# On the Virtual Cell Transmission in Ultra Dense Networks
## 2 System Model
Denoting the transmitted signal vector form the merged virtual cell cluster \(m\) to its user \(k\) as \(\mathbf{x}_{k,\mathcal{B}_{m}}\in\mathbb{C}^{N_{m}\times 1}\), the received signal vector at user \(k\) can then be modeled as:  
\[\begin{split} y_{k}&=\sum_{l=1}^{M}\sum_{j\in \mathcal{K}_{l}}\sqrt{p_{j}}\mathbf{h}_{k,\mathcal{B}_{l}}\mathbf{x}_{j, \mathcal{B}_{l}}+n_{k}\\ &=\sqrt{p_{k}}\mathbf{h}_{k,\mathcal{B}_{m}}\mathbf{x}_{k, \mathcal{B}_{m}}+\sum_{j\in\mathcal{K}_{m,j}\
eq k}\sqrt{p_{j}}\mathbf{h}_{k, \mathcal{B}_{m}}\mathbf{x}_where \(n_{k}\sim\mathcal{CN}(0,\sigma^{2})\) is the additive white Gaussian noise (AWGN) observed by user \(k\), and \(p_{j}\) is the transmitting power of user \(j\), which is assumed to be the same for any different users in this paper, i.e., \(p_{j}\equiv P,\forall j\in\mathcal{K}\). The first term on the right-hand side of the second equal sign in (2) is the useful signal received by user \(k\), while the second term and the third term denote the intra-cluster interference caused by other users in the \(m\)-th merged virtual cell cluster and inter-cluster interference caused by users in other merged virtual cell clusters, respectively. Without loss of generality, we assume the transmitted signal vector \(\mathbf{x}_{k,\mathcal{B}_{m}}\) to be the precoded symbols of user \(k\), which can be written as:  
\[\mathbf{x}_{k,\mathcal{B}_{m}}=\mathbf{w}_{k,\mathcal{B}_{m}}s_{k} \tag{3}\]  
where \(\mathbf{w}_{k,\mathcal{B}_{m}}\in\mathbb{C}^{N_{m}\times 1}\) is the precoding vector of user \(k\), and \(s_{k}\) is the information symbol of user \(k\) with zero mean and unit variance. Information symbols are assumed to be independent and uncorrelated between different users. To meet the equal total transmitting power constraint of each user, we have \(\mathbf{w}_{k,\mathcal{B}_{m}}^{H}\mathbf{w}_{k,\mathcal{B}_{m}}=1\). Substituting (3) into (2), we can write the received useful signal power and interference power as (4) and (5), respectively. \[S_{k}=P*\mathbf{h}_{k,\mathcal{B}_{m}}\mathbf{w}_{k,\mathcal{B}_{m}}\mathbf{w }_{k,\mathcal{B}_{m}}^{H}\mathbf{h}_{k,\mathcal{B}_{m}}^{H} \tag{4}\]  
\[I_{k}=\sum_{j\in\mathcal{K}_{m},j\
eq k}P*\mathbf{h}_{k,\mathcal{B}_{m}} \mathbf{w}_{j,\mathcal{B}_{m}}\mathbf{w}_{j,\mathcal{B}_{m}}^{H}\mathbf{h}_{k, \mathcal{B}_{m}}^{H}+\sum_{l=1,l\
eq m}^{M}\sum_{j\in\mathcal{K}_{l}}P*\mathbf{ h}_{k,\mathcal{B}_{l}}\mathbf{w}_{j,\mathcal{B}_{l}}\mathbf{w}_{j,\mathcal{B}_{l}}^{H }\mathbf{h}_{k,\mathcal{B}_{l}}^{H} \tag{5}\]  
In a typical UDN, it is often the case that there are a large number of interfering users. Therefore, the interference received by user \(k\) can be well approximated as a zero mean complex Gaussian random variable with variance \(I_{k}\). As a result, the ergodic spectral efficiency of user \(k\) can be then expressed as (6), according to Shannon Equation [20],  
\[R_{k}=\mathbb{E}_{\mathbf{H}_{K_{m},\mathbf{B}_{m}}}\left[1+\frac{S_{k}}{I_{k }+\sigma^{2}}\right] \tag{6}\]  
where the expectation is taken over the channel matrix between user set \(\mathcal{K}_{m}\) and its serving RBAs set \(\mathcal{B}_{m}\). Figure 1: Illustration of the downlink ultra dense network (UDN) based on virtual cell merging. Red dashed ellipses denote the virtual cells before virtual cell merging, and blue solid ellipses are the ones after virtual cell merging. User equipments (UEs) in the same blue solid ellipse are jointly served by the remote base station antennas (RBAs) therein.