# On the Virtual Cell Transmission in Ultra Dense Networks
## 3 Transmission Schemes Based on Virtual Cell Merging
### Virtual Cell Merging with ZF and MVC-MMSE Transmission  
Let \(\mathbf{w}_{k,\mathcal{B}_{m}}\) denote any precoding vector of any user \(k\in\mathcal{K}_{m}\), i.e., \(\mathbf{w}_{k,\mathcal{B}_{m}}\) may be any one of \(\mathbf{w}_{k,\mathcal{B}_{m}}^{\text{ZF}}\) and \(\mathbf{w}_{k,\mathcal{B}_{m}}^{\text{MVC-MMSE}}\). Five different transmission schemes are simulated in this paper, including MRT, ZF with virtual cell merging based on virtual cell overlapping (ZF-VCMO) [11], ZF with virtual cell merging based on the interference graph (ZF-VCMG), MVC-MMSE with virtual cell merging based on virtual cell overlapping (MVC-MMSE-VCMO) and MVC-MMSE with virtual cell merging based on the interference graph (MVC-MMSE-VCMG). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Network layout & Uniform \\ Number of BSs & 10,000/1000 \\ Number of UEs & 50 \\ Carrier frequency & 2 GHz \\ Bandwidth & 20 MHz \\ Path loss factor & 4 \\ Transmitting SNR & \(P/\sigma^{2}=10\) dB \\ Number of UE antennas & 1 for ZF and MVC-MMSE transmission and 2 for IA transmission \\ Channel model & Rayleigh channel \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. MVC-MMSE, multi-virtual-cell minimum mean square error; IA, interference alignment. Figure 4 compares the average user spectral efficiency \(\overline{R}\) of MRT, ZF-VCMO and MVC-MMSE-VCMO at different initial virtual cell sizes. It can be seen clearly that different from the no virtual cell merging case (MRT) where the highest \(\overline{R}\) is achieved when the initial virtual cell size \(N_{0}\) is small, the ZF-VCMO scheme can markedly improve \(\overline{R}\) as the increase of \(N_{0}\). When \(N_{0}\) is large enough, \(\overline{R}\) becomes saturated. This is because by including more RBAs into each user's initial virtual cell, more users would overlap with each other. It is inevitable that all of the users and RBAs would be grouped together at some \(N_{0}\), after which no more increase of \(\overline{R}\) is observed. We can also draw the conclusion from Figure 4 that MVC-MMSE-VCMO can significantly improve \(\overline{R}\) at the same \(N_{0}\) compared with ZF-VCMO. As a result, MVC-MMSE-VCMO can reach a targeted \(\overline{R}\) at a much smaller \(N_{0}\), hence at a much smaller merged virtual cell cluster size, although inter-cluster CSI exchange is required. This implies that MVC-MMSE-VCMO can significantly reduce the signal processing complexity, which scales with the sizes of the merged virtual cell clusters, at the price of heavier signaling overhead. Figure 5 compares \(\overline{R}\) of MRT, ZF-VCMG and MVC-MMSE-VCMG at different relative interference graph thresholds \(\varphi\), which is defined as \(\varphi=\left(g_{th}-\min\gamma_{k,j}\right)/\left(\max\gamma_{k,j}-\min\gamma _{k,j}\right)\). Intuitively, \(\varphi\) represents the connection degree of \(\mathcal{G}_{b}\). By tuning \(\varphi\), we can easily control the size of merged virtual cell clusters. For ZF-VCMG and MVC-MMSE-VCMG, the initial virtual cell size \(N_{0}\) is set to be two; that is to say, every user primarily selects two nearest RBAs to form its initial virtual cell before virtual cell merging is performed. It can be seen clearly that ZF-VCMG can achieve significantly higher \(\overline{R}\) than MRT at different \(\varphi\). When \(\varphi\) is large enough, \(\overline{R}\) also becomes saturated. This is because when \(\mathcal{G}_{b}\) is dense enough, there would be only one MCC, resulting in a single merged virtual cell cluster, containing all of the initial virtual cells. Different from the case in Figure 4, MVC-MMSE-VCMG only shows a marginal performance increase compared to ZF-VCMG. This implies that the virtual cell merging based on the interference graph can effectively avoid major interference in UDN without inter-cluster CSI exchange.