, we take the value of \(B=1000\) and \(K=50\) directly from [11] for the sake of comparison. However, it is pointed out in [7] that the number of BSs in the UDN is likely to be significantly more than the number of User equipments (UEs). Hence, we make an even more aggressive assumption that \(B=10{,}000\) in this part to catch a glimpse of the performance of the proposed schemes in a network that is really \"ultra dense\". Then, the network density can be described as the ratio of the number of RBAs and the number of users \(\rho=\frac{B}{K}\). The path loss between any user \(k\in\mathcal{K}\) and any RBA \(b\in\mathcal{B}\) is modeled as \(\lambda_{k,b}=d_{k,b}^{\mu/2}\), where \(d_{k,b}\) is the distance between user \(k\) and RBA \(b\), the minimum of which is set to be 0.1; \(\mu=-4\) is the path-loss exponent factor. In fact, large-scale fading is equal to the product of shadowing fading and path loss. If shadowing fading is considered in this system model, it may be equivalent to the distance in path loss calculation being different. In this paper, we used Monte Carlo simulation in the simulation process; the number of simulations is quite large, including almost all kinds of distances; so here, we can ignore the shadowing fading. This paper aims to handle the intricate interference in UDN, so an interference limited scenario is considered, where \(P/\sigma^{2}\) is set to 10 dB. The simulation parameters are listed in Figure 3:1  
We focus on the performance of average user spectral efficiency, which can be formulated as \(\overline{R}=\frac{1}{K}\sum_{k\in\mathcal{K}}R_{k}\). We now demonstrate the advantages of the proposed transmission schemes based on different virtual cell merging schemes, in terms of average user spectral efficiency and the corresponding signal processing embodied by the average cluster size.