## 2 System Model  
The specific two-cell relay HetNet with two communication phases is illustrated in Figure 1. Both the relay and CCU are connected directly to the BS. The relay is located at \(d_{R}\) away from BS, while the CCU is randomly distributed in the center area within the radius of \(d_{U1},(d_{U1}\leq\tau_{1})\), the distance between them being \(d_{F1}\). The CEU communicates with the base station through the relay, \(d_{U2}\) away from it. Meanwhile, the CEU is interfered by the relay in the adjacent cell at a distance of \(d_{F2}\). We assume that the number of antennas configured at the transmitting (both BSs and relays) and receiving nodes (CCUs and CEUs) are \(M\) and \(N\), respectively. Without loss of generality, all the nodes work on the same spectrum for each phase of downlink transmission. The AF-mode relay only forwards the received signals through a processing matrix without decoding them. As shown in Figure 1, the whole transmission period is partitioned into two sub-time slots and the \(i\)-th \((i=1,2)\) user in the \(l\)-th cell is denoted as \(\text{UE}[i,l]\). In particular, we assume that the Rayleigh channel model is static during a whole transmission period and the global CSI is obtained by the BSs and CEUs in the network. Because the transmit beamforming matrix at each BS is obtained through the closed-form method as in Reference [5; 6], it requires the inter-cell and intra-cell interference channelsto be known to the BS. Furthermore, note that we define the local CSI as the channel information shared only in the same cell, which is not available in adjacent cells. For relays and CCUs, only local CSI needs to be known, since the inter-cell interference has no influence on them, and they are able to align intra-cell interference to the desired signal subspace without inter-cell CSI. Taking the communications in Cell 1 as an example, in sub-time slot 1, when BS1 transmits \(N_{\text{s}}\) data streams to its users (i.e., UE[1, 1], UE[2, 1]), the intra-cell interference will be generated between UE[1, 1] and Relay 1. The received signals at these two nodes can be expressed as  
\[\mathbf{y}_{t1}^{[1,1]}=\sqrt{g_{1}^{[1,1]}}\mathbf{H}_{1}^{[1,1]}(\mathbf{V} ^{[1,1]}\mathbf{s}_{t1}^{[1,1]}+\mathbf{V}^{[2,1]}\mathbf{s}_{t1}^{[2,1]})+ \mathbf{n}_{t1}^{[1,1]}, \tag{1}\]  
\[\mathbf{y}_{t1}^{R1}=\sqrt{g_{1}^{R1}}\mathbf{H}_{1}^{R1}(\mathbf{V}^{[2,1]} \mathbf{s}_{t1}^{[2,1]}+\mathbf{V}^{[1,1]}\mathbf{s}_{t1}^{[1,1]})+\mathbf{n}_ {t1}^{R1}, \tag{2}\]  
where \(\mathbf{H}_{j}^{[i,l]}\in\mathbb{C}^{N\times N},i,j,l\in\{1,2\}\) and \(\mathbf{H}_{j}^{Ri}\in\mathbb{C}^{N\times N},i,j\in\{1,2\}\) stand for the channel matrices from BS \(j\) to UE[\(i,l\)] and relay \(i\), respectively. \(\sqrt{g_{j}^{[i,l]}}\) and \(\sqrt{g_{j}^{Ri}}\) denote the long term path gain from BS \(j\) to UE[\(i,l\)] and to relay \(i\). \(\mathbf{V}^{[i,l]}\in\mathbb{C}^{M\times N_{s}},\mathbf{V}^{[i,l]H}\mathbf{V} ^{[i,l]}=\mathbf{I}_{N_{i}}\) is the transmit beamforming matrix towards UE[\(i,l\)]. The desired signal vector and additive white Gaussian noise (AWGN) vector with variance \(\sigma^{2}\) in sub-time slot \(tj\) at UE[\(i,l\)] are denoted as \(N_{s}\times 1\) vectors \(\mathbf{s}_{tj}^{[i,l]}\) and \(\mathbf{n}_{tj}^{[i,l]}\). In sub-time slot 2, Relay 1 forwards the combined signal to UE[2,1]. However, it should be noticed that UE[2,1] will also receive the inter-cell interference from Relay 2 in Cell 2, which is given by  
\[\mathbf{y}_  
\[ \begin{split}\mathbf{y}_{12}^{[1,1]}&=\sqrt{\varsigma_{1}^{[1,1]}}\mathbf{H}_{1}^{[1,1]}\mathbf{V}^{[1,1]}\mathbf{s}_{12}^{[1,1]}\!\!+\! \sqrt{\varsigma_{R1}^{[1,1]}}\rho_{(R_{1})}\mathbf{H}_{R1}^{[1,1]}\mathbf{G}_{1 }\mathbf{y}_{t1}^{R1}\\ &+\mathbf{n}_{t2}^{[1,1]}.\end{split}  \tag{4}\]  
In order to mitigate the intra-cell and inter-cell interference, we use the receive beamforming matrix at each user to receive the combined signal.