# Efficient Power Control Framework for Small-Cell Heterogeneous Networks
## 2 System Model  
We consider a heterogeneous network having a single macro-cell and multiple small cells, as shown in Figure 1. In total, we assume that there are \(L\) small cells, whereby, each small-cell BS is equipped with \(N_{l}\) antennas. We also consider that each small-cell is serving \(K\) UEs that are equipped with single antennas. The UEs are divided into \(M\) nonoverlapping multicast groups based on their quality of service requirements. For instance, the UEs could be categorized into groups based on their application and data rate needs. However, since the main purpose of this work is to provide a solution for power control problem in small-cell network, the issue of multicast group formation is beyond the scope of this work. The channel from the \(l\)-th small-cell to the \(k\)-th UE is represented as \(h_{kl}\in C^{N_{l}}\). The transmitted signal from the small-cell is written as  
\[x_{l}=\sum_{m=1}^{M}v_{lm}s_{m}, \tag{1}\]  
where \(v_{lm}\in C^{N_{l}}\) is the transmit beamforming vector to the UEs from the \(l\)-th small-cell. Moreover, \(s_{m}\in C\) is the symbol, where it is considered that \(E[|s_{m}|^{2}]=1\). The transmit power of the \(l\)-thsmall-cell is constrained by the maximum power, i.e., \(P_{l}\). This constrained on the beamforming vector is represented as  
\[V=\{v_{lm}\in C^{N_{l}}:\sum_{m=1}^{M}|v_{lm}|_{2}^{2}\leq P_{l}\} \tag{2}\]  
As a result of the transmission, the received signal at the \(k\)-th UE in \(m\)-th group can be written as  
\[y_{km}=\sum_{l=1}^{L}h_{kl}^{H}v_{lm}s_{m}+\sum_{i\
eq m}\sum_{l=1}^{L}h_{kl}^{H }v_{li}s_{i}+n_{k} \tag{3}\]  
where all the users are considered to apply single user detection, thus, \(s_{m}\) and \(n_{k}\) are mutually independent. Moreover, \(n_{k}\) is the additive white Gaussian noise with \(\lambda_{k}^{2}\) as variance and 0 mean. Based on this, the signal-to-intereference-and-noise ratio (SINR) is written as  
\[\Omega_{k,m}(v)=\frac{v_{m}^{H}\delta_{k}v_{m}}{\sum_{i\
eq m}v_{l}^{H}\delta_ {k}v_{i}+\lambda_{k}^{2}} \tag{4}\]  
where \(N=\sum_{l=1}^{L}N_{l}\) is the vector dimension for \(\delta_{k}=h_{k}h_{k}^{H}\). Also, the channel vectors and beamforming vectors having beamforming coefficients are, respectively, given as  
\[h_{k}=[h_{k1}^{H},h_{k2}^{H},\dots,h_{kL}^{H}]^{H}, \tag{5}\]  
and  
\[v_{m}=[v_{m1}^{H},v_{m2}^{H},\dots,v_{mL}^{H}]. \tag{6}\]  
We now focus our attention on the power-control problem in heterogeneous networks. One way for controlling the power of the network for macro-cell BS is to reduce the activity of small-cell BSs. Specifically, based on the variations in data traffic and quality of service demands, the macro-cell BS can switch off the small-cell BSs. This can put the inactive small-cell BSs in dormant mode, thereby, reducing the consumed power at the BSs. Let us represent the power consumption from small-cell BSs  
Figure 1: System Model. to UEs as \(p_{1}\) and from macro-cell BS to small-cell BS as \(p_{2}\). Then, the total consumed power can be denoted as  
\[p(v)=p_{1}(v)+p_{2}(v) \tag{7}\]  
For any value of drain efficiency coefficient, \(\mu_{l}>0\), the consumed power \(p_{1}(v)\) is given as  
\[p_{1}(v)=\sum_{l=1}^{L}\sum_{m=1}^{M}\frac{1}{\eta_{l}}|v_{lm}|_{2}^{2}. \tag{8}\]  
Similarly, \(p_{2}(v)\) is defined as  
\[x_{l}=\sum_{m=1}^{M}v_{lm}s_{m}, \tag{1}\]0  
It is worth mentioning that \(I(S(v)\cap V_{l}\
eq\emptyset)\) represents whether the \(l\)-th small-cell BS is switched off or on and \(S(v)\) is the support vector of \(v\). This is called an indicator function and takes a value of 0 when a particular small-cell BS is off. Additionally, \(P_{l}^{c}\) is the passive optical power consumption for the \(l\)-th small-cell BS. Moreover, \(V_{l}\) is defined as  
\[x_{l}=\sum_{m=1}^{M}v_{lm}s_{m}, \tag{1}\]1  
For a pre-specified quality of service requirements of different UEs in the network, i.e., \((\omega_{1},\omega_{2},\ldots\omega_{K})\), it is important to control the power subject to these constraints.