# 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
#### MVC-MMSE Transmission  
Note that the ZF transmission scheme can only eliminate the intra-cluster interference. As far as a practical UDN scenario is considered, it is often the case that multiple virtual cell clusters coexist. The consequent inter-cluster interference may also affect the system performance prominently. In order to further mitigate the inter-cluster interference, an MVC-MMSE precoding scheme is proposed, where not only intra-cluster CSI, but also inter-cluster CSI are required. This indicates that there are CSI exchange between virtual cell clusters, which contribute to the signaling overhead. However, it is still much milder than the fronthaul compared to the global optimum scheme, where user data exchange is required among clusters. For any user group \(\mathcal{K}_{m}\), the MVC-MMSE precoding matrix \(\mathbf{M}_{\mathcal{K}_{m},\mathcal{B}_{m}}\) can be obtained by solving the following optimization problem:  
\[\mathbf{M}_{\mathcal{K}_{m},\mathcal{B}_{m}}=\text{argmin}_{\{\{\mathbf{G}\}}} \mathbb{E}\bigg{[}\|(\mathbf{H}_{\mathcal{K}_{m},\mathcal{B}_{m}}\mathbf{G} \mathbf{s}_{\mathcal{K}_{m}}+\mathbf{n}_{\mathcal{K}_{m}}-\mathbf{s}_{ \mathcal{K}_{m}}\|^{2}+\sum_{l=1,\underline{J}\
eq l}^{M}\|\mathbf{H}_{ \mathcal{K}_{l},\mathcal{B}_{m}}\mathbf{G}\mathbf{s}_{\mathcal{K}_{m}}\|^{2} \bigg{]} \tag{10}\]  
where \(\mathbf{H}_{\mathcal{K}_{l},\mathcal{B}_{m}}\in\mathbb{C}^{|\mathcal{K}_{l}| \times|\mathcal{B}_{m}|}\) is the downlink channel matrix between the RBAs in the \(m\)-th cluster and the users in the \(l\)-th cluster, \(\mathbf{G}\) is the objective variable, \(\mathbf{s}_{\mathcal{K}_{m}}\in\mathbb{C}^{|\mathcal{K}_{m}|\times 1}\) is the information symbol vector transmitted for the users in the \(m\)-th cluster and \(\mathbf{n}_{\mathcal{K}_{m}}\in\mathbb{C}^{|\mathcal{K}_{m}|\times 1}\) is the AWGN whose entries are independently and identically drawn from \(\mathcal{CN}(0,\sigma^{2})\). The objective function in (10) is to jointly minimize the signal receiving error in the \(m\)-th cluster and the amount of interference it pours into the other clusters. With a constrained total transmission power, the solution of (10) is given as follows. Firstly, simplify the objective function in (10) as:  
\[ \begin{split}\text{J}(\mathbf{G})=&\mathbb{E}\bigg{[} \|(\mathbf{H}_{\mathcal{K}_{m},\mathcal{B}_{m}}\mathbf{G}-\mathbf{I}_{\mathcal{ K}_{m}})|\mathbf{s}_{\mathcal{K}_{m}}\|^{2}+\sum_{l=1,\underline{J}\
eq m}^{M}\| \mathbf{H}_{\mathcal{K}_{l},\mathcal{B}_{m}}\mathbf{G}\mathbf{s}_{\mathcal{K }_{m}}\|^{2}\bigg{]}+\sigma^{2}\left|\mathcal{K}_{m}\right|\\ =&\text{tr}\bigg{\{}\mathbb{E}\Big{[}(\mathbf{H}_{ \mathcal{K}_{m},\mathcal{B}_{m}}\mathbf{G}-\mathbf{I}_{\mathcal{K}_{m}})^{H}( \mathbf{H}_{\mathcal{K}_{m},\mathcal{B}_{m}}\mathbf{G}-\mathbf{I}_{\mathcal{ K}_{m}})+\sum_{l=1,\underline{J}\
eq m}^{M}\mathbf{G}^{H}\mathbf{H}_{ \mathcal{K}_{l},\mathcal{B}_{m}}^{H}\mathbf{H}_{\mathcal{K}_{l},\mathcal{B}_{ m}}\mathbf{G}\Big{]}\bigg{\}}+\sigma^{2}\left|\mathcal{K}_{m}\right|\\ =&\text{tr}\bigg{\{}\mathbf{G}^{H}\Big{(}\mathbf{H}_{ \mathcal{K}_{m},\mathcal{B}_{m}}^{H}\mathbf{H}_{\mathcal{K}_{m},\mathcal{B}_{ m}}+\sum_{l=1,\underline{J}\
eq m}^{M}\mathbf{H}_{\mathcal{K}_{l},\mathcal{B}_{ m}}^{H}\mathbf{H}_{\mathcal{K}_{l},\mathcal{B}_{m}}\Big{)}\mathbf{G}-\mathbf{G}^{H} \mathbf{H}_{\mathcal{K}_{m},\mathcal{B}_{m}}^{H}-\mathbf{H}_{\mathcal{K}_{m}, \mathcal{B}_{m}}\mathbf{G}\bigg{\}}+\sigma^{2}\left|\mathcal{K}_{m}\right| \end{split}  \tag{11}\]  
Then, we can solve the optimization problem in (11) with the Lagrange multiplier method.