# Application Research of Multi-Mode Relay in Future Heterogeneous Networks
## 2 System Model  
The HHWN system max SINR downlink coverage regions are shown in Figure 2. Every base station in the macrocell, picocell, and femtocell can be represented as the relay in MMR. Because of the different transmission powers in the base station of different layers, the relay must have more than two modes if it must represent the BSs. In other words, the relay is not only the picocell BS but also the femtocell BS, which is why we call it MMR. The MMR helps node A and node B complete the communication. The MMR model is defined in Figure 3. The communication inside the MMR is not considered here. The emission powers of the two nodes in MMR equal to \(p_{s}\), the relay power of MMR is \(p_{r}\), and the noises of two source nodes are AWGN with zero mean and unit variance. There are limited N interferences at MMR. In this condition, the received signal at MMR is written in Equation (1), according to [14; 15]. \[\mathbf{y_{r}}=\sqrt{p_{s}}\mathbf{x_{A}}\mathbf{H}_{A}+\sqrt{p_{s}}\mathbf{x_ {B}}\mathbf{H}_{B}+\sum_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}}\mathbf{H}_{I,n} +\mathbf{n_{r}} \tag{1}\]  
where \(\mathbf{H}_{A}\) and \(\mathbf{H}_{B}\) are the channel fading \(p_{I,n}\), \(\mathbf{s_{n}}\) is the power of nth interference signal, and \(\mathbf{H_{I,n}}\) is the channel fading of nth interference signal. \(\mathbf{n_{r}}\) is Gaussian noise with zero mean and variance of \(\sigma_{r}^{2}\). As the TDD system has channel symmetry, the channel fading between MMR and i are identical. Figure 3: System model of multi-mode relay \(p_{r1}\
eq p_{r2}\). Figure 2: 3-tier HHWN max SINR downlink coverage regions with macrocells, picocells, and femtocells. In the first time slot to communicate, two nodes of A and B transmit their signal to MMR respectively, so the received signal at MMR is \(y_{r}\); in the second slot, after processing, MMR relay signals back to A node and B node in two ways, and due to the multi-mode of relay, the emission power of A and B is different, respectively \(p_{r1}\) and \(p_{r2}\). In fact, node A and B cannot achieve the best CSI, so the channel fading is modeled as  
\[\mathbf{H_{A}}=\mathbf{H_{A}^{\sim}}+\mathbf{e_{A}} \tag{2}\]  
\[\mathbf{H_{B}}=\mathbf{\widetilde{H_{B}}}+\mathbf{e_{B}} \tag{3}\]  
here \(\mathbf{e_{A}}\) and \(\mathbf{e_{B}}\) are deviation of the channel estimation, and make \(E\left(\left|\mathbf{e_{A}}\right|\right)=E\left(\left|\mathbf{e_{B}}\right| \right)=\delta_{E}^{2}\), provided that \(\mathbf{\hat{H}}_{A}\) and \(\mathbf{\hat{H}}_{B}\) both are independent of \(\mathbf{e_{A}}\) and \(\mathbf{e_{B}}\) with the variances \(\mu^{2}=1-\delta_{E}^{2}\). Equations (2) and (3) are extensively used in the literature [16]. Considering the difference of the transmission power of MMR, the amplifying-and-forward (AF) factors \(\beta_{s}\) which are used in MMR could be different from the traditional amplifying factor. Combining the amplifying factor in [14], the new normalized \(\beta_{s}\) can be denoted by  
\[\beta_{1}=\sqrt{p_{r1}}\sqrt{\left[ \begin{array}{c}p_{s}\left(\left|\mathbf{ \hat{H}}_{A}\right|^{2}+\left|\mathbf{\hat{H}}_{B}\right|^{2}\right)\\ +2p_{s}\sigma_{E}^{2}+\sum\limits_{n=1}^{N}p_{I,n}\left|\mathbf{H}_{I,n}\right| ^{2}\end{array} \right]} \tag{4}\]  
\[\beta_{2}=\sqrt{p_{r2}}\sqrt{\left[ \begin{array}{c}p_{s}\left(\left|\mathbf{ \hat{H}}_{A}\right|^{2}+\left|\mathbf{\hat{H}}_{B}\right|^{2}\right)\\ +2p_{s}\sigma_{E}^{2}+\sum\limits_{n=1}^{N}p_{I,n}\left|\mathbf{H}_{I,n}\right| ^{2}\end{array} \right]} \tag{5}\]  
Note that in [17], it is complex for MMR to identify between source signals and interferences. Consequently, the MMR normalizes the received signals and relays them, then the two sources A and B receive signals, which are expressed by Equations (6) and (7).